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+{"text":"\\section{Introduction}\n\nThe algebra of symbols of classical pseudodifferential operators $\\Psi \\mathrm{DO}_{cl}(X,E)$ on \na closed manifold $X$ acting on sections of a vector bundle $E$ can be defined as the \nquotient of $\\Psi \\mathrm{DO}_{cl}(X,E)$ by the ideal of smoothing operators. Since pseudodifferential  \noperators are smooth off the diagonal the symbol algebra is localized on the diagonal and \nit therefore can also be defined locally, using the product expansion formula and the change \nof charts formula for pseudodifferential operators. That the local heat kernel coefficients \nand the index of elliptic pseudodifferential operators are locally computable relies on the \nfact that the index and asymptotic spectral properties of pseudodifferential operators depend \nonly on their class in the symbol algebra. Note that the principal symbol of a \npseudodifferential operator is a section of the bundle of endomorphisms of $\\pi^* E$, \nwhere $\\pi: T^* X \\to X$ is the canonical projection.\n\nThe bundle of endomorphisms of a complex hermitian vector bundle is a bundle of simple \nmatrix algebras with $*$-structure. However, not all bundles of simple matrix algebras with \n$*$-structure, so-called Azumaya bundles, are isomorphic to endomorphism bundles of hermitian \nvector bundles. The obstruction is the so-called Dixmier-Douady class in $H^3(X,\\mathbb{Z})$. \nGiven an Azumaya bundle $\\mathcal{A}$ it is possible to construct algebras of symbols whose \nprincipal symbols takes values in the space of sections of the Azumaya bundle \n$\\pi^*\\mathcal{A}$ (see for example \\cite{MMS05} and the discussion in \\cite{MMS06}). \nFollowing \\cite{MMS05} we refer to such a symbol algebra as the algebra of symbols of \nprojective pseudodifferential operators. For such symbol algebras one can define an index \nand Mathai, Melrose and Singer \\cite{MMS06} proved an index formula for projective \npseudodifferential operators, analogous to the Atiyah-Singer index formula. The topological \nindex in this case is a map from twisted $K$-theory to $\\mathbb{R}$. It has also been shown \nin \\cite{MMS06} that any oriented manifold admits a projective Dirac operator even if the \nmanifold does not admit a spin structure. In this case its index may fail to be an integer.\n\nAnother important quantity that depends only on the class of the symbol of a \npseudodifferential operator is the so-called Wodzicki residue or noncommutative residue. \nUp to a factor it is the \nunique trace on the algebra of pseudodifferential operators. The Wodzicki residue appeared \nfirst as a residue of a zeta function measuring spectral asymmetry (\\cite{APS76,Wo84}). \nWodzicki showed that the regularity of the $\\eta$-function of a Dirac-type operator at \nzero -- a necessary ingredient to define the $\\eta$-invariant -- follows as a special case \nfrom the vanishing of the Wodzicki residue on pseudodifferential projections (as remarked \nby Br\\\"uning and Lesch \\cite{BL99} the regularity of the $\\eta$-function at zero for any \nDirac type operator and the vanishing of the Wodzicki residue on pseudodifferential \nprojections are actually equivalent). The regularity of the $\\eta$-function was proved by \nAtiyah, Patodi and Singer in \\cite{APS76} in the case when $X$ is odd dimensional and \nlater by Gilkey (\\cite{G81}) in the general case using $K$-theoretic arguments. Note that \nwhereas the Wodzicki residue can be locally computed its vanishing on pseudodifferential \nprojections is not a local phenomenon. Gilkey \\cite{G79} constructed a pseudodifferential \nprojection whose residue density is non-vanishing but integrates to zero. \n\nIn our paper we show that the Wodzicki residue can also be defined for projective \npseudodifferential operators (this has already been observed in \\cite{MMS06}) and show \nthat it vanishes on projections in case the dimension of the manifold is odd. \nOur proof is based on the  Leray-Hirsch theorem in twisted $K$-theory and a purely \nalgebraic result on `residue-traces' in filtered rings. \n\nIf $L$ is a filtered ring then we call a linear functional $\\tau: L \\to \\mathbb{C}$ a \nresidue trace if $\\tau(L^{-N})=\\{0\\}$ for $N$ large enough. We prove that the value of \n$\\tau$ on projections depends only on their class in $L^{(0)}:=L\/L^{-1}$. Thus, if the \nmap $K^0_{alg}(L) \\to K^0_{alg}(L^{(0)})$ is surjective the map $\\tau$ descends to a map \nfrom the algebraic $K$-theory of $L^{(0)}$ to $\\mathbb{C}$. This result can be applied \nto the Wodzicki residue showing that it descends to a map from twisted $K$-theory  \n$K^0(S^*X,\\pi^*\\mathcal{A})$ to $\\mathbb{C}$. We then use the Leray-Hirsch theorem to \nshow that this map actually vanishes. We reduce the \nproblem to positive spectral projections of generalized Dirac operators for which it \nis known \\cite{BG92} that the residue density vanishes.\n\n\\section{Convolution bundles and Azumaja bundles}\n\nPseudodifferential operators on a smooth closed Riemannian manifold $X$ acting on sections of a vector \nbundle $E$ can be understood as co-normal distributional sections in the vector bundle \n$E \\boxtimes E^*$\\footnote{$E \\boxtimes E^*$ denotes the external tensor product of $E$ \nand its dual bundle $E^*$, i.e., the fibre over a point $(x,y)$ is $E_x\\otimes E^*_y$.} \nover the space $X \\times X$, by identifying the operators with their distributional kernel. \nThe bundle $E \\boxtimes E^*$ has the following structure that allows to convolve kernels of \nintegral operators: any element in the fibre over ${(x,y)}$ may be multiplied by an element \nin the fibre over ${(y,z)}$ to give an element in the fibre over ${(x,z)}$. \nMoreover, this multiplication satisfies natural conditions such as associativity. \nIn order to define projective pseudodifferential operators it is \nconvenient to formalize this structure, as we shall do in this section.\n\n\\subsection{Convolution bundles}\n\nLet $\\mathcal{U}$ denote an open neighborhood of the diagonal $\\Delta(X)$ \nin $X \\times X$ which is symmetric under the reflection map $s:(x,y)\\mapsto(y,x)$. \nLet $p_{ik}:X \\times X \\times X \\to X \\times X$ be defined by \n$p_{ik}(x_1,x_2,x_3)=(x_i,x_k)$ and set \n$\\widetilde{\\mathcal{U}}\n   :=p_{12}^{-1}(\\mathcal{U}) \\cap p_{23}^{-1}(\\mathcal{U})  \n   \\cap p_{13}^{-1}(\\mathcal{U})$. \nDenote by $\\widetilde p_{ik}$ the restriction of the map $p_{ik}$ to $\\widetilde{\\mathcal{U}}$.\n\n\\begin{definition}\\label{def:bundle}\nLet $\\pi: F \\to \\mathcal{U}$ be a locally trivial vector bundle with typical fibre \n$\\mathrm{Mat}(k)$, the complex $k\\times k$-matrices. We call $F$ a \\emph{convolution bundle} \nif there exists a homomorphism of vector bundles \n$m: \\widetilde{p}_{12}^* F \\otimes \\widetilde{p}_{23}^* F \\to F$ \nsuch that the following conditions are satisfied:\n\\begin{itemize}\n \\item[(i)] The following diagram is commutative: \n   \\begin{equation*\n    \\xymatrix@1@=1.2cm@M=5pt{{\\widetilde{p}_{12}^* F \\otimes \\widetilde{p}_{23}^* F} \n    \\ar[d] \\ar[r]^{\\qquad m}  & F \\ar[d] \\\\\n    \\widetilde{\\mathcal{U}}   \\ar[r]^{\\widetilde{p}_{13}} & \\mathcal{U}}\n   \\end{equation*}\n \\item[(ii)] $m$ is associative, i.e., whenever $f_{ij}$ belong to the fibre \n   $F_{(x_i,x_j)}$ then \n   \\begin{align*}\n    m \\left( m(f_{12}\\otimes f_{23}) \\otimes f_{34}\\right)= \n    m \\left( f_{12}\\otimes m(f_{23} \\otimes f_{34})\\right).\\footnotemark  \n   \\end{align*} \n \\item[(iii)] There\\footnotetext{Here we implicitely assume that $(x_1,x_2,x_3)$, \n   $(x_1,x_3,x_4)$, and $(x_1,x_2,x_4)$ belong to $\\widetilde{\\mathcal{U}}$.} \n   is an atlas $\\{\\mathcal{O}_\\alpha\\}$ of $\\mathcal{U}$ together with local trivializations \n    $$\\phi_\\alpha: \\pi^{-1}\\mathcal{O}_\\alpha \\to \\mathcal{O}_\\alpha \\times \n      \\mathrm{Mat}(k),$$ \n   such that\n    $$\\phi_\\alpha(m(f_{12} \\otimes f_{23})) = \\phi_\\alpha(f_{12}) \\cdot \\phi_\\alpha(f_{23})$$\n   whenever $f_{ij}\\in F_{(x_i,x_j)}$ with \n   $(x_1,x_2,x_3)\\in \\widetilde{p}_{12}^{-1}(\\mathcal{O}_\\alpha)\n   \\cap\\widetilde{p}_{23}^{-1}(\\mathcal{O}_\\alpha)$. \n \\end{itemize}\n\\end{definition}\n\n\\begin{definition}\\label{def:star}\nA \\emph{$*$-structure} on $F$ is a conjugate linear map $*: F \\to F$ of vector bundles \nsuch that \n\\begin{gather*}\n \\xymatrix@1@=1.2cm@M=5pt{\n     F  \\ar[d] \\ar[r]^{*} & F \\ar[d] \\\\\n    \\mathcal{U} \\ar[r]^{s} & \\mathcal{U} \n }\n\\end{gather*}\ncommutes, such that\n$\n (m(f \\otimes g))^*=m(g^* \\otimes f^*), \n$\nand such that the above local trivializations additionally satisfy  \n$$\n \\forall f \\in \\pi^{-1}(\\mathcal{O}_\\alpha \\cap s(\\mathcal{O}_\\alpha)): \\qquad \\phi_\\alpha(f^*)=\\phi_\\alpha(f)^*, \n$$\nwhere the star on the right hand side denotes the hermitian conjugation of matrices.\nWe will refer to a convolution bundle with $*$-structure as a \\emph{$*$-convolution bundle}.\n\\end{definition}\n\nNote that $E \\boxtimes E^*$ is a particular example for a $*$-convolution bundle; in this \ncase we can choose $\\mathcal{U}=X \\times X$. The restriction of a $*$-convolution \nbundle $F$ to the diagonal in $X \\times X$ is a bundle $\\mathcal{A}$ of finite dimensional \nsimple $C^*$-algebras. Following the literature \nwe refer to such bundles of matrix algebras as Azumaja bundles. \n\nAs shown in \\cite{MMS05} any Azumaja bundle $\\mathcal{A}$ \non $X$ gives rise to a convolution bundle near the diagonal in the following way, using an \natlas of local trivializations with respect to a good \ncover\\footnote{A cover is good if finite intersections of elements therein are either empty \nor contractible.} \n$\\{{U}_\\alpha\\}$ of $X$: The transition functions $\\sigma_{\\alpha \\beta}$ are smooth functions \non ${U}_{\\alpha \\beta}={U}_{\\alpha} \\cap {U}_{\\beta}$ with values in the automorphisms of \n$\\mathrm{Mat}(k)$. Since all automorphims are inner we \ncan choose local functions $\\varphi_{\\alpha \\beta}: {U}_{\\alpha \\beta} \\to SU(k)$\nthat implement $\\sigma_{\\alpha \\beta}$, i.e., \n$\\sigma_{\\alpha \\beta}(x)(A) = \\varphi_{\\alpha \\beta}^{-1}(x) A \\varphi_{\\alpha \\beta}(x)$. \nIn general, the functions $\\varphi_{\\alpha \\beta}$ may violate the co-cycle condition \nand therefore are not the transition functions of a vector bundle. The cocycle condition \nfor the $\\sigma_{\\alpha\\beta}$ together with the condition that the $\\varphi_{\\alpha \\beta}$ \nare chosen in $SU(k)$ show that any \n$\\varphi_{\\alpha \\beta} \\varphi_{\\beta \\gamma} \\varphi_{\\gamma \\alpha}$\nmust be a constant function on ${U}_\\alpha \\cap {U}_\\beta \\cap {U}_\\gamma$, \nequal to an $k$-th root of unity times the identity \nmatrix.\\footnote{On different triple intersections, the resulting unit-root can be different. \nThis induces a torsion element in $H^3(X,\\mathbb{Z})$, the Dixmier-Douady class.} \nThen we obtain a convolution bundle $F$ with typical fibre $\\mathrm{Mat}(k)$ \non a neighborhood of the diagonal by choosing the transition functions\n $$\\phi_{\\alpha\\beta}(x,y)(A)=\\varphi_{\\alpha \\beta}(x) A\\, \\varphi_{\\alpha \\beta}(y)^{-1},\n   \\qquad A\\in \\mathrm{Mat}(k),$$ \non ${U}_{\\alpha\\beta} \\times {U}_{\\alpha\\beta}$. \nThere are also other possible extensions of $\\mathcal{A}$, cf.\\ \\cite{MMS06}, and Proposition \n\\ref{prop:transition}, below. \n\n\\begin{remark}\\label{rem:atlas}\nIn the sequel it will be occasionally convenient to choose an atlas for $F$ consisting of sets \n$\\mathcal{O}_\\alpha:=U_\\alpha\\times U_\\alpha$, where $\\{U_\\alpha\\}$ is a good cover of $X$; \nthe corresponding trivialisations we shall denote by $\\phi_\\alpha$ $($so we use the same notation \nas in Definition {\\rm\\ref{def:bundle}.(iii)} above, but possibly have changed the atlas$)$. \n\\end{remark}\n\\subsection{Transition functions} \n\nIn the previous section we have seen how an Azumaja bundle leads to a convolution bundle by \nchoosing certain transition functions. Let us now have a closer look to the transition functions \nof an arbitrary $*$-convolution bundle. Fix an atlas as explained in Remark \\ref{rem:atlas} and let \n$\\phi_{\\alpha\\beta}:\\mathcal{O}_\\alpha\\cap \\mathcal{O}_\\beta=:\\mathcal{O}_{\\alpha\\beta}\n\\longrightarrow GL(\\mathrm{Mat}(k))$\nbe the transition functions defined by \n\\begin{equation*}\\label{eq:trans0}\n \\phi_\\beta\\circ\\phi_\\alpha^{-1}\\big((x,y),A\\big)=\\big((x,y),\\phi_{\\alpha\\beta}(x,y)(A)\\big),\n\\end{equation*}\nThen condition iii$)$ of Definition \\ref{def:bundle} is equivalent to \n $$\\phi_{\\alpha\\beta}(x,y)(A)\\phi_{\\alpha\\beta}(y,z)(B)=\\phi_{\\alpha\\beta}(x,z)(AB).$$\nIn particular, \n\\begin{equation}\\label{eq:trans1}\n (x,x)\\mapsto\\phi_{\\alpha\\beta}(x,x):\\mathcal{O}_{\\alpha\\beta}\\cap\\Delta(X)\n \\longrightarrow\\mathrm{Aut}(\\mathrm{Mat}(k)).\n\\end{equation} \nMoreover, Definition \\ref{def:star} on the level of the transition functions means that \n\\begin{equation}\\label{eq:trans2a}\n \\phi_{\\alpha\\beta}(x,y)(A^*)=\\phi_{\\alpha\\beta}(y,x)(A)^*.\n\\end{equation}\n\n\\begin{proposition}\\label{prop:transition}\nLet $F$ be a $*$-convolution bundle with transition functions $\\phi_{\\alpha\\beta}$ as described above. \nThen\n\\begin{equation}\\label{eq:trans2}\n \\phi_{\\alpha\\beta}(x,y)(A)\n =\\lambda_{\\alpha\\beta}(x,y)\\varphi_{\\alpha\\beta}(x)A\\,\\varphi_{\\alpha\\beta}(y)^{-1}\n\\end{equation}\nwith mappings \n $$\\varphi_{\\alpha\\beta}:\\mathcal{O}_{\\alpha\\beta}\\longrightarrow SU(k), \\qquad \n   \\lambda_{\\alpha\\beta}:\\mathcal{O}_{\\alpha\\beta}\\longrightarrow {\\mathbb C},$$\nsatifying  \n $$\\lambda_{\\alpha\\beta}(x,x)=1,\\qquad \n   \\lambda_{\\alpha\\beta}(x,y)\\lambda_{\\alpha\\beta}(y,z)=\\lambda_{\\alpha\\beta}(x,z),\\qquad\n   \\lambda_{\\alpha\\beta}(x,y)=\\overline{\\lambda_{\\alpha\\beta}(y,x)},$$\nand such that all $\\varphi_{\\alpha\\beta}\\varphi_{\\beta\\gamma}\\varphi_{\\gamma\\alpha}$  \nare constant functions on their domain of definition, equal to a $k$-th root of unity times \nthe identity matrix.\n\\end{proposition} \n\\begin{proof}\nCombining \\eqref{eq:trans1} with \\eqref{eq:trans2a} we find $\\varphi_{\\alpha\\beta}$ with \n\\begin{equation*}\n \\phi_{\\alpha\\beta}(x,x)(A)=\\varphi_{\\alpha\\beta}(x)A\\,\\varphi_{\\alpha\\beta}(x)^{-1}, \n\\end{equation*}\nsince all automorphisms of $\\mathrm{Mat}(k)$ are inner. \nNow let us define\n $$\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\n   =\\varphi_{\\alpha\\beta}(x)^{-1}\\phi_{\\alpha\\beta}(x,y)(A)\\varphi_{\\alpha\\beta}(y).$$\nWe then have \n $$\\phi_{\\alpha\\beta}^\\prime(x,x)(A)=A,\\qquad \n   \\phi_{\\alpha\\beta}^\\prime(A)(x,y)\\phi_{\\alpha\\beta}^\\prime(y,z)(B)=\\phi_{\\alpha\\beta}^\\prime(x,z)(AB).$$\nIt follows that \n$\\phi_{\\alpha\\beta}^\\prime(x,y)(AB)=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\\phi_{\\alpha\\beta}^\\prime(y,y)(B)\n=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)B$\nand, analogously, $\\phi_{\\alpha\\beta}^\\prime(x,y)(AB)=A\\phi_{\\alpha\\beta}^\\prime(x,y)(B)$. \nTherefore, for all matrices $A$, \n $$\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1})A=\\phi_{\\alpha\\beta}^\\prime(x,y)(A)\n   =A\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1}),$$\nwhere $\\mathbf{1}$ is the identity matrix. \nThis shows $\\phi_{\\alpha\\beta}^\\prime(x,y)(\\mathbf{1})$ is a multiple of the identity matrix. \nDenoting the corresponding factor by $\\lambda_{\\alpha\\beta}(x,y)$, the claim follows. \n\\end{proof}\n\n\\section{Projective Pseudodifferential Operators}\n\nProjective pseudodifferential operators have been defined in \\cite{MMS05}. We adapt this definition to fit \nin our setting of convolution bundles. \n\n\n\\subsection{Pseudodifferential operators}\n\nTo clarify notation let us briefly recall the definition of classical (or polyhomogeneous) \npseudodifferential operators on an open subset $\\Omega$ of ${\\mathbb R}^n$. Let $V\\cong{\\mathbb C}^k$ be a \n$k$-dimensional vector space.\n\nA symbol of order $m\\in{\\mathbb R}$ is a smooth function \n$a:\\Omega\\times\\Omega\\times{\\mathbb R}^n\\to\\mathrm{End}(V)=V\\otimes V^*$ satisfying estimates \n $$\\left\\|\\partial^\\alpha_\\xi\\partial^\\beta_{(x,y)} a(x,y,\\xi)\\right\\|\\le \n   C_{\\alpha\\beta K}(1+|\\xi|)^{m-|\\alpha|}$$\nfor any multi-indices $\\alpha,\\beta$ and any compact subset $K$ of $\\Omega\\times\\Omega$, \nand having an asymptotic expansion $a\\sim\\sum\\limits_{j=0}^\\infty \\chi a_{m-j}$ with a\nzero-excision function $\\chi=\\chi(\\xi)$ and homogeneous components $a_{m-j}$, i.e., \n $$a_{m-j}(x,y,t\\xi)=t^{m-j}a_{m-j}(x,y,\\xi)$$\nfor all $(x,\\xi)$ with $\\xi\\not=0$ and all $t>0$. The pseudodifferential operator \n$\\mathrm{op}(a):C^\\infty_{0}(\\Omega,V)\\to C^\\infty(\\Omega,V)$ associated with $a$ is \n $$[\\mathrm{op}(a)\\varphi](x)=\\iint e^{i(x-y)\\xi}a(x,y,\\xi)\\varphi(y)\\,dyd\\hspace*{-0.08em}\\bar{}\\hspace*{0.1em}\\xi,\n   \\qquad \\varphi\\in C^\\infty_0(\\Omega,V).$$ \nAn operator $R:C^\\infty_{0}(\\Omega,V)\\to C^\\infty(\\Omega,V)$ is called smoothing if it has \na smooth integral kernel $k\\in C^\\infty(\\Omega\\times\\Omega,\\mathrm{End}(V))$, i.e., \n $$(R\\varphi)(x)=\\int_\\Omega k(x,y)\\varphi(y)\\,dy,\\qquad \\varphi\\in C^\\infty_0(\\Omega,V).$$\nA pseudodifferential operator of order $m\\in{\\mathbb R}$ on $\\Omega$ is an operator of the form \n$A=\\mathrm{op}(a)+C$, where $a$ is a symbol of order $m$ and $R$ is smoothing. \n\nAny pseudodifferential operator $A=\\mathrm{op}(a)+R$ of order $m$ can be represented in the form \n$\\mathrm{op}(a_L)+R^\\prime$, where $a_L(x,\\xi)$ is a $y$-independent `left-symbol' of order $m$; up to \norder $-\\infty$ the left-symbol is uniquely determined by the asymptotic expansion \n $$a_L(x,\\xi)\\sim \\sum_{|\\alpha|=0}^\\infty\\frac{1}{\\alpha!}\\partial^\\alpha_\\xi D^\\alpha_y \n   a(x,y,\\xi)\\Big|_{x=y}.$$\nThe homogeneous components of $A$ are by definition those of $a_L$,\n $$\\sigma_{m-j}(A)(x,\\xi):=(a_L)_{m-j}(x,\\xi).$$  \nBy the Schwarz kernel theorem, we can identify $A$ with its distributional \nkernel \n $$K_A\\in \\mathscr{D}^\\prime(\\Omega\\times\\Omega,V\\otimes V^*),$$\nthe topological dual of $C^\\infty_0(\\Omega\\times\\Omega;V^*\\otimes V)$. \nIt is uniquely defined by the relation \n $$\\skp{K_A}{\\psi\\otimes\\varphi}=\\skp{\\psi}{A\\varphi},\\qquad \n   \\psi\\in C^\\infty_0(\\Omega,V^*),\\quad\n   \\varphi\\in C^\\infty_0(\\Omega,V).$$\nDenoting by $\\mathrm{tr}:V^*\\otimes V\\to{\\mathbb C}$ the canonical contraction map, we have explicitly \n $$\\skp{K_A}{u}=\\int_\\Omega \\mathrm{tr}[A u(x,\\cdot)](x)\\,dx,\\qquad \n   u\\in C^\\infty_0(\\Omega\\times\\Omega;V^*\\otimes V).$$\nBy pseudo-locality, \n$K_A\\in C^\\infty(\\Omega\\times\\Omega\\setminus\\Delta(\\Omega),V\\otimes V^*)$. \n\nIf $U\\subset X$ is a coordinate neighborhood, we can pull-back the local operators \nunder the coordinate map. The resulting space of operators we shall denote by \n$\\Psi \\mathrm{DO}^m_{cl}(U;\\mathrm{End}(V))$, the subspace of smoothing operators by  \n$\\Psi \\mathrm{DO}^{-\\infty}(U;\\mathrm{End}(V))$. \n\n\\subsection{Projective pseudodifferential operators}\n\nIn the following choose an atlas as explained in Remark \\ref{rem:atlas}. \n\n\\begin{definition}\\label{def:pseudo}\n Let $F$ be a $*$-convolution bundle over $\\mathcal{U}$. A distribution \n $A \\in \\mathcal{D}^\\prime(\\mathcal{U}, F)$ \n is called a projective pseudodifferential operator of order $m\\in{\\mathbb R}$ if \n \\begin{itemize}\n  \\item[(i)] $A$ is smooth outside the diagonal, \n  \\item[(ii)] for any $\\alpha$ the distribution \n   $\\left( \\phi_\\alpha^{-1} \\right)^* A\\big|_{U_\\alpha\\times U_\\alpha}$ is the distributional kernel of a \n   pseudodifferential operator $A_\\alpha\\in\\Psi \\mathrm{DO}^m_{cl}(U_\\alpha;\\mathrm{End}({\\mathbb C}^{k}))$. \n \\end{itemize}\n We denote the vector space of $m$-th order projective pseudodifferential \n operators by $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$, the subspace of smoothing elements by \n $\\Psi \\mathrm{DO}^{-\\infty}(\\mathcal{U};F)$.  \n\\end{definition}\n\nThe subspace $\\mathrm{Diff}^m(\\mathcal{U};F)$ of projective differential operators consists \nof all projective pseudodifferential operators which are supported on the diagonal. \n\n\\begin{remark}\nIf $\\mathcal{U} = X \\times X$ and $F=E\\boxtimes E^*$ for a bundle $E$ over $X$ \nthen $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$ coincides with $\\Psi \\mathrm{DO}^m_{cl}(X;E,E)$, the \npseudodifferential operators of order $m$ acting on sections into $E$. \n\\end{remark}\n\nThough projective pseudodifferential operators, in general, are not operators in the \nusual sense (i.e., acting between sections of vector bundles) all elements of \nthe standard calculus can be generalized to this setting. In particular, \nthe $*$-structure gives rise to a conjugation on $\\Psi \\mathrm{DO}^m_{cl}(\\mathcal{U};F)$, \ndefined by $A^*(x,y):=(A(y,x))^*$ in the distributional sense. \n\nLet $A$ projective pseudodifferential operator with local representatives $A_\\alpha$ and $A_\\beta$, \ncf.\\ Definition \\ref{def:pseudo}, where $\\mathcal{O}_{\\alpha\\beta}$ is not empty. \nBy passing to local coordinates on $U_\\alpha\\cap U_\\beta$, we can associate with $A_\\alpha$ \nand $A_\\beta$ local symbols $a_\\alpha(x,\\xi)$ and $a_\\beta(x,\\xi)$, respectively. These symbols are \nthen related by \n\\begin{align}\\label{eq:asymptotic}\n\\begin{split}\n a_\\beta(x,\\xi)\n &=\\sum_{|\\gamma|=0}^\\infty \n   \\frac{1}{\\gamma!}\\partial^\\gamma_\\xi D^\\gamma_y\\Big|_{y=x}\n   \\phi_{\\alpha\\beta}(x,y)\\big(a_\\alpha(x,\\xi)\\big)\\\\ \n &=\\sum_{|\\gamma|=0}^\\infty \n   \\frac{1}{\\gamma!}\\partial^\\gamma_\\xi D^\\gamma_y\\Big|_{y=x}\n   \\big[\\lambda_{\\alpha\\beta}(x,y)\\varphi_{\\alpha\\beta}(x)a_\\alpha(x,\\xi)\\varphi_{\\alpha\\beta}(y)^{-1}\\big],\n\\end{split}\n\\end{align}\nwhere the transition function $\\phi_{\\alpha\\beta}$ is as described in \\eqref{eq:trans0} and \nProposition \\ref{prop:transition}. Note that this behaviour, in general, differs from the standard case, \ndue the factor $\\lambda_{\\alpha\\beta}(x,y)$. \nHowever, \\eqref{eq:asymptotic} together with $\\lambda_{\\alpha\\beta}(x,x)=1$ shows that with $A$ we can \nassociate a well-defined homogeneous principal symbol \n $$\\sigma_m(A)(x,\\xi)\\in C^\\infty(S^*X,\\pi^*\\mathcal{A}),$$\nwhere $\\pi:S^*X\\to X$ is the canonical co-sphere bundle over $X$. Vice versa, any given such \nsection can be realized as the principal symbol of a projective pseudodifferential operator. \n\nIf the projective pseudodifferential operators $A_1$ and $A_2$ are supported in a \nsufficiently small neighborhood of the diagonal in $\\mathcal{U}$ their usual composition \n$$ \n (A_1 \\circ A_2)(x,z)= \\int_{X} m\\left(A_1(x,y) \\otimes A_2(y,z)\\right)\\, dy\n$$\nis a distribution. By passing to local coordinates and using the composition theorems for \npseudodifferential operators one can see that $A_1 \\circ A_2$ is a projective pseudodifferential \noperator. The homogeneous principal symbol behaves multiplicative under composition. Of course, \nany projective pseudodifferential operator can be written as a sum of two operators, where one \nis smoothing and the other is supported near the diagonal. Summarizing, the coset space \n\\begin{equation}\\label{eq:coset}\n L^*_{cl}(\\mathcal{U},F):=\\Psi \\mathrm{DO}^*_{cl}(\\mathcal{U},F) \/ \\Psi \\mathrm{DO}^{-\\infty}(\\mathcal{U},F)\n\\end{equation}\nis a filtered $*$-algebra. As in the standard case, asymptotic summations of sequences \nof projective operators of one-step decreasing orders are possible and parametrices \n(i.e., inverses modulo smoothing remainders) to elliptic elements can be constructed.  \n\n\\begin{theorem}\\label{wres}\n Let $F$ be a $*$-convolution bundle and let $A$ be a projective pseudodifferential \n operator. For $x\\in X$ define\n $$\n  \\mathrm{WRes}_x(A) := \\int_{S^*_xX} \\mathrm{tr}\\, a_{-n}(x,\\xi) \\,d\\sigma(\\xi),\n $$ \n where $a_{-n}(x,\\xi)$, $n=\\mathrm{dim}\\,X$, is the homogeneous component of \n order $-n$ of a symbol of a local representative $A_\\alpha$ with $x\\in\\mathcal{O}_\\alpha$, \n cf. Definition {\\rm\\ref{def:pseudo}}. Then  $\\mathrm{WRes}_x(A)$ is well-defined and \n defines a global density on $X$. Moreover,\n $$ \n  \\mathrm{WRes}(A) := \\int_X \\mathrm{WRes}_x(A) \\,dx\n $$\n defines a trace functional on the algebra $L^*_{cl}(\\mathcal{U},F)$, the so-called \n \\emph{noncommutative residue} or Wodzicki residue.\n\\end{theorem}\n\\begin{proof}\n Let $A_\\beta$ be another local representative and $x\\in\\mathcal{O}_\\beta$. Fixing local coordinates on \n $\\mathcal{O}_\\alpha\\cap\\mathcal{O}_\\beta$, the local symbols $a_\\alpha$ and $a_\\beta$ are related by \n the asymptotic expansion \\eqref{eq:asymptotic}. Following the proof in \\cite{FGLS} terms containing \n a derivative $\\partial_\\xi^\\gamma$, $|\\gamma|\\ge1$, vanish under integration. We thus obtain the same \n value for $\\mathrm{WRes}_x(A)$ using either $a_\\alpha(x,\\xi)$ or $a_\\beta(x,\\xi)$. \n That $\\mathrm{WRes}_x(A)$ transforms as density under changes of \n coordinates is seen as in the standard case, cf.\\ \\cite{FGLS}. \n \n To see that the integral of the residue density defines a  trace functional we need to show that it \n vanishes on commutators $[A,B]$. To this end fix a cover $\\{U_\\sigma^\\prime\\}$ of $X$ by coordinate maps \n together with a subordinate partition of unity, such that $U_\\sigma^\\prime\\cup U_\\rho^\\prime$ is contained \n in some $U_\\alpha$ whenever $U_\\sigma^\\prime\\cap U_\\rho^\\prime$ is not empty. We then \n can write $A= \\sum_\\alpha A_{\\sigma}$ and $B= \\sum_\\alpha B_{\\sigma}$ modulo smoothing operators, \n where the $A_\\sigma$ and $B_\\sigma$ are supported in \n $\\mathcal{O}_{\\sigma}^\\prime:=U_\\sigma^\\prime \\times U_\\sigma^\\prime$. Then the commutator\n $[A,B]$ can be written as a sum of terms $[A_\\sigma,B_\\rho]$. Such a commutator is smoothing \n if $\\mathcal{O}_{\\sigma}^\\prime\\cap\\mathcal{O}_{\\rho}^\\prime$ is empty. Otherwise it \n is contained in some set $\\mathcal{O}_\\alpha$. Therefore the calculation reduces to a local one, which \n is not different from the one for usual pseudodifferential operators that can be found in \\cite{FGLS}. \n\\end{proof}\n\nFor purposes below let us establish the following result: \n\n\\begin{proposition}\\label{projection}\nLet $F$ be a $*$-convolution bundle and $\\mathcal{A}$ be the Azumaja \nbundle obtained by restricting $F$ to the diagonal. Moreover, let \n$p\\in C^\\infty(S^*X,\\pi^*\\mathcal{A})$ with $p^2=p$. Then there exists a \nprojective pseudodifferential operator $P\\in L^0_{cl}(\\mathcal{U},F)$ which is a projection, \ni.e., $P^2=P$, and which has $p$ as its principal symbol. If, additionally, $p^*=p$ \nthen $P$ can be chosen such that $P^*=P$. \n\\end{proposition}\n\\begin{proof}\nThe proof follows \\cite{Schu01}. Using local coordinates together with a partition of \nunity we can construct a $P_1\\in L^0_{cl}(\\mathcal{U},F)$ having $p$ as principal symbol. \nIf $V:={\\mathbb C}\\setminus\\{0,1\\}$ then $\\lambda-P_1$ is elliptic for any \n$\\lambda\\in V$. Thus there is a parametrix $Q(\\lambda)$, depending holomorphically \non the parameter $\\lambda\\in V$ (i.e., the local symbols depend holomorphically on $\\lambda$). \nThen one takes   \n $$P=\\int_\\mathscr{C} Q(\\lambda)\\,d\\lambda,$$\nwhere $\\mathscr{C}=\\partial U_{1\/2}(1)$ is the counter-clockwise oriented boundary of the disc of \nradius $1\/2$ centred in $1$ $($more precisely, we decompose $Q(\\lambda)$ in a sum of \nlocal terms and integrate each of these terms seperately over $\\mathscr{C})$. \nIf also $p^*=p$ we repeat the above construction, replacing $P_1$ by $P^*P$. \n\\end{proof}\n\n\\section{The noncommutative residue in twisted $K$-theory}\n\n\\subsection{Twisted K-theory}\n\nSuppose that $\\mathcal{A}$ is an Azumaja bundle over a compact manifold $X$.\nThe twisted K-theory is defined to be the K-theory of the $C^*$-algebra\nof continuous sections $C(X; \\mathcal{A})$ of $\\mathcal{A}$.\n\nIf $Y \\subset X$ is a closed subset then the set of sections $C(X,Y;\\mathcal{A})$\nvanishing on $Y$ is a closed two-sided ideal in $C(X,Y;\\mathcal{A})$\nand the quotient by this ideal can be identified with the space of continuous\nsections $C(Y;\\mathcal{A})$ of the Azumaja bundle $\\mathcal{A}|_Y$. We therefore have\nthe six term exact sequence as a consequence of the six term exact sequence in the theory\nof $C^*$-algebras.\n\\begin{gather*}\n  \\xymatrix@1@=1.2cm@M=2pt{\n     K^0(X,Y;\\mathcal{A}) \\ar[r] & K^0(X;\\mathcal{A}) \\ar[r] & K^0(Y;\\mathcal{A}) \\ar[d] \\\\\n    K^1(Y;\\mathcal{A})  \\ar[u] \\ar[r] & K^1(X;\\mathcal{A}) \\ar[r] & K^1(X,Y;\\mathcal{A}) }\n\\end{gather*}\nwhere the relative $K$-groups $K^*(X,Y;\\mathcal{A})$ are defined as\n$K_*(C(X,Y;\\mathcal{A}))$.\n\nThere is a natural map\n\\begin{gather*}\n K_*(C(X,Y;\\mathcal{A})) \\otimes_\\mathbb{Z} K_*(C(X)) \\mapsto K_*(C(X,Y;\\mathcal{A}) \\hat \\otimes C(X)).\n\\end{gather*}\nHere $\\hat \\otimes$ is the tensor product of $C^*$-algebras which is well defined in this case as $C(X)$\nis nuclear.\nThe usual multiplication\n\\begin{gather*}\n C(X,Y;\\mathcal{A}) \\hat \\otimes C(X) \\to C(X,Y;\\mathcal{A})\n\\end{gather*}\ninduces a map $K_*(C(X,\\mathcal{A}) \\hat \\otimes C(X)) \\to K_*(C(X,\\mathcal{A}))$.\nThe composition of these two maps makes $K^*(X,Y;\\mathcal{A})$\na module over the $\\mathbb{Z}_2$-graded ring $K^*(X)$.\nChoosing $Y=\\emptyset$ defines a $K^*(X)$ module structure\non $K^*(X;\\mathcal{A})$.  Note that the morphisms in the six term exact sequence are module\nhomomorphisms.\n\nThese observations can be used to prove the following Leray-Hirsch theorem: \n\n\\begin{theorem}\\label{leray}\n Let $R$ be a commutative torsion-free ring.\n Suppose that $\\pi: M \\xrightarrow{F} X$ is a compact smooth fibre bundle with fibre $F$ \n over $X$ and let $\\mathcal{A}$ be an Azumaja bundle over $X$. Assume that $K^*(F) \\otimes_{{\\mathbb Z}} R$ is a \n free $R$-module and suppose there exist elements\n $c_1,\\ldots,c_N \\in K^*(M) \\otimes_{{\\mathbb Z}} R$ such that the $c_j|_{M_x}$ form a basis \n for $K^*(M_x)\\otimes_{{\\mathbb Z}} R$ for every $x \\in X$. Then the following map is an isomorphism$:$\n \\begin{equation*}\\label{eq:leray}\n   K^*(X;\\mathcal{A})\\otimes_{{\\mathbb Z}}  R^N \\longrightarrow K^*(M,\\pi^*(\\mathcal{A})) \\otimes_{{\\mathbb Z}} R,\\qquad \n   (p,\\alpha) \\mapsto \\sum_{j=1}^N \\alpha_j \\pi^*(p) \\cdot c_j. \n \\end{equation*}\n\\end{theorem}\n\nIndeed, the usual proof  of the Leray-Hirsch theorem in topological $K$-theory (see e.g. \\cite{hatcher}, Theorem 2.25)\ncan be adapted to our setting in the following way.\nIf $Y \\subset X$ is a closed subset of $X$, we have the following diagram:\n\\begin{gather*}\n  \\xymatrix@=0.5cm@1@=0.5cm@=0.5cm@M=2pt{\n     \\ar[r] & K^*(X,Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R^N \\ar[d]^\\Phi  \\ar[r] &  K^*(X;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R^N  \n     \\ar[d]^\\Phi \\ar[r] & K^*(Y;\\mathcal{A})\\otimes_{{\\mathbb Z}} R^N   \\ar[d]^\\Phi  \\ar[r] &  \\\\\n     \\ar[r] &K^*(\\pi^{-1}X,\\pi^{-1}Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R \\ar[r] & K^*(\\pi^{-1}X;\\mathcal{A})\\otimes_{{\\mathbb Z}} R \\ar[r] & K^*(\\pi^{-1}Y;\\mathcal{A}) \\otimes_{{\\mathbb Z}} R\\ar[r] & }\n\\end{gather*}\nHere  $\\Phi$ is defined as in the theorem,\n$\\Phi(p,\\alpha) = \\sum\\limits_{j=1}^N \\alpha_j \\pi^*(p) \\cdot c_j$. The rows of this diagram are exact\nsince tensoring with $R^N$ and $R$ is an exact functor. All maps in the six term exact sequence are natural\nand therefore the pull back $\\pi^*$ commutes with them. Moreover, the maps in the six term exact sequence for\nthe pair $(\\pi^{-1}X,\\pi^{-1}Y)$ are $K^*(M)$ module homomorphisms. Thus, the diagram commutes.\nSince $X$ is a finite cell complex one can proceed in the usual way using the $5$-lemma and induction\nin the number of cells and the dimension  to prove the theorem.\n\n\\subsection{Residue-traces on filtered rings}\n\nLet $L$ be a ring with filtration, i.e., $L=L^0\\supset L^{-1}\\supset L^{-2}\\supset\\ldots$ \nwith sub-rings $L^{-j}$ and the multiplication induces maps $L^{-i}\\times L^{-j}\\to L^{-i-j}$ \nfor any choice of $i,j$. \n\nA trace functional on $L$ is a map $\\tau:L\\to V$ for some vector space $V$ having the following properties: \n\\begin{itemize}\n \\item[$(1)$] $\\tau$ is linear, $\\tau(A+B)=\\tau(A)+\\tau(B)$ for all $A,B\\in L$, \n \\item[$(2)$] $\\tau$ vanishes on commutators, $\\tau([A,B])=\\tau(AB-BA)=0$ for all $A,B\\in L$.\n\\end{itemize}\nWe call $\\tau$ a residue-trace if, additionally,    \n\\begin{itemize}\n \\item[$(3)$] there exists a $N$ such that $\\tau(A)=0$ for all $A\\in L^{-N}$. \n\\end{itemize}\nWe shall now show that a residue-trace restricted to the set of projections in $L$ is insensible \nfor lower order terms. The proof is elementary and purely algebraic. \n\n\\begin{theorem}\\label{prop1}\nLet $\\tau$ be a residue-trace on $L$ and $P,\\widetilde{P}\\in L$ be two projections, i.e., \n$P^2=P$ and $\\widetilde{P}^2=\\widetilde{P}$. If $P-\\widetilde{P}\\in L^{-1}$ then $\\tau(P)=\\tau(\\widetilde{P})$. \n\\end{theorem}\n\\begin{proof}\nSet $R=\\widetilde{P}-P$ and then define  \n $$A=P R P,\\quad B=P R(1-P),\\quad C=(1-P)R P,\\quad D=(1-P)R(1-P).$$\nObviously then $\\widetilde{P}=P+R$ and $R=A+B+C+D$. Using that $P(1-P)=(1-P)P=0$ we obtain \n\\begin{align*}\n (P+R)(P+R)=&P+2A+B+C+A^2+AB+BC+ \\\\\n &+BD+CA+CB+DC+D^2.\n\\end{align*}\nOn the other hand, using that $\\widetilde{P}$ is a projection, \n $$(P+R)(P+R)=(P+R)=P+A+B+C+D.$$\nEquating these two expressions and rearranging of terms yields \n $$A^2+A+BC+D^2-D+CB+AB+BD+CA+DC= 0.$$\nMultiplying this identity from the left and the right with $P$ and $1-P$, respectively, \nyields \n\\begin{equation*}\n A^2+A+BC = 0,\\qquad D^2-D+CB =0.\n\\end{equation*}\nLet us rewrite these equations as \n\\begin{equation*}\n A(1+A)=-BC,\\qquad (-D)\\big(1+(-D)\\big)=-CB.  \n\\end{equation*}\nMultiplying the first equation by $(1-A)$ yields $A\\equiv -BC$ modulo $L^{-3}$. Multiplying it \nwith $(1-A+A^2)$ then yields $A\\equiv-BC-(BC)^2$ modulo $L^{-4}$. Proceeding by induction we obtain \n $$A\\equiv \\sum_{k=1}^\\ell c_{k\\ell}(BC)^{k}\\mod L^{-(2+\\ell)}$$\nfor any $\\ell\\in{\\mathbb N}$ with suitable constants $c_{k\\ell}$. In the same way, with the same \nconstants $c_{k\\ell}$,  \n $$-D\\equiv \\sum_{k=1}^\\ell c_{k\\ell}(CB)^{k}\\mod L^{-(2+\\ell)}.$$\nTherefore we have \n $$A+D=\\sum_{k=1}^\\ell c_{k\\ell}\\left[B,(CB)^{k-1}C\\right]\\mod L^{-(2+\\ell)}.$$\nChoosing $\\ell$ large enough, we deduce that $\\tau(A)+\\tau(D)=0$. Furthermore, \n $$\\tau(B)=\\tau(P R(1-P))=\\tau((1-P)P R)=0$$ \nand, analogously, $\\mathrm{res}(C)=0$. Altogether we obtain \n $$\\tau(\\widetilde{P})=\\tau(P)+\\tau(A)+\\tau(D)+\\tau(B)+\\tau(C)=\\tau(P)$$\nwhich is the claim we wanted to prove. \n\\end{proof}\n\n\\subsection{The noncommutative residue}\n\nWe shall show that the noncommutative residue induces a map on twisted $K$-theory. \n\n\\begin{proposition}\nLet $\\mathcal{A}$ be the Azumaja bundle obtained by restricting a $*$-con\\-vo\\-lution bundle \n$F$ to the diagonal. The noncommutative residue from Theorem {\\rm\\ref{wres}} descends to a \ngroup homomorphism\n\\begin{equation}\\label{eq:wres}\n \\mathrm{WRes}:K^0(S^*X,\\pi^*\\mathcal{A}) \\to \\mathbb{C},\n\\end{equation}\nwhere $\\pi:S^*X\\to X$ denotes the co-sphere bundle over $X$. \n\\end{proposition}\n\\begin{proof}\nA typical element in $K^0(S^*X,\\pi^*\\mathcal{A})$ can be represented by a section  \n$p\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_N(\\mathcal{A}))$ which is $($pointwise$)$ a projection. \nThis is possible, since the natural inclusion of the $K$-theory of the local $C^*$-algebra \n$C^\\infty(S^*X,\\pi^*\\mathcal{A})$ into that of $C(S^*X,\\pi^*\\mathcal{A})$ is an isomorphism, \ncf. \\cite{Bl98}. By Proposition \\ref{projection} each such section is the principal symbol \nof a projective \npseudodifferential operator $P\\in L^0(X;\\mathrm{Mat}_N(F))$ which is a projection. \nThe noncommutative residue of the $K$-group element is then defined as $\\mathrm{WRes}(P)$ \nin the sense of Theorem \\ref{wres}. \n\nWe have to show that this map is well-defined. So let \n$\\widetilde{p}\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_M(\\mathcal{A}))$ \nrepresent the same element as $p$ does. Let $\\widetilde{P}\\in L^0(X;\\mathrm{Mat}_M(F))$ \nbe associated with $\\widetilde{p}$. \nSince $p$ and $\\widetilde{p}$ are equivalent there exists a unitary \n$u\\in C^\\infty(S^*X,\\pi^*\\mathrm{Mat}_{M+N}(\\mathcal{A}))$ such that \n$u(p\\oplus 0_{M})u^{-1}$ coincides with $0_{N}\\oplus\\widetilde{p}$. \nLet $U\\in L^0(X;\\mathrm{Mat}_{M+N}(F))$ have $u$ as its principal symbol. Then \n $$\\mathrm{WRes}(U(P\\oplus 0_{M})U^{-1})=\\mathrm{WRes}(P\\oplus 0_{M})=\\mathrm{WRes}(P).$$\nOn the other hand $U(P\\oplus 0_{M})U^{-1}$ is a projection having the same principal \nsymbol as $0_{N}\\oplus\\widetilde{P}$. Thus, by Theorem \\ref{prop1},  \n $$\\mathrm{WRes}(U(P\\oplus 0_{M})U^{-1}) =\\mathrm{WRes}(0_{N}\\oplus\\widetilde{P})=\\mathrm{WRes}(\\widetilde{P}).$$\nThis shows that the noncommutative residue is independent of the choice of the representative. \n\\end{proof}\n\n\n\\section{Twisted Dirac operators and connections}\n\nLet $\\mathcal{A}$ be the Azumaja bundle obtained by restricting a $*$-convolution bundle \n$F$ to the diagonal. \n\n\\begin{definition}\\label{def:connection}\nA projective connection $\\nabla=\\nabla^F$ on $F$ is a linear map \n $$Y\\mapsto\\nabla_Y:\\quad C^\\infty(X;TX)\\longrightarrow \\mathrm{Diff}^1(\\mathcal{U};F)$$\nsatisfying, for any vector field $Y\\in C^\\infty(X,TX)$ and any function \n$f\\in C^\\infty(X)$,  \n\\begin{itemize}\n \\item[$(1)$] $\\nabla_{fY}=f\\nabla_Y$, \n \\item[$(2)$] $[ \\nabla_Y , f] = Yf$ for any $f\\in C^\\infty(X)$.\n\\end{itemize}\nIt is called a hermitian connection if additionally \n\\begin{itemize}\n \\item[$(3)$] $\\nabla_Y^* + \\nabla_Y + \\mathrm{div}\\, Y =0$\n\\end{itemize}\n$($here, $f$ and $\\mathrm{div}\\,Y$ are considered as elements of $\\mathrm{Diff}^0(\\mathcal{U};F))$. \n\\end{definition}\n\nNote that in case $\\mathcal{U}=X\\times X$ and $F=E \\boxtimes E^*$ for a vector bundle \n$E$ over $X$ we just recover a usual hermitian connection on $E$. One can always construct \na projective hermitian connection from local hermitian connections \nby gluing with a partition of unity.\n\nIf $\\nabla=\\nabla^F$ is a projective connection and $\\phi_\\alpha$ is a local trivialization\nof $F$ over $U_\\alpha\\times U_\\alpha$ as described in Remark \\ref{rem:atlas}, the corresponding \nlocal differential operator \n $$\\nabla^\\alpha_Y\\in\\mathrm{Diff}^m(U_\\alpha,\\mathrm{End}({\\mathbb C}^k))$$ \nis of the form  \n $$\\nabla^\\alpha_Y=Y+\\Gamma_Y^\\alpha(x),\\qquad \n   \\Gamma_Y^\\alpha\\in C^\\infty(U_\\alpha,\\mathrm{Mat}(k)).$$ \nIf we use another trivialisation $\\phi_\\beta$ of $F$ on $U_\\beta\\times U_\\beta$, \nwe have the relation \n $$\\Gamma_Y^\\beta(x)=\\phi_{\\alpha\\beta}(x,x)(\\Gamma_Y^\\alpha(x))\n   +Y_y\\phi_{\\alpha\\beta}(x,y)(\\mathbf{1})\\big|_{y=x},\\qquad x\\in U_\\alpha\\cap U_\\beta,$$\nwhere $\\mathbf{1}$ is the identity matrix. Thus, analogous to the theory of standard connections, \nwe may describe projective connections by `connection matrices' $\\Gamma_Y^\\alpha$ associated to \na covering $X=\\mathop{\\mbox{\\large$\\cup$}}_\\alpha U_\\alpha$ satisfying the above compatibility \nrelations. For a hermitian connection the connection matrices also have to be skew-symmetric, \n$\\Gamma_Y^\\alpha(x)^*=-\\Gamma_Y^\\alpha(x)$.\n\nSuppose now $S$ is a Clifford module over $X$ and let $\\gamma$ denote the Clifford \nmultiplication. Moreover, let $\\nabla^{S}$ be a connection on $S$ which is compatible \nwith the Clifford structure. Writing $\\widetilde{F}:=S\\boxtimes S^*$ it is easy to see that \n$F\\otimes\\widetilde{F}$ is a $*$-convolution bundle over $\\mathcal{U}$ and we can define \nthe projective hermitian connection \n $$\\nabla:=\\nabla^F\\otimes 1+1\\otimes\\nabla^{S}$$ \nby choosing the corresponding connection matrices as  \n $$\\Gamma_Y^{F,\\alpha}(x)\\otimes 1+1\\otimes \\Gamma_Y^{S,\\alpha}(x),\\qquad x\\in U_\\alpha,$$\nwhere the $U_\\alpha$ are chosen in such a way that both $F$ and $\\widetilde{F}$ are locally trivial \nover $U_\\alpha\\times U_\\alpha$. \nThen we can define the twisted Dirac operator \n $$D:=(1\\otimes\\gamma)\\circ \\nabla\\in \\mathrm{Diff}^1(\\mathcal{U};F\\otimes\\widetilde{F});$$\nin fact, in each local trivialisation $\\nabla$ is a usual hermitean connection and we can compose \nit locally with $1\\otimes\\gamma$. \n\n\\section{Vanishing of the Wodzicki residue}\n\n\\begin{theorem}\\label{thm:vanish}\n If $X$ is an odd dimensional oriented manifold \n the map $\\mathrm{WRes}$ of \\eqref{eq:wres} vanishes identically.\n\\end{theorem}\n\nAs a direct consequence, $\\mathrm{WRes}(P)=0$ for any projection \n$P\\in L^0(\\mathcal{U},F)$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:vanish}]\nSuppose the dimension of $X$ is $n=2 \\ell -1$.\nLet $S=\\mathop{\\mbox{$\\oplus$}}\\limits_{k\\text{ even}}\\Lambda^k(T^*X)$ denote the bundle of \neven-degree forms over $X$ and let $*: \\Lambda^k(T^*X) \\to \\Lambda^{n-k}(T^*X)$ the Hodge star\noperator and denote by $d$ and $\\delta$ the exterior differential and the co-differential respectively.\nDefine the operator $D^S$ acting on sections of $S$ as\n $$D^S=i^{\\ell} * (\\delta+(-1)^{k+1}d)\\quad\\text{on $k$-forms}.$$\nThen by Proposition 1.22 and 2.8 in \\cite{BGV04} this is a generalized Dirac operator, \nwhere the Clifford action on $S$ is given by \n$\\gamma(\\xi)=i^{\\ell} * \\left(\\mathrm{int}(\\xi)+(-1)^{k+1}\\mathrm{ext}(\\xi)\\right)$ for $\\xi\\in T^*X$\nand the compatible connection is the Levi-Civita connection.\nThe principal symbol of $D^S$ restricted to the co-sphere bundle is a self-adjoint involution and the\nprojection $\\sigma_+(D^S)=\\frac{1}{2}(\\sigma(D^S)+1)$ onto its $+1$ eigenspace defines an element\nin $K^0(S^*X)$. It is well known (see for instance \\cite{APS76}) that restriction of this element to each\nco-sphere $S^*_x X$ equals $2^\\ell$ times the Bott element on $S^{n-1}$ which together with the class of the trivial\nline bundle generates $K^0(S^{n-1})$.\n\nFor notational convenience denote by $K_{{\\mathbb R}}^*(X)$ the groups $K^*(X) \\otimes_{{\\mathbb Z}} {\\mathbb R}$.\nBy Theorem \\ref{leray} applied to the co-sphere bundle of $X$, any element of \n$K^0_{{\\mathbb R}}(S^*X,\\pi^*\\mathcal{A})$ can be represented in the form \n $$\\alpha_0\\pi^*(p)\\cdot [\\mathbf{1}]+\\alpha_1\\pi^*(p)\\cdot [\\sigma_+(D^S)]$$\nfor some $\\alpha_0,\\alpha_1 \\in {\\mathbb R}$ and some $p\\in K^*(X,\\mathcal{A})$. Here both\nthe class $[\\mathbf{1}]$ of the trivial line bundle and the class $[\\sigma_+(D^S)]$ are understood\nas elements in $K^0_{\\mathbb R}(S^*X)$.\nThe elements in $\\alpha_0\\pi^*(p)\\cdot [\\mathbf{1}]$ can be represented by projections in \n$C^\\infty(X;\\mathrm{Mat}_N(\\mathcal{A}))$.\nTherefore, the noncommutative residue of these elements vanishes. \nIt remains to show that this is also true for the second summand. \n\nTo this end let $p$ be a projection in $\\mathrm{Mat}_N(C^\\infty(X;\\mathcal{A}))$. \nLet us define the new convolution bundle $F_p$ \nhaving fibre $p(x)\\mathrm{Mat}_N(F)_{(x,y)}p(y)\\subset \\mathrm{Mat}_N(F)_{(x,y)}$ in $(x,y)$. \nWe now apply the above construction and build a twisted Dirac operator $D_p$ with respect to \n$F_p\\otimes\\widetilde{F}$, $\\widetilde{F}=S\\boxtimes S^*$. Then $\\sigma_+(D_p)$ represents the class \n$\\pi^*([p])\\cdot[\\sigma_+( D^S)]$ in $K^0(S^*X,\\pi^*\\mathcal{A})$. \n\nThe projective differential operator $D_p$ can now be used to construct a certain \nprojection $Q \\in L^0_{cl}(X,F_p)$  which has principal symbol \nas $\\sigma_+(D_p)$ on $S^* X$.\nIn the case of a Dirac type operator $D$ acting on a vector bundle the projection would just be the\noperator $\\frac{1}{2}(|D|^{-1} D+1)$. The symbol of this projection can be constructed from the a parametrix\nof $D$ and this construction is local modulo smoothing operators. That is the full symbol of $\\frac{1}{2}(|D|^{-1} D+1)$\nmodulo smoothing terms in local coordinates depends only on the full symbol of $D$ in these local coordinates.\nThus, the construction can be repeated for the operator $D_p$ to yield an element in $L^0_{cl}(X,F_p)$\nwhich we denote by $Q$ or formally $\\frac{1}{2}(|D_p|^{-1} D_p+1)$. By construction $[\\sigma(Q)] \\in K^0_{\\mathbb R}(S^*X;\\mathcal{A})$ \nis equal to  $\\pi^*([p]) \\cdot [\\sigma_+(D^S)]$.\n\nIn  \\cite{BG92} (Theorem 3.4) Branson and Gilkey have used invariant theory to show that the residue \\emph{density} \nof the positive spectral projection for any generalized Dirac operator vanishes identically. \nLocally, $D_p$ is a generalized Dirac operator\nand since the construction of the residue density is local the residue density of  $Q$\nvanishes as well.\nSo we can conclude that the noncommutative residue of  $Q$ vanishes which\ncompletes our proof. \n\\end{proof}\n\n\\section{Discussion of even dimensional manifolds}\n\nThe vanishing of the Wodzicki residue on projective pseudodifferential projections is very likely\nto hold also in even dimensions.\nTo compute the residue as a map from $K^0(S^* X;\\pi^* \\mathcal{A})$ it is enough to calculate it\non the generators of $$K^0_{{\\mathbb R}}(S^* X;\\pi^* \\mathcal{A}) \/ \\pi^* K^0_{\\mathbb R}(X;\\mathcal{A})$$ which by\nthe Leray-Hirsch theorem is isomorphic to $K^1_{\\mathbb R}(X;\\mathcal{A})$. Of course the result then also holds\nif $K^1_{\\mathbb R}(X;\\mathcal{A})=\\{0\\}$.\n\n\\noindent{\\bf Acknowledgements.} The authors would like to thank\nThomas Schick for comments and for pointing out a gap in an earlier version\nof this paper. \n\n\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\\label{sec:conclusion}\n\\noindent\nWe provide a systematic study of error correction techniques that address leakage faults where qubit states leave the computational space. We developed a simple model that captures realistic elements of leakage phenomena. Using this model we studied the performance of various circuits that suppress leakage in the toric code, ranging from a simple circuit that only adds one $CNOT$ gate per syndrome extraction, to a scheme that reduces leakage after the application of every gate. To accurately compare the performance of these circuits, we considered two scenarios, one in which measurements distinguish leaked qubits from qubits in the computational state, and another in which they do not. For each of these cases we designed an optimized decoding algorithm and recorded its success rate in Monte Carlo simulations. Our results show that the Partial-LRU and Quick circuits are effective at reducing leakage and reduce the accuracy threshold of the toric code by less than a factor of $4$ in a realistic scenario where leakage was added to depolarizing noise. We see a factor of $2$ threshold improvement over Standard decoders for the HL Decoder that uses information about leakage detection events. Finally, there is evidence that the Full-LRU, which reduces leakage after every gate, may achieve the lowest logical error rates of all of the circuits when the physical error rate is less than $2\\times 10^{-4}$. Future work could explore the logical error rates of these circuits at higher distances and at error rates that are far below threshold, perhaps by applying the splitting method \\cite{bravyi13}.\n\n\\nonumsection{Acknowledgements}\n\\noindent\nWe thank Sergey Bravyi, Ken Brown, Oliver Dial, David DiVincenzo, Easwar Magesen, Graeme Smith, and John Smolin for helpful discussions, Chris Lirakis and Mark Ritter for their support, and Karen Bard for making Blue Gene available to us. AWC and JMG acknowledge support from ARO under contract {\\em W911NF-14-1-0124}.\n\n\\section{Heralded Leakage Reduction}\\label{sec:heralded}\n\\noindent\n\nUp to this point in the discussion, we have assumed that measurements cannot distinguish a leaked qubit from a qubit in the state $|1\\rangle$, and therefore there was no mechanism to detect leakage faults. We now assume that measurements produce a third outcome ``L'' given a leaked qubit and optimistically assume that they output ``L'' if and only if the input qubit is leaked. Taking the same circuits considered in Sec.~\\ref{sec:reduction} (specifically the Partial-LRU and Quick circuits), we modify the decoders to use this additional information. This approach is relatively simple, requiring no change to the quantum circuits, and leads to significantly improved logical error rates and threshold (see Sec.~\\ref{sec:results}). A 3-outcome measurement is not equivalent to a simple erasure model because the precise space-time location of the leakage fault is not available, and leakage faults induce regular errors that depend on the leakage fault location.\n\nThe \\emph{Heralded Leakage (HL) decoder} ingests results from 3-outcome measurements. Using these 3-outcome measurements provides an advantage because in our model leakage is likely to produce regular errors on specific locations around the leaked qubit, and the decoder is then able to match defects using correct prior probabilities in the space-time neighborhood of the ``L'' event in the decoding graph. Such events do not indicate exactly which gate suffered a leakage fault, since ``L'' events can only be detected by measuring a qubit, which only happens in an LRU or when measuring a syndrome. Therefore, upon registering an ``L'' event, the decoder only ``knows'' that the measured qubit leaked at some location between its initialization and measurement.\n\nConditioned on the observed ``L'' events, the HL decoder generates a new decoding graph from the Standard decoding graph ${\\cal G}$ whose edge weights are given by Table~\\ref{table:weights}. The ``L'' events are independent in our model, so suppose that a set of these events ${\\cal L}$ occurs. Because our leakage model independently depolarizes qubits when they interact with a leaked qubit, we can construct a conditional decoding graph ${\\cal G}_{\\cal L}$, for bit-flips say, whose edges $\\varepsilon$ again correspond to independent faults and whose edge weights are computed from conditional probabilities $p(\\varepsilon|{\\cal L})$. These conditional probabilities are computed in the same way as explained in Section~\\ref{sec:reduction}. ${\\cal G}_{\\cal L}$ is generally not translation invariant and contains additional edges not present in Fig.~\\ref{fig:cube}. There will be edges in ${\\cal G}_{\\cal L}$ with very low weight associated to highly probable errors on all of the qubits that interacted with leaked qubits.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=quickCircuit.pdf, width=.88\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:quickCircuit} (Color online) A measurement detects a leakage event in the Quick circuit at time $t+1$. This circuit depicts all of the gates in error-correction cycles $t$, $t+1$, and $t+2$. Each qubit that interacts with the potentially leaked qubit may be depolarized; subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $p_1$ in Fig.~6 (b)) where an ancilla leaks and introduces an $X$ error on qubit $U$ that propagates to other qubits. The notation ``$X\\mapsto I$'' indicates a cancellation of errors.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=quickKey,width=.23\\textwidth}\n   \\label{subfig:quickKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=quickXCube,width=.36\\textwidth}\n   \\label{subfig:quickXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=quickZCube,width=.27\\textwidth}\n   \\label{subfig:quickZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:quick} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Quick circuit. (b,c) Upon leakage detection at time $t$ on the plaquette ancilla *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nIf an ``L'' event occurs when measuring a syndrome, the corresponding syndrome bit is absent from the syndrome history and the usual approach for computing defect locations is somewhat modified. Suppose that ``L'' events occur for some check operator at $m$ contiguous times $t+1$, $\\dots$, $t+m$. In this case, each of the corresponding ``$a$'' edges $e_{t+1}$, $\\dots$, $e_{t+m}$ in the decoding graph are assigned weight $0$. Let $s_t$ be value of the check operator's syndrome bit at time $t$. If $s_t$ differs from $s_{t+m+1}$, we place a defect on the earliest vertex in $\\partial\\{e_{t+1}\\}$.\n\nTo compute the edge weights in the conditional decoding graph, we proceed as follows, examining each ``L'' event independently. When an ``L'' event occurs on some qubit $q$, we consider each of the $n$ fault locations between and including $q$'s initialization and measurement. Since each gate suffers leakage faults independently and with equal probability, the probability that $q$ is leaked when interacting at the $i$-th location is approximately $i\/n$. An $X$ and $Z$ error each occurs on the qubit that interacts with $q$ at location $i$ with probability $p_i = i\/2n$. We consider each of these errors separately, find the defect pair they cause, and modify the weight of the edge connecting this defect pair. Since $p_i$ can be large, we take any probability $p_0$ previously associated to the edge and set the new probability equal to $p_0+p_1-2p_0p_i$.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=circuitAncillaCircuit.pdf, width=.67\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitAncillaCircuit} (Color online) A plaquette ancilla measurement detects a leakage event at time $t$ in the \\emph{Partial-LRU} circuit. This circuit depicts all of the gates on this set of qubits in the $t$th and $t+1$th error-correction cycle. Each data qubit that interacts with the ancilla may be depolarized, and subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $p_3$ in Fig.~8 (c)) where an ancilla leaks and introduces a $Z$ error on qubit $R$ that propagates to other qubits.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=circuitAncillaKey,width=.23\\textwidth}\n   \\label{subfig:circuitAncillaKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitAncillaXCube,width=.36\\textwidth}\n   \\label{subfig:circuitAncillaXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitAncillaZCube,width=.27\\textwidth}\n   \\label{subfig:circuitAncillaZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitAncilla} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Partial-LRU scheme. (bc) Upon leakage detection at time $t$ on the plaquette ancilla *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nThe added edges and their weights are obviously specific to each leakage reduction circuit, so we now describe how to construct conditional decoding graphs in greater detail for the Quick and Partial-LRU circuits.\n\nFig.~\\ref{fig:quickCircuit} shows three rounds of plaquette measurements using the Quick circuit. Leakage is detected at time $t+1$ on the ancilla qubit labeled with $*$. Fig.~\\ref{fig:quick} shows the relative position of the qubits in the circuit, and the low weight edges that are added to or modified in the $X$- and $Z$-error conditional decoding graphs. The edges are shown relative to the position of the leaked qubit.  Consider the following example in which we show that the long edge labelled $p_1$ in Fig.~\\ref{subfig:quickXCube} is associated to an event that occurs with probability $\\approx 1\/22$ conditioned on the ``L'' event shown in Fig.~\\ref{fig:quickCircuit}. This edge is not present in the standard decoding graph and is added to the conditional decoding graph for $X$ errors. The error labels in Fig.~\\ref{fig:quickCircuit} show the events responsible for addition of the edge. A total of $11$ gates act on the leaked qubit between its initialization and measurement, and the qubit is leaked directly after initialization with probability $1\/11$. Therefore, an $X$ error spreads to qubit $U$ at time $t$ with probability $1\/2 \\times 1\/11$. This $X$ error spreads to qubit $\\alpha$ at time $t$ and continues to be detected by the $\\alpha$ plaquette from time $t$ onward. An $X$ error also spreads to the ancilla that replaces the ancilla $*$ at time $t+1$, which in turn spreads to qubit $\\delta$ starting at time $t+2$, as indicated. Therefore, this error introduces the long edge labeled $p_1$ connecting the $\\alpha$ and $\\delta$ qubits in the $X$ error decoding graph. Other edges in the figure are obtained by considering other errors.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n\\begin{minipage}[c]{\\textwidth}\n\\centering\n\\epsfig{file=circuitHDataCircuit.pdf, width=.61\\textwidth}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitHDataCircuit} (Color online) A data qubit's trailing LRU detects a leakage event at time $t$. This circuit depicts all of the gates that act on the data qubit in the $t$th error-correction cycle, together with the preceding LRU. Each ancillary qubit that interacts with the data may be depolarized, as well as the outgoing qubit of the trailing LRU, and subsequent gates in the cycle potentially spread the errors. The error labels show an example fault path (corresponding to the edge labeled $P_3$ in Fig.~10 (c)) where the data leaks and introduces a $Z$ error on an ancilla qubit.}\n\\vspace*{13pt}\n\\end{minipage}\n\\begin{minipage}[c]{\\textwidth}\n \\subfloat[short for lof][]{\n \\epsfig{file=circuitHDataKey,width=.23\\textwidth}\n   \\label{subfig:circuitHDataKey}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitHDataXCube,width=.36\\textwidth}\n   \\label{subfig:circuitHDataXCube}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitHDataZCube,width=.30\\textwidth}\n   \\label{subfig:circuitHDataZCube}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:circuitHData} (Color online) (a) The location of the detected leakage (labeled by $*$) and the relative locations of other qubits in the Partial-LRU scheme. (bc) Upon leakage detection at time $t$ on the data qubit *, the edge weights in the decoding graphs are updated as indicated.}\n\\vspace*{13pt}\n\\end{minipage}\n\\end{figure}\n\nUnlike the Quick circuit, the Partial-LRU circuit can detect leakage on ancillas, which are directly measured, and data qubits, which are measured by LRUs. First consider the ancillas. Two rounds of plaquette measurements are shown in Fig.~\\ref{fig:circuitAncillaCircuit} with an ``L'' event at the ancilla $*$ at time $t$. The modified edges of the decoding graph, conditioned on the ``L'' event, are shown in Fig.~\\ref{fig:circuitAncilla}. The error labels in Fig.~\\ref{fig:circuitAncillaCircuit} show the events responsible for addition of the edge labeled $p_3$ in Fig.~\\ref{subfig:circuitAncillaZCube}. The probability that the ancilla is leaked after the second $CNOT$ gate is $3\/5$ and a $Z$ error spreads to qubit $R$ with probability $p_3=3\/10$. The $Z$ error spreads to the qubits labeled $2$ and $4$ at time $t$ and $t+1$, and hence we modify the weight of the diagonal edge labeled $p_3$ connecting these two qubits in the decoding graph. Next, consider Fig.~\\ref{fig:circuitHDataCircuit}, which shows LRUs acting on a data qubit. If the second LRU detects leakage, the edges shown in Fig.~\\ref{fig:circuitHData} are added or modified to construct the conditional decoding graphs. The error labels in Fig.~\\ref{fig:circuitHDataCircuit} correspond to the error responsible for addition of edge labeled $p_3$ in the $Z$ error decoding graph.\n\n\n\\section{Introduction}\n\nLarge-scale quantum computers require a fault-tolerant architecture based on quantum error-correcting codes. While there are many approaches to quantum fault-tolerance, topological codes~\\cite{kitaev03,bravyi98} stand out due to their many favorable properties, including local check operators, simple syndrome extraction circuits~\\cite{dennis02}, and flexible fault-tolerant logic based on transversal gates, code deformation~\\cite{rh07,raussendorf07}, or lattice surgery~\\cite{horsman12}. These properties endow topological codes with a high accuracy threshold. The value of the surface code's accuracy threshold depends on assumptions regarding the noise model and error-correction approach, but estimates vary from $0.67\\%$ \\cite{stephens14} to above $1\\%$ \\cite{wang11}.\n\nThese threshold estimates assume a depolarizing noise model that approximates realistic noise, but the model does not include so-called leakage faults. Leakage faults map quantum states out of the 2-dimensional qubit subspace and into a higher-dimensional Hilbert space. Many physical realizations of qubits are multilevel systems that we expect to suffer from leakage faults. Leakage errors can be mitigated by constructing quantum circuits with suitable properties. Broadly speaking, such gadgets convert leakage errors into ``regular'' errors and may or may not simultaneously raise a flag to indicate the location of the leakage error. Gadgets that detect leakage convert it into a located loss error that is easier to correct~\\cite{gottesman97,preskill98,knill98}. Leakage reduction units (LRUs) convert leakage errors into regular errors but do not give an indication that leakage has occurred~\\cite{aliferis07, fong11}. Leakage mitigation with LRUs was rigorously analyzed by Aliferis and Terhal~\\cite{aliferis07}, who showed existence of a threshold to leakage in concatenated codes. Subsequent work has shown that circuits can be further simplified while remaining fault-tolerant \\cite{fortescue14}.\n\nMost studies of quantum codes do not consider errors due to leakage, and the cost of leakage mitigation has not been thoroughly understood, particularly in the surface code. When leakage errors are converted to detected loss, topological codes are known to have a 50\\% loss threshold in an idealized model~\\cite{stace09,fujii12}. However, the strategy of~\\cite{stace09} measures high weight ``super'' check operators. While this may be natural for measurement-based quantum computing \\cite{barrett10}, it is more difficult with a fixed planar array of qubits, as is the case with some superconducting architectures \\cite{chow14}. Furthermore, if leakage errors instead remain undetected, or are detected but could have originated from multiple space-time points in the circuit, they may lead to additional error spread. Leakage of this form has been studied for a quantum repetition code~\\cite{fowler13} and for specific types of gates applied to superconducting qubits~\\cite{ghosh13, ghosh14}. Despite this work, the impact of leakage on the accuracy threshold of topological codes remains unclear. To address this problem, we systematically study several leakage reducing circuits and develop a new syndrome processing strategy that further enhances the benefits of these circuits.\n\nThe paper is organized as follows. Section~\\ref{sec:toriccode} reviews the toric code and fault-tolerant error-correction using bare stabilizer measurements. Section~\\ref{sec:leakagemodel} introduces a new model of leakage faults that is amenable to simulation within the stabilizer formalism, much like the standard depolarizing noise model. Section~\\ref{sec:reduction} reviews circuit-based leakage reduction techniques, introduces the leakage mitigation circuits we study and why they work, and explains how matching-based syndrome processing algorithms are modified for these circuits. Section~\\ref{sec:heralded} then proposes a model where we are given 3-outcome projective measurements for detecting leaked qubits and shows how to further modify syndrome processing algorithms to take advantage of this extra information. Section~\\ref{sec:simulation} explains our simulation approach and relevant details of the implementation. Finally, Section~\\ref{sec:results} presents our numerical results and Section~\\ref{sec:conclusion} concludes.\n\n\n\\section{Leakage Errors}\\label{sec:leakagemodel}\n\nLeakage errors are possible because the two-dimensional Hilbert space of a qubit is a subspace of a larger physical Hilbert space and interactions with the system may populate states in that larger physical space. The two-dimensional Hilbert space ${\\cal H}_{C[j]}$ of the $j$th qubit is encoded into a Hilbert space ${\\cal H}_{S[j]}={\\cal H}_{C[j]}\\oplus {\\cal H}_{L[j]}$ where ${\\cal H}_{L[j]}$ is an auxiliary leakage space associated to the qubit. We say that the $j$th qubit is {\\em leaked} (resp. {\\em contained}) if its state is supported entirely on ${\\cal H}_{L[j]}$ (resp. ${\\cal H}_{C[j]}$). The total Hilbert space of the system is ${\\cal H}=\\bigotimes_j{\\cal H}_{S[j]}={\\cal H}_S\\oplus {\\cal H}_L$ where ${\\cal H}_S=\\bigotimes_j{\\cal H}_{C[j]}$ is the computational subspace and ${\\cal H}_L$ is the remaining leakage subspace. States supported on ${\\cal H}_L$ have at least one leaked qubit.\n\nConsider a quantum computation implemented by one- and two-subsystem unitary gates that act on the total Hilbert space. For convenience we refer to these as one- and two-qubit gates. A {\\em sealed single qubit gate} maps states in ${\\cal H}_{C[j]}$ back to ${\\cal H}_{C[j]}$ and therefore has the form $U_j = U_{{\\cal H}_{C[j]}}\\oplus U_{{\\cal H}_{L[j]}}$, i.e. it does not leak. Similarly, a {\\em sealed two-qubit gate} acting on states in ${\\cal H}_{S[i]}\\otimes {\\cal H}_{S[j]}$ is a direct sum of unitary gates\n\\begin{equation}\nU = U_{{\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}} \\oplus\nU_{{\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}} \\oplus\nU_{{\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}} \\oplus\nU_{{\\cal H}_{L[i]}\\otimes {\\cal H}_{L[j]}}.\\label{eq:idealgate}\n\\end{equation}\nIt is clear that the subspaces corresponding to each block are invariant under these gates. In this sense, sealed gates do not propagate leakage errors. In many physical settings, where gates are applied by a resonant or off-resonant drive or by tuning subsystems into resonance \\cite{chow11,strauch13}, all blocks except ${\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}$ are approximately diagonal in the eigenbasis of the undriven system Hamiltonian and hence the physical gates are approximately sealed.\n\nLeakage errors are generally coherent operations that are challenging to simulate for large quantum codes. Physical leakage noise may not be accurately captured by a Markovian noise model and generally needs to be treated as a coherent process \\cite{aliferis07}. For simplicity and tractability, we define and simulate a stochastic leakage model, analogous to the standard approach of using depolarizing faults as a proxy for realistic decoherence and control faults. Although such a model does not accurately capture all conceivable leakage phenomena, it captures some aspects and yet remains amenable to stabilizer simulation.\n\nOur stochastic model assumes ideal gates are sealed and that each subsystem is a 3-level system (a qutrit). We add three new elements to define faulty gates. The first two elements are rather straightforward. First, discrete leakage events occur independently with probability $p_\\uparrow$ on each output qutrit of an ideal gate. These events are described by the map\n\\begin{equation}\n{\\cal E}_\\uparrow(\\rho_j)=(1-p_\\uparrow)\\rho_j + p_\\uparrow |2\\rangle\\langle 2|\n\\end{equation}\nwhere $\\rho_j$ is the state of the $j$th qutrit. Second, there is a relaxation process with probability $p_\\downarrow$, analogous to amplitude damping, that acts independently on each output qutrit of an ideal gate. The stochastic map for this process is given by\n\\begin{equation}\n{\\cal E}_\\downarrow(\\rho_j)=(1-p_\\downarrow)\\rho_j+p_\\downarrow {\\cal E}_{\\mathrm{decay}}(\\rho_j)\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal E}_{\\mathrm{decay}}(\\rho_j)=A\\rho_j A^\\dagger + \\sum_{k=0,1} A_{k}\\rho A_{k}^\\dagger\n\\end{equation}\nis a decay map with elements\n\\begin{equation}\nA = |0\\rangle\\langle 0| + |1\\rangle\\langle 1|,\\ A_{k} = \\frac{1}{\\sqrt{2}}|k\\rangle\\langle 2|.\n\\end{equation}\nThese elements are chosen so that when a leaked qubit decays, it is replaced by a maximally mixed contained qubit. Let $Z^{(2)}=|0\\rangle\\langle 0|+|1\\rangle\\langle 1|-|2\\rangle\\langle 2|$ and suppose that an initial state $\\rho_0$ is a fixed point of\n\\begin{equation}\n\\Lambda(\\rho) = \\frac{1}{2}\\left( \\rho + Z^{(2)}\\rho Z^{(2)} \\right).\n\\end{equation}\nThen it is clear that any sequence of operations ${\\cal E}_\\uparrow$, ${\\cal E}_\\downarrow$, and sealed single qubit gates applied to $\\rho_0$ produces a state that is also a fixed point of $\\Lambda$, i.e. states are mixtures where qubits are leaked or contained. For convenience we call such mixtures on any number of qubits $|2\\rangle$-{\\em dephased}.\n\nThe final element of our model for faulty gates concerns two-qubit gates. Sealed two-qubit gates apply the intended operation to the ${\\cal H}_{C[i]}\\otimes {\\cal H}_{C[j]}$ block and act arbitrarily on the remaining blocks. The ${\\cal H}_{L[i]}\\otimes {\\cal H}_{L[j]}$ block consists of a single matrix element that applies what amounts to a harmless global phase in our model. The ${\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}$ and ${\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}$ blocks of a sealed gate merely apply a single qubit gate to qubit $i$ or $j$, respectively. As a final simplifying assumption, these blocks each implement Haar distributed random unitary gates, so that the resulting process completely depolarizes the contained qubit. Noisy two-qubit gates are followed by independent excitation and relaxation maps on each output qutrit. This is motivated by the assumption that matrix elements coupling computational states to $|22\\rangle$ are significantly smaller than other matrix elements of the drive Hamiltonian. Like before, $|2\\rangle$-dephased states remain so under the action of these two-qubit gates.\n\nIt is important to observe that this two-qubit gate model destroys correlations that could otherwise arise from interacting with leaked qubits. When multiple contained qubits interact with a leaked qubit, those contained qubits will experience independent depolarizing noise rather than some collective process. This would not be the case if, for example, the ${\\cal H}_{C[i]}\\otimes {\\cal H}_{L[j]}$ and ${\\cal H}_{L[i]}\\otimes {\\cal H}_{C[j]}$ blocks are fixed unknown single qubit unitaries. In this sense, our model is not a worst-case stochastic model. However, for topological codes, each leaked qubit interacts with a small set of neighbors and persists for at most a few syndrome extraction cycles before being reset by leakage reduction circuits. This would limit the impact of even a worst-case stochastic model but quantifying that impact we leave to future work.\n\nStabilizer circuits constructed from one- and two- qubit gates under the influence of our noise model can be simulated within the stabilizer formalism. The main observation is that $|2\\rangle$-dephased states remain $|2\\rangle$-dephased in our model, so we can introduce a classical {\\em leakage indicator bit} for each qubit whose value indicates if the qubit is contained or leaked. We assume all qubits are initially contained. Quantum operations now act on an $n$-qubit stabilizer state and an $n$-bit leakage indicator array in the following way. The map ${\\cal E}_\\uparrow$ acts as identity with probability $1-p_\\uparrow$ and otherwise traces out the corresponding qubit and sets the leakage indicator bit. Likewise ${\\cal E}_{\\mathrm{decay}}$ resets the leakage indicator bit and replaces the corresponding qubit, setting its state to be completely mixed. How sealed gates act on the $n$-qubit stabilizer state depends on the value of the leakage indicator array. Roughly speaking, the future quantum circuit is rewritten depending on the state of the leakage indicators. For our model, sealed one-qubit gates act as intended on contained qubits and as identity on leaked qubits. Likewise, sealed two-qubit gates also act as intended when both inputs are contained. However, a sealed two-qubit gate depolarizes one of the input qubits when the other input is leaked. When both inputs are leaked, a sealed two-qubit gate acts as identity. Taken together, and given a suitable definition of measurement for leaked qubits, one can sample from the output distribution of faulty stabilizer circuits built from these elements.\n\nOften a full-blown stabilizer simulation is not needed for the study of fault-tolerant quantum error-correction and it suffices to track the Pauli error for each qubit. This works because we assume that the code qubits are always in an eigenstate of the stabilizer, and we can use the error operator to compute the eigenvalues of the stabilizers and normalizers and to predict deviations from ideal measurement outcomes without maintaining a description of the current quantum state as a set of stabilizer generators. For our leakage model, can we similarly divest ourselves of the full stabilizer framework and simply propagate for each qubit a Pauli error operator and a leakage indicator bit? The main concern is that the stabilizer circuit we simulate may not always measure operators in the current stabilizer when one or more leakage indicators are set. For example, leaked code qubits cause subsequent syndrome extraction circuits to measure check operators that are supported only on contained qubits. Likewise, leaked syndrome qubits ``turn off'' the associated check operator measurement. Without knowing the full stabilizer, we cannot predict how the measurement outcomes are correlated through parity constraints enforced by the stabilizer. The key observation that enables us merely to work with Pauli error and leakage indicator labels is that our model completely depolarizes syndrome qubits when code qubits are leaked, destroying any correlations that might otherwise have been present. Furthermore, when leaked qubits relax or pass through circuits to reset them, they are replaced by completely depolarized qubits. This guarantees that past syndrome history is uncorrelated with the particular state of the reset qubit (although it is correlated with the fact that the qubit has leaked). Reassured by this rough argument, we do not retain a complete set of stabilizer generators for the quantum state in the simulation. Instead, we simulate the model by maintaining and propagating a label ($I$, $X$, $Y$, $Z$ or $L$) for each qubit, where ``L'' denotes that the leakage indicator bit is set.\n\nNow that we have explained our model and justified our simulation approach, here follows a summary of the details of the model. The model has parameters $p$ (depolarizing error probability), $q$ (syndrome measurement error probability, we set $q=p$), $p_\\uparrow$ (probability of excitation outside of the computational state), and $p_\\downarrow$ (probability of relaxation back to the computational state). For convenience, we also define the relative excitation rate $r = p_\\uparrow \/ p$ and relaxation rate $s = p_\\downarrow \/ p$. Elementary quantum gates behave as follows. Idle qubits are subject to depolarizing noise with probability $p$, where we apply one of $X$, $Y$, or $Z$ uniformly at random. They undergo free evolution and do not leak. However, a leaked qubit may relax to the computational space with probability $p_\\downarrow$. The $|0\\rangle$ or $|+\\rangle$ state preparation succeeds with probability $1-p$, otherwise the orthogonal state is prepared. With probability $p_\\uparrow$ the freshly prepared qubit leaks. If the input qubit is in the computational space, measurements report the incorrect outcome with probability $p$ and otherwise report the correct outcome. When the input qubit is leaked, we consider two scenarios: either the measurement cannot distinguish higher levels and reports ``1'', or the measurement has a third outcome ``L''. Noisy CNOT gates are subject to joint depolarizing noise with probability $p$, where we apply one of the non-identity two-qubit Pauli errors uniformly at random. In addition, if any input to the CNOT is leaked, all non-leaked qubits are completely depolarized, i.e. one of the $4$ single-qubit Pauli operators is applied uniformly to each non-leaked qubit. Finally, each output qubit leaks with probability $p_\\uparrow$ and relaxes with probability $p_\\downarrow$.\n\nUnlike a similar model in~\\cite{fowler13}, we assume that qubits reach a steady state leakage error rate at the beginning of any simulation, which more faithfully models the conditions of a long computation with the code. Consider the error-correction circuits in the standard toric code. Unlike ancillas, the data qubits are never re-initialized, and therefore they gradually accumulate leakage according to the transition probabilities $p_\\uparrow$ and $p_\\downarrow$ until an equilibrium is reached. It is easy to show from direct calculation of eigenstates of the transition matrix that the equilibrium distribution for data qubits in the toric code is given by $p_{eq}\\approx\\frac{4p_\\uparrow}{4p_\\uparrow+6p_\\downarrow}$ where the factor $4$ is due to the four $CNOT$ gates, and the factor $6$ is due to the four $CNOT$s and two idle time steps. The second eigenvalue of the transition matrix is $(1-p_\\downarrow)^6(1-p_\\uparrow)^4$, so the non-equilibrium component of the distribution for each qubit decays as $\\mathrm{exp}(-\\zeta n)$ where $\\zeta=-6\\ln\\left[(1-p_\\downarrow)(1-p_\\uparrow)\\right]$. This suggests that starting from an equilibrium distribution may not be an overly pessimistic assumption. Under the reasonable assumptions $p_\\uparrow \\approx p_\\downarrow$, the fraction of leaked data qubits reaches $40\\%$ at which point the threshold is far exceeded and computation is clearly impossible (without leakage reduction, of course).\n\n\n\\section{Leakage Reduction}\\label{sec:reduction}\n\\noindent\n\nLeakage reduction is a process that converts leakage errors into regular errors that can be later corrected by quantum codes. Prompt leakage reduction is desirable as a single leaked qubit can cause many other errors (e.g., a leaked data qubit can damage ancillas which then spread more errors to other data qubits). A circuit that converts leakage on a single qubit into a regular error is a leakage reduction unit (LRU). LRUs satisfy two properties \\cite{aliferis07}: (a) if their input is in the computational space then the identity operation is performed, and (b) if their input is in the leakage space then their output is some state in the computational space. We use LRUs as building blocks to construct several possible leakage suppressing error-correction circuits in the toric code.\n\n\\begin{figure}[b]\n\\vspace*{13pt}\n\\centering\n\\epsfig{file=LRU,width=.19\\textwidth}\n\\fcaption{\\label{fig:lru} A leakage reduction unit (LRU) implemented by one-bit teleportation.}\n\\end{figure}\n\nAn LRU based on one-bit teleportation \\cite{zhou00} is shown in Fig.~\\ref{fig:lru}. This LRU differs from canonical LRUs based on full quantum teleportation \\cite{mochon04,aliferis07}. It does not satisfy the properties of an LRU if ideal two-qubit gates can output a pair of leaked qubits when one of the inputs is leaked. In our model, ideal two-qubit gates are sealed and therefore the simpler one-bit teleportation circuit is an LRU and uses fewer resources than full quantum teleportation. Note that reinitializing qubits with most of their population in the computational space is essential to implementing any LRU in practice.\n\nWe use this LRU to build several error-correction circuits that represent a tradeoff between the circuit complexity and effectiveness of leakage reduction. We list the circuits from the most complicated one with the best leakage suppression capability to the simplest one with the fewest extra ancillas:\n\n\\begin{enumerate}\n\\item \\textbf{Full-LRU}: We call an error-correction circuit that replaces each gate by a rectangle containing the gate followed by LRUs on each output qubit (as in \\cite{aliferis07}) the \\emph{Full-LRU} circuit; see Fig~\\ref{fig:LRcircuits}~(a). This circuit removes leakage immediately after each gate but uses a large number of additional gates and qubits per cycle. The Full-LRU circuit uses $16$ additional gates (in $4$ LRUs) per data and ancilla qubit per cycle. No LRU is applied to ancillas prior to measurement.\n\n\\item \\textbf{Partial-LRU}: The \\emph{Partial-LRU} error-correction circuit uses LRUs less frequently than the Full-LRU circuit. Once per error correction cycle, LRUs act on each data qubit while the ancillas are measured, as shown in Fig.~\\ref{fig:LRcircuits}~(b). No additional effort is made for the ancillas, which are periodically measured and reinitialized anyway. While this strategy applies LRUs less frequently, which leads to greater spread of errors relative to the \\emph{Full-LRU}, it only uses $4$ additional gates (in $1$ LRU) per data qubit per syndrome extraction and no additional gates applied to the ancillas.\n\n\\item \\textbf{Quick}: The \\emph{Quick} error-correction circuit further reduces the frequency of leakage reduction; see Fig.~\\ref{fig:LRcircuits}~(c). At the end of each cycle, the circuit swaps each data qubit with an ancilla. A similar construction was proposed in~\\cite{ghosh14} and shown to reduce leakage for a particular model relevant to superconducting qubits. In our case we choose to swap the ancilla with the data qubit $d_D$ immediately below the ancilla. Three $CNOT$ gates implement a $SWAP$ and, due to gate cancellation, there is only one additional $CNOT$ gate per data qubit compared to the standard circuit. After the swap, the physical qubits representing data and ancilla have traded roles. Each physical qubit is measured and reset every other cycle, so qubits do not remain leaked for many cycles.\n\n\\item \\textbf{No LRU}: The last and simplest circuit we study is the standard syndrome extraction circuit depicted in Fig.~\\ref{fig:toric}~(b,c), which we call \\emph{No LRU} for convenience. The circuit is not suitable for error correction in the presence of leakage since data qubits are never reset. This circuit provides a point of reference from which to assess the effectiveness of the other circuits.\n\\end{enumerate}\n\nWhile the Partial- and Full-LRU circuits are relatively straightforward to understand since they incorporate LRUs directly, the Quick circuit does not use LRUs. If there are no leakage errors, it is clear that the Quick circuit is functionally equivalent to the standard circuit. However, if an input ancilla or data qubit has leaked, one can calculate the effective circuit within our leakage model. Referring to Fig.~\\ref{fig:LRcircuits}~(c), a leaked ancilla $a_z$ depolarizes each data qubit in the corresponding check, and the leaked qubit takes the role of $d_D$. The original $d_D$ is depolarized and measured, producing a corrupted syndrome bit. On the other hand, a leaked data qubit, which we take to be $d_D$, depolarizes the ancilla $a_z$. The data and ancilla exchange roles and the leaked qubit is measured. Meanwhile, $d_U$, $d_L$, and $d_R$ have interacted with $a_z$ before it was depolarized. One can check that these gates merely re-encode the plaquette operators and do not produce additional errors.\n\nThe realization of the Quick circuit in the toric code does not require extra ancillas. In the surface code, which does not have periodic boundary conditions, one could alternate swapping the ancillas with the $D$ and $U$ qubits in every odd and even error-correction cycle respectively. This approach only requires $O(d)$ additional qubits at the boundary of the lattice.\n\n\\begin{figure}[t]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=fullLRU,width=.37\\textwidth}\n   \\label{subfig:full}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=circuitModel,width=.27\\textwidth}\n   \\label{subfig:partial}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=quickModel,width=.32\\textwidth}\n   \\label{subfig:quick}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:LRcircuits} (a) The Full-LRU circuit replaces each gate with the gate followed by LRUs on each output qubit. (b) In the partial-LRU scheme, we insert an LRU on each data qubit at the end of each error-correction cycle. (c) The Quick scheme uses a circuit that is equivalent to those in Fig.~\\ref{fig:toric}. The final CNOT gate acting on the ancilla and the $d_U$ data qubit has been replaced by a CNOT followed by a SWAP. This simplifies to a pair of CNOT gates.}\n\\end{figure}\n\nTo decode syndromes produced by the leakage reducing circuits, we adjust the edge weights in the decoding graph so that the corresponding decoders again find a correction from among the most likely errors consistent with the syndrome. The decoder paired with the standard error-correction circuit (our No LRU circuit) uses a decoding graph depicted in Fig.~\\ref{fig:cube}. The prior probability of an edge $\\varepsilon$ (restricted to $\\{a,b,c,d,e,f\\}$ due to translation invariance) is taken as $p(\\varepsilon)=\\sum_j p_j(\\varepsilon)$ where $p_j(\\varepsilon)$ is the probability that a fault occurs at location $j$ and yields the syndrome $\\partial\\{\\varepsilon\\}$. The sum is taken over all locations in the circuit although not all locations contribute positive probability. If the physical error rates are sufficiently small, $p(\\varepsilon)$ is approximately proportional to the probability that $\\varepsilon$ carries an error. The edge weights are then given by $-\\log p(\\varepsilon)$. Edge weights for the standard circuit were approximated in this way in~\\cite{wang11} by counting single-location faults, and we use an analogous approach to calculate the participating edges of the decoding graph and their weights for the other circuits introduced in this section. Specifically, we iterate over regular faults placed at single fault locations $j$, compute the corresponding syndrome and associated pair of defects, identify an edge $\\varepsilon$ with the same pair of defects $\\partial\\{\\varepsilon\\}$, and add the probability of the fault $p_j(\\varepsilon)$ to $p(\\varepsilon)$.\n\n\\begin{table}[t]\n\\tcaption{Edges weights for each leakage reduction circuit}\n\\centerline{\\footnotesize\\smalllineskip\n\\begin{tabular}{r c c c c c c}\\\\\n\\hline\n{} & $\\mathrm{exp}(-w_a)$ & $\\mathrm{exp}(-w_b)$ & $\\mathrm{exp}(-w_c)$ & $\\mathrm{exp}(-w_d)$ & $\\mathrm{exp}(-w_e)$ & $\\mathrm{exp}(-w_f)$ \\\\\n\\hline\n{No LRU} & $31\/15p+q$ & $28\/15p$ & $16\/15p$ & $52\/15p$ & $8\/15p$ & $8\/15p$ \\\\\n{Quick} & $7\/3p+q$ & $32\/15p$ & $4\/3p$ & $4p$ & $8\/15p$ & $8\/15p$ \\\\\n{Full-LRU} & $103\/15p+q$ & $52\/15p$ & $88\/15p$ & $172\/15p$ & $32\/15p$ & $32\/15p$ \\\\\n{Partial-LRU} & $31\/15p+q$ & $52\/15p$ & $16\/15p$ & $76\/15p$ & $8\/15p$ & $8\/15p$ \\\\\n\\hline\\\\\n\\end{tabular}}\n\\label{table:weights}\n\\end{table}\n\nThe calculated edge weights in the decoding graph are summarized in Table~\\ref{table:weights}. Here $p$ is the probability of gate, initialization, and idle error and $q$ is the probability of measurement error, which we set equal to $p$. The edge labels correspond to labels shown in Fig.~\\ref{fig:cube}. The symmetry between the weights of edge pairs $b$-$d$ and $c$-$e$ is broken because the stabilizer measurements are not atomic operations but involve $CNOT$ gates applied in a particular order (see \\cite{wang11}). Although the leakage reducing circuits differ from the No LRU circuit, there are no new edges in the decoding graphs of these circuits. One can see that, for example, the weights for the Full-LRU circuit on edges $e$ and $f$ are $4$ times the corresponding weights of the other circuits due to the $4$ locations in the LRU. The edge weights for the No LRU circuit agree with the weights in~\\cite{wang11} except for edge $a$. Our probability for edge $a$ is higher because our error-correction circuits include ancilla initializations. We call a decoder that uses these edge weights the \\emph{Standard decoder}. Note that the Standard decoder does not account for leakage events, and hence we expect it will be suboptimal.\n\n\\section{Results}\\label{sec:results}\n\\noindent\n\nA systematic exploration of the accuracy thresholds for the Standard and HL decoders is depicted in Fig.~\\ref{fig:thresholds}~(a). The plot shows the accuracy threshold as a function of the amount of leakage (the relative excitation rate $r$). The plot was obtained by recording the failure rates for code distance $d=7$ and $d=9$ and by varying $r$ in increments of $0.1$ and $p$ in increments of $0.01\\%$. We chose a relative relaxation rate $s = 1$. For each parameter choice we found the crossover of the failure rates for the two code distances, which approximates the threshold.\n\n\\begin{figure}[b!]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=allThresholds,width=.595\\textwidth}\n   \\label{subfig:all}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=pdExploration,width=.365\\textwidth}\n   \\label{subfig:pdexplore}\n}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:thresholds} (Color online) (a) Summary of thresholds for the HL and Standard decoders. The HL decoder offers a substantially improved threshold. (b) The threshold, show here for the Quick circuit with HL decoder, does not change with the relative relaxation rate $s$.}\n\\end{figure}\n\nBased on Fig.~\\ref{fig:thresholds}~(a) we make the following observations. In the regime with no leakage ($r = 0$), simpler circuits, i.e. circuit with fewer locations per syndrome extraction, have higher thresholds than more complicated circuits, in agreement with expectations. With no leakage, the No LRU circuit has a threshold of about $0.70\\%$ whereas the Full-LRU circuit, which is the most complicated circuit we studied, has a threshold of about $0.22\\%$. We also observe that the Quick and Partial-LRU circuits give an almost identical threshold, differing visibly only for $r<0.3$. The Quick circuit benefits from having fewer gates and qubits, whereas the Partial-LRU benefits from more frequent qubit reinitializations and therefore more effective leakage suppression. These features appear to have comparable effects on the threshold. The HL decoder significantly improves the threshold compared to the Standard decoder when $r$ exceeds about $0.3$.\n\nAs the $r$ increases, the threshold decreases monotonically for all circuits and decoders. To understand this, consider an idealized setting where we assume that leaked qubits are immediately replaced by completely depolarized qubits, causing errors $X$, $Y$, $Z$, and $I$ each with equal probability. This makes a leakage event equivalent to a regular depolarizing error with probability $\\beta=3\/4$. Suppose that physical errors occur at an effective rate $\\tilde{p}=(1+\\beta r)p$ that is the sum of two terms where the first one is due to regular depolarizing errors and the second one due to leakage. Let's also assume that there is some minimum number of faults $m$ that can produce a logical error. The logical error rate is then given by\n\\begin{equation}\np' = A\\tilde{p}^m+O(\\tilde{p}^{m+1}) = A(1+\\beta r)^{m}p^m + O(p^{m+1}).\n\\end{equation}\nNeglecting the higher order terms and solving for the fixed point gives us a very rough approximation for the threshold as\n\\begin{equation}\np_{th}^{(r)} \\approx \\left[A(1+\\beta r)^m\\right]^{-1\/(m-1)}.\n\\end{equation}\nWe are interested in the ratio of thresholds when $r=0$ versus $r>0$, which is\n\\begin{equation}\n\\frac{p_{th}^{(r)}}{p_{th}^{(0)}} = (1+\\beta r)^{-m\/(m-1)} \\approx \\frac{1}{1+\\beta r}.\n\\end{equation}\n\n\\begin{figure} [b!]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=threshCombined.pdf,width=.585\\textwidth}\n   \\label{subfig:linear}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=threshCombinedLog.pdf,width=.375\\textwidth}\n   \\label{subfig:log}\n}\n\\fcaption{\\label{fig:thCombined} (Color online) Comparisons of decoding success probabilities for the Full-LRU circuit and the Quick circuit with the Standard and HL decoder. (a) Linear scale on the left, (b) log scale on the right.}\n\\end{figure}\n\nThe threshold then decays as $\\frac{\\alpha}{1+\\beta r}$ where $\\alpha$ is a constant equal to the numeric value of the threshold in the regime without leakage. Assuming $\\beta=3\/4$ and choosing $\\alpha$ such that the threshold at $r=0$ intersects the threshold of the Quick circuit with HL decoder, we plot the threshold of the ``idealized decoder'' in Fig.~\\ref{fig:thresholds}~(a). A Standard decoder's threshold will not exceed that of the idealized one we just described, but a decoder that uses results from three-outcome measurements could in principle. Table~\\ref{table:fitparameters} lists some extracted values of $\\alpha$ and $\\beta$ from the data in Fig.~\\ref{fig:thresholds}~(a).\n\nFig.~\\ref{fig:thresholds}~(b) shows that the threshold does not depend much on the relaxation rate $s$, likely because relaxation is much slower than the leakage reduction occuring in the circuits. This plot was obtained by recording thresholds for the Quick circuit with the HL decoder using the same technique as for Fig.~\\ref{fig:thresholds}.\n\nIn our last set of simulations, we chose $r = s = 1$ and recorded the failure rate of the decoders as a function of $p$. Recall that $p_\\uparrow = r p$ and $p_\\downarrow = s p$. In Fig.~\\ref{fig:thCombined} (a) we show failure rates for the Full-LRU circuit and the Quick circuit with both the Standard and HL Decoder. The thresholds of each circuit\/decoder are visible as crossing points of the curves. The Full-LRU circuit has the lowest threshold, followed by the Quick circuit with the Standard decoder and the Quick circuit with the HL decoder. The success rate for the Partial-LRU circuit is not plotted because it was indistinguishable from the Quick circuit.\n\nThe same plot on the log scale in Fig.~\\ref{fig:thCombined} (b) shows that when $p$ is well below threshold, the success rate of the Full-LRU circuit improves faster than for the Quick circuit with Standard decoder, which suggests that for low enough physical error rates the Full-LRU circuit will be better than the other circuits. We have a sufficient number of samples to observe at least $100$ failures at each data point shown in the figure. Table~\\ref{table:fitparameters} presents the degree of error suppression $\\gamma$ for each circuit based on fitting the logarithm of the logical error rate. For the HL decoder, we give the fit for each distance $d=7$, $9$, and $11$ since we appear to be observing the effects of higher order terms in the logical error rate and have not reached a sufficiently low physical error rate to capture the lowest order term in the polynomial. Since numeric simulations at extremely low physical error rates are inefficient and may require different techniques \\cite{bravyi13}, we leave further exploration of this question for future work.\n\n\\begin{table}[t]\n\\tcaption{Threshold $\\sim \\frac{\\alpha}{1+\\beta r}$ and sub-threshold logical error rate $\\sim Ap^{\\gamma d}$}\n\\centerline{\\footnotesize\\smalllineskip\n\\begin{tabular}{r c c c}\\\\\n\\hline\n{} & $\\alpha$ & $\\beta$ & $\\gamma$ \\\\\n\\hline\n{idealized} & $0.65$ & $3\/4$ & n.a. \\\\\n{Full-LRU} & $0.22$ & $1.72$ & $~0.67$\\\\\n{Quick} & $0.65$ & $3.59$ & $~0.45$ \\\\\n{Partial-LRU} & $0.55$ & $2.55$ & $~0.46$ \\\\\n{Quick (HL)} & $0.62$ & $1.23$ & $~\\{0.45,0.52,0.64\\}$ \\\\\n{Partial-LRU (HL)} & $0.55$ & $0.92$ & $~\\{0.48,0.53,0.65\\}$ \\\\\n\\hline\\\\\n\\end{tabular}}\n\\label{table:fitparameters}\n\\end{table} \n\\section{Simulation Methods}\\label{sec:simulation}\n\\noindent\n\nWe implemented simulations of the No LRU, Partial-LRU, Full-LRU, and Quick leakage reduction strategies. We evaluate both the Standard and HL decoders for the Partial-LRU and Quick strategies, and the Standard decoder for the No LRU and Full-LRU strategies. The simulation propagates labels $I$, $X$, $Y$, $Z$ or $L$ for each ancilla and data qubit in a toric code of distance $d$. We simulate the propagation of regular and leakage errors according to the error model described in Section~\\ref{sec:leakagemodel}. Fault-tolerant error-correction in the toric code of distance $d$ requires $O(d)$ rounds of syndrome measurements so we simulate $d$ rounds. In addition, we include a round of perfect syndrome measurement at the end of each simulation where leaked qubits are replaced by completely depolarized qubits and the syndromes are measured without errors. All $d+1$ syndromes are processed together, a correction is found, and a failure is registered if the state of the code qubits after correction anticommutes with a logical operator of the code. Syndrome processing is performed such that the final state of the code qubits commutes with the stabilizer.\n\nWe assume that the $d$ rounds of fault-tolerant error-correction take place in the course of a longer computation. Therefore, each qubit is initially assigned the $L$ label with probability that is calculated based on the parameters of the model ($p_\\uparrow$ and $p_\\downarrow$) and on the frequency of the qubit re-initializations for the particular circuit. Specifically, persistent qubits that are never re-initialized, such as data qubits in the No LRU circuit, start in the $L$ state with the equilibrium probability derived in Section~\\ref{sec:leakagemodel}.\n\nThe Standard decoding graph is built using the unit cell in Fig.~\\ref{fig:cube}. The translation invariance of the decoding graph allows us to calculate the distance of two defects directly from their $(x, y, t)$ coordinates where the $x$ and $y$ specify the location of the defect on the torus and $t$ is the location in the time dimension. The decoding graph of the HL decoder is not translation invariant due to the presence of the low weight edges introduced in Section~\\ref{sec:heralded}. Since calculating distance of the defects based on their coordinates would be difficult, we build the entire decoding graph and use a shortest path algorithm to find the distance between defects.\n\nThe implementation uses the Boost~\\cite{Boost} libraries to find shortest paths, and the Blossom V~\\cite{kolmogorov09} library to perform minimum weight perfect matching. Our results were generated by Monte Carlo simulations which were repeated until we reached at least $10,000$ iterations and at least $1,000$ failures for each configuration. These simulations were run on an IBM Blue Gene\/Q~\\cite{BGTeam11} using about 30,000 CPU-hours.\n\\section{Toric Code}\\label{sec:toriccode}\n\nThe toric codes \\cite{kitaev03} are the prototypical example of topological stabilizer codes. Each toric code is defined on a $d$ by $d$ square array whose left-right and top-bottom boundaries are associated. The vertices of the array are connected to form a graph with $d^2$ vertices, $2d^2$ edges, and $d^2$ faces. Each of the $n=2d^2$ edges carries a physical qubit of the code, called a code or data qubit, and each vertex and face carries an ancillary qubit used for error-correction, called a syndrome or ancilla qubit (see Fig.~\\ref{subfig:lattice}). The stabilizer $S$ of the quantum code is generated by a set of check operators $\\{A_v\\}$ and $\\{B_f\\}$ that belong to the $n$-qubit Pauli group and are attached to each vertex $v$ and face $f$ of the graph. These operators are tensor products of single qubit Pauli operators\n\n\\begin{figure}[t]\n\\vspace*{13pt}\n\\centering\n \\subfloat[short for lof][]{\n \\epsfig{file=lattice,width=.31\\textwidth}\n   \\label{subfig:lattice}\n }\n \\subfloat[short for lof][]{\n\\epsfig{file=ZZZZMeas,width=.33\\textwidth}\n   \\label{subfig:ZZZZ}\n}\n \\subfloat[short for lof][]{\n\\epsfig{file=XXXXMeas,width=.33\\textwidth}\n   \\label{subfig:XXXX}\n }\n\\vspace*{13pt}\n\\fcaption{\\label{fig:toric} (Color online) (a) The toric code has a natural two-dimensional layout on the surface of a torus. Qubits function as either data qubits, used to store the encoded quantum state, or ancilla qubits, used to measure check operators (stabilizers) of the quantum code. Z-type check operators associate to faces (plaquettes) and X-type check operators associate to vertices (stars). Each check operator involves four data qubits and is measured using an ancilla qubit. The torus encodes a pair of qubits whose representative logical Pauli operators $X_1$, $Z_1$, $X_2$, and $Z_2$ are shown. (b,c) These circuits measure (b) plaquette and (c) star operators using four CNOT gates together with preparations and measurements. In each circuit, data qubits $d_{D,R,L,U}$ interact with an ancilla qubit $a_{z\/x}$ that is prepared in the state $|0\\rangle$ or $|+\\rangle=\\frac{1}{\\sqrt{2}}\\left(|0\\rangle+|1\\rangle\\right)$. This ancilla is then measured in the basis of eigenstates of Z or X, respectively.}\n\\end{figure}\n\n\\begin{equation}\nX = \\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right)\\ \\textrm{and}\\ Z = \\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right).\n\\end{equation}\nThe vertex (or star) operators\n\\begin{equation}\nA_v =\\bigotimes_{\\varepsilon\\in N(v)}X_\\varepsilon\n\\end{equation}\nare $X$-type checks that apply a Pauli $X$ operator to the qubit on each edge $\\varepsilon$ in the neighborhood $N(v)$ of vertex $v$. The neighborhood $N(v)$ of a vertex is the set of edges incident to $v$. The face (or plaquette) operators\n\\begin{equation}\nB_f =\\bigotimes_{\\varepsilon\\in N(f)}Z_\\varepsilon\n\\end{equation}\nare $Z$-type checks that apply a Pauli $Z$ operator to the qubit on each $\\varepsilon$ in the neighborhood $N(f)$ of a face $f$. The neighborhood $N(f)$ of a face is the set of edges on the boundary of $f$. Each check operator involves only four code qubits since each vertex is incident to four edges and each face is bounded by four edges. The set of generators clearly commutes since any star shares zero or two edges with any face. Due to the periodic boundary conditions, any set of $d^2-1$ faces and $d^2-1$ vertices associate to an independent set of check operators that generate the stabilizer of the toric code. This implies that the toric code encodes a pair of logical qubits. Representatives for each class of logical Pauli $X$ and $Z$ operators are shown in Fig.~\\ref{subfig:lattice}.\n\nThe overcomplete set of $2d^2$ check operators are measured simultaneously \\cite{dennis02} using the circuits shown in Fig.~\\ref{subfig:ZZZZ} and Fig.~\\ref{subfig:XXXX}. In the first step, we prepare all plaquette ancillas in $|0\\rangle$ and all site ancillas in $|+\\rangle\\propto |0\\rangle+|1\\rangle$. In the next four steps, CNOT gates act between each ancilla and the data qubit above, left, right, and below the ancilla, in that order. This corresponds to the gate order used in \\cite{wang11,stephens14}. Finally, we measure each plaquette ancilla in the $Z$ eigenbasis and each star ancilla in the $X$ eigenbasis. These six steps constitute an error-correction cycle. One cycle produces a single noisy syndrome given by $2d^2$ bits.\n\nA single error-correction cycle cannot be fault-tolerant in the toric code since local errors in the syndrome can lead to macroscopic errors after error-correction. However, $O(d)$ cycles suffice to improve confidence in the syndrome so that error-correction becomes fault-tolerant. A processing algorithm ingests these $O(d)$ syndromes and infers a corrective Pauli operator. The classic approach \\cite{dennis02}, which we follow in this work, processes plaquette and site syndromes independently to find bit and phase error corrections. The problem of inferring the most probable error given the observed syndrome is mapped to a minimum weight perfect matching problem that can be solved with Edmond's algorithm \\cite{edmonds65}.\n\n\\begin{figure} [b]\n\\vspace*{13pt}\n\\centerline{\\epsfig{file=noLeakCube.pdf, width=.30\\textwidth}}\n\\vspace*{13pt}\n\\fcaption{\\label{fig:cube} (Color online) The unit cell of the decoding graph has six distinctly-weighted edges shown here. Edges $b$ and $d$ correspond to qubit errors, edge $a$ to measurement error, edges $c$ and $e$ to correlated qubit-measurement errors, and edge $f$ to correlated qubit-qubit-measurement errors. The particular edges that appear are determined by the error-correction circuits.}\n\\end{figure}\n\nThe concept of a decoding graph ${\\cal G}=({\\cal V},{\\cal E})$ is a useful abstraction for relating errors and corresponding syndromes \\cite{bravyi13}. There is a separate decoding graph for bit flip ($X$) and phase flip ($Z$) errors. With the exception of Sec.~\\ref{sec:heralded}, the graphs we consider are translation invariant, generated by translating the unit cell shown in Fig.~\\ref{fig:cube} in each of three orthogonal directions. The edges ${\\cal E}$ correspond to error events in the error-correction circuits, and, with the exception of diagonal edges $c$, $e$, and $f$ that correspond to correlated errors, are identified with qubits (horizontal edges ``$b$'' and ``$d$'') and measurement outcomes (vertical edges ``$a$'') for each error-correction cycle. An error on an edge $\\varepsilon$ of ${\\cal G}$ creates a pair of defects at the two vertices incident to $\\varepsilon$. Errors on a subset of edges $E\\subseteq {\\cal E}$, called an error chain, create defects on $\\partial E$ where $\\partial E\\subseteq {\\cal V}$ is the set of vertices with an odd number of edges incident from $E$. Operationally, syndrome bit outcomes label the vertical ``$a$'' edges, and a defect occurs on vertex $v$ if the two incident ``$a$'' edges to $v$ have different syndrome labels.\n\nThe syndrome processing algorithm, or decoder, uses the minimum weight matching algorithm to match pairs of defects on the decoding graph. The error is then corrected by applying the $X$ or $Z$ correction on an error chain connecting each pair of matched defects. A chain $E$ is closed if $\\partial E$ is empty, which occurs if the chain commutes with the stabilizer. A closed chain is contractible if the corresponding error belongs to the stabilizer. All error chains with the same $\\partial E$ are equivalent up to closed chains, which include harmless contractible chains but also include non-contractible chains that are logical operators. The weight of each edge in the decoding graph is chosen to approximate the negative logarithm of the probability of the corresponding error. Therefore, independently for $X$ and $Z$ errors, a minimum weight matching decoder finds a correction from among the most likely errors and fails when the product of the correction and the error chain is not in the stabilizer.\n\nIn this work we restrict our attention to fault-tolerant quantum error-correction but our results apply to both active memory and computation \\cite{rh07,horsman12}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nWater masers arising from thin disks around massive black holes provide \nhigh brightness temperature non-thermal dynamical tracers of gas in Keplerian \norbits.  As such, water maser disks viewed edge-on provide tracers of the\nKeplerian potential and enable measurement of the black hole mass, provided a distance \nis known in order to translate the apparent angular disk size into a physical \nsize \\citep[e.g.,][]{miyoshi1995}.\nMaser accelerations and proper motions can also be observed, and because\nthe circular velocity is known from Doppler shifts, the geometric distance \nto the black holes (and host galaxies) can be determined \\citep[e.g.,][]{herrnstein1998}.  \nGeometric distances obtained from water masers provide a crucial independent measurement of the Hubble\nconstant and can be used to calibrate other distance indicators such as \nthe period-luminosity relation of Cepheids \\citep{riess2016}.\n\nThese measurements require maser disks that are viewed within a few degrees of edge-on:\notherwise, maser beaming directs emission away from the observer because masers \npropagate along velocity-coherent paths through the disk.  For thin disks, this propagation \noccurs along the radial path along the line of sight toward the black hole and along the disk \ntangent points.  For warped disks, such as that found in NGC 4258, the picture is\nmore nuanced because a warped disk provides numerous sightlines and inclinations that intersect velocity-coherent \nparts of the disk \\citep{humphreys2013}.  Nonetheless, inclined maser disks are generally not seen in water maser\nsurveys.  Or are they?\n\n\\begin{figure*}[ht!]\n\\epsscale{1.1}\n\\plotone{lensed_masers.eps}       \n\\caption\nMaser amplification pathways for Keplerian disks orbiting massive black holes.  These diagrams are schematic and not to scale.\nLeft:  the geometry of an egde-on rotating disk showing systemic velocity masers (green) and high-velocity redshifted (red) and blueshifted\n(blue) masers at the tangent points of the disk.  \nCenter:  in the black hole rest frame, all disk motion is perpendicular to the radial direction, providing velocity-coherent amplification \npathways at the systemic velocity (or rest frame).  In-going (and out-going) radial masers will populate the disk in this reference frame.\nRight:  In-going masers may be gravitationally lensed\/deflected by the central black hole into the sightline of an observer who would \nnot otherwise see maser\nemission from an inclined disk.   Only the systemic velocity masers would be seen by this observer, and they will appear to \narise from the location of the black hole.  High-resolution imaging might reveal a ring or arc of water maser emission, \nproviding a black hole mass measurement.  \n\\label{fig:schematic}}\n\\end{figure*}\n\nIt seems likely that inclined water maser disks have already been detected by single dish surveys, \nbut they have been discarded because they show no high velocity lines that have canonically been\nused to identify maser disks.  \nIn this paper, we propose a mechanism to produce detectable maser emission from inclined disks \nthat may also be used to obtain black hole masses\n(Section \\ref{sec:thought}), we present a method to detect inclined maser disks based on extant surveys \n(Section \\ref{sec:candidates}), we present Karl G. Jansky Very Large Array \n(VLA)\\footnote{The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.} observations of candidate inclined disks \n(Sections \\ref{sec:obs} and \\ref{sec:results}), and we \nprovide a list of inclined disk candidates for further study (Sections \\ref{sec:analysis} and \\ref{sec:discussion}).  \nIn what follows, we assume a flat cosmology with parameters  $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$, $\\Omega_M = 0.27$, and \n$\\Omega_\\Lambda = 0.73$, and we calculate distances from redshifts in the CMB rest frame.\n\n\n\\section{A Thought Experiment} \\label{sec:thought}\n\nConsider a typical 22 GHz water maser thin disk in Keplerian orbit around a massive black hole.\nThis thin disk shows systemic velocity masers at the center of the observed disk, along a radial amplification path, \nand it shows high-velocity red- and blue-shifted masers at the tangent points of the disk, along azimuthal paths, \namplifying spontaneous emission and background host galaxy continuum (Figure \\ref{fig:schematic}, left).  \nIf a thin disk is inclined by more than a few degrees, the masers are \nno longer beamed toward the observer and the disk would not be detected.  \nThe known water maser disks show an inclination typically within $\\sim$5$^\\circ$ of edge-on \\citep[e.g.,][]{kuo2011}, \nsuggesting that an order of magnitude additional maser disks exist, and they are simply beamed in directions we cannot observe.  \nNote that the famous water maser disk in NGC 4258 \\citep[e.g.,][]{miyoshi1995,herrnstein1998} is atypical, with \na disk inclination of 72$^\\circ$, but the disk warp provides sightlines that are nearly edge-on, and \nthis is where the masers are seen \\citep{humphreys2013}.\n\nFor the following discussion we will use fiducial parameters similar to those of many of the known maser disks \n\\citep{kuo2011,greene2016}.  We assume a distance\nof 10 Mpc, a black hole mass of $M_{BH} = 10^7\\ M_\\odot$ (and therefore a Schwarzschild radius of $R_s\\sim1\\ \\mu$pc), and\na circular rotation speed of 1000 km s$^{-1}$ in the maser-active part of the disk located 0.5 pc from the black hole and \nspanning 0.2 pc.  We \nassume a disk thickness equal to that of NGC 4258: $5.5\\times10^{14}$ cm or 0.18 mpc \\citep[1$\\sigma$;][]{argon2007}. \n\nConsider how this system appears to an observer at the location of the black hole:\nthe disk describes a plane in the sky.  Water masers reaching this observer must be purely radial (propagating inward) but will have a\ncoherent amplification path in all radial directions (Figure \\ref{fig:schematic}, center).  All Keplerian Doppler shifts\nwill be across the line of sight, so all maser emission reaching this location will have zero velocity in this reference frame (ignoring\nthe transverse Doppler shift, \nwhich is negligible at $\\sim$2 km~s$^{-1}$).  The black hole-frame observer will thus see a ring of maser light \nor maser spots with low velocity spread.  The maser-active part of the disk would subtend $\\sim$2.5 arcminutes (1$\\sigma$). \nThe expected maser opening angle would be similar, $\\sim$3 arcminutes, assuming a maser path-to-size ratio of \n$\\sim$0.2 pc\/0.2 mpc $\\simeq 1000$, implying maser light spanning $\\sim$500 $R_s$ at the black hole.   \n \nThe black hole Kerr metric would thus be bathed in maser light with a continuous distribution of impact parameters, \nimplying gravitational deflection of maser light \ninto nearly all angles (the deflection at 100 $R_s$ is 1.1$^\\circ$ and grows inversely with impact parameter $R$ as $\\theta = 2 R_s\/R$;\n\\citet{einstein1915}).  \nOne would therefore\nexpect incoming maser light to be scattered into many angles, representing a gravitational lensing de-focusing of a given incoming\nbeam.  But portions of multiple maser beams from around the disk may be deflected into a given direction, making the \nnet flux as a function of angular direction uncertain and in need of numerical modeling. \n\nNGC 4258 has a warped disk \\citep[e.g.,][]{humphreys2013}, which means that the in-going radial masers can be misaligned and thus be\nbeamed above or below the dynamical center of the disk, and that the continuous distribution of disk inclination angles will \nfully populate the black hole metric volume with beams from multiple angles, \nfurther enhancing the sampling of impact parameters and deflection angles. \nIf warped disks are common, \nthen maser disks with high velocity lines will be detected more often than one would assume based on random inclination alone \n(in other words, parts of many \ndisks will have inclinations appropriate for detectable maser amplification), and the black hole will be illuminated by in-going \nmasers from above and below\nthe nominal disk plane, producing a range of incoming maser impact parameters and angles with respect to the disk,\nthereby enhancing the gravitational deflection probability toward any given observer.  \nRegardless of whether disks are warped or not, maser light will be \nscattered away from the maser disk plane, making inclined disks potentially detectable, but likely faint compared to edge-on disk\nmasers.\n\nMasers amplify seed photons, and these seed photons can be the portion of a radio continuum that matches the maser line \nfrequency, appropriately Doppler-shifted, or they can be spontaneously emitted maser line photons.  In known maser disks, the\nsystemic radial masers typically amplify the AGN radio continuum, and the high-velocity tangential masers amplify spontaneous \nemission or possibly radio continuum from the host galaxy itself.  In-going radial masers do not have access to AGN seed photons\nand must amplify either continuum from the host galaxy or inward-directed spontaneous emission from the outer parts of the disk.  \nIn either case, it is reasonable to expect in-going masers to be weaker than the out-going systemic masers from \nedge-on disks.  It is unclear how the amplification of in-going masers would compare to the amplification of the \nhigh-velocity masers in edge-on disks because \nthe tangential amplification pathway may or may not have a physically longer or higher column density \nvelocity-coherent path for amplification.  \n\nIf one were to observe a water maser disk from a direction other than edge-on, would water maser emission be seen?  \nThe thought experiment \ndescribed here suggests that it would, and the inclination of the disk with respect to the observer would select the deflection angle\n(or equivalently the impact parameter) of the observed maser light.  For example, deflection by 10$^\\circ$ would allow observation of masers\nfrom the back side of a disk with an inclination of 80$^\\circ$, requiring an impact parameter of 12 $R_s$.  Other parts of the disk could be viewed if light reaches smaller impact parameters.  If, as expected, there is a continuous range of impact parameters, then a continuous distribution of maser light from an extended portion of the disk will contribute to a spectrally narrow maser line complex seen by the observer at nearly any inclination (but perhaps with higher likelihood and intensity at higher inclinations).  \n\n\\subsection{Observable Signatures}  \n \nThe observational signatures of an inclined water maser disk would be:\\\\\n\\indent 1.  A narrow line or line complex\\\\\n\\indent 2.  at the systemic velocity\\\\\n\\indent 3.  at the apparent black hole location.\\\\\nThe black hole location would be indicated by the radio continuum core, ideally observed at the same frequency as the water\nmaser.\nThe observational signatures of an inclined water maser disk, however, may also arise from other mechanisms.\nWater masers can be produced in radio jet-molecular cloud interactions \\citep[e.g.,][]{gallimore1996,claussen1998,peck2003,henkel2005}, in \nstar-forming regions \n\\citep[e.g.,][]{tarchi2002a,tarchi2002b,henkel2005,hofner2006, darling2008,brogan2010,darling2011,tarchi2011,amiri2016}, \nand in outflows \\citep[e.g.,][but note that some objects in the latter survey may be \ncandidates for inclined disk masers]{greenhill2003b,kondratko2005,tarchi2011a}.\nThese are likely to be the main contaminant among an inclined disk\nsurvey sample.  VLBI identification of the maser with an AGN via spatial coincidence with \nthe core radio continuum --- identified by the spectral index --- can resolve the ambiguity (Section \\ref{sec:results}).\n\n\nIn contrast to edge-on disk systemic masers, it is unclear whether inclined disk maser lines will show proper motion or acceleration.  This may depend on the clumpiness of maser-emitting regions, on the maser beam sizes, on the physical extent of the \ndisk that is sampled by the observed line, and on the deflection angle to the observer.\nOne might expect lensed masers to show little or no time variability, but the substructure seen in known water maser disks \nand the natural variability of water masers suggests that this may be a bad assumption.  \n\nMany extragalactic water masers detected in previous surveys meet some or all of the above observational criteria:  \ninclined water maser disks may have already been detected!  In most cases, when \na single systemic velocity line is detected in water maser surveys, there is no interferometric follow-up\nbecause distance or black hole mass measurements (traditionally) require edge-on disks, the signature of which are \nthe high velocity lines emitted from the tangent points of the disk.  \nHigh resolution observations may also be frustrated by weak or variable water masers.\n\n\nIn Section \\ref{sec:obs} we present interferometric mapping of a sample of narrow-line systemic velocity water masers that appear to be inclined disk maser candidates.  If observations show that the masers remain unresolved and are centered at the location of the \ncentral massive black hole (as identified by simultaneous radio continuum observations at 20 GHz), then they remain candidates and should be mapped with VLBI.\n\n\n\n\n\\subsection{Black Hole Masses}\n\nThe Einstein radius of a strong gravitational lens is \n\\begin{equation}\n   \\theta_E = \\sqrt{2 R_s {D_{LS}\\over D_L D_S}} \n                 = 9.2 \\sqrt{{R_s \\over {\\rm mpc}} \\, {D_{LS} \\over {\\rm pc}}} \\left( D_L \\over {\\rm Mpc} \\right)^{-1} {\\rm mas}, \n\\end{equation}\nwhere $D_L$, $D_S$, and $D_{LS}$ are the angular diameter distances to the lens, to the source, and between the lens and the source,\nrespectively, and $D_L \\simeq D_S$ for the black hole-maser disk configuration \\citep[after][]{einstein1936}.\nFor the fiducial parameters listed above ($R_s = 1$ $\\mu$pc, $D_{LS} = 0.5$ pc, and $D_L = 10$ Mpc),\n$2 \\theta_E = 0.041$ mas. \nSince the angular resolution (HPBW) of the Very Long Baseline Array (VLBA) is 0.3 milliarcseconds at 22.2 GHz (which should be \ncompared to twice the Einstein radius), \nthe Einstein radius for the fiducial maser disk and black hole would require space-based VLBI to resolve.  \nOn the other hand, terrestrial VLBI could resolve \na water maser Einstein ring for a more massive black hole with a physically larger maser disk:  for \n$R_s = 100$ $\\mu$pc ($M_{BH} = 10^9\\ M_\\odot$), $D_{LS} = 2$ pc, and $D_L = 10$ Mpc,\n$2 \\theta_E = 0.82$ mas.  Unfortunately, water maser disks have yet to be identified orbiting $10^9\\ M_\\odot$ black holes, \nbut it is unclear whether is this a selection effect or a consequence of physics \\citep{vandenbosch2016}.  The most \nmassive black hole measured using a water maser disk to date is in NGC 1194 with $M_{BH} = 10^{7.85\\pm0.05}\\ M_\\odot$ \\citep{kuo2011,greene2016}.\n\nWere an Einstein ring observable from a back side in-going maser in an edge-on disk or from an in-going maser from \nan edge-on portion of an inclined warped disk, then one can infer a black hole mass from the angular size of the ring.  \nOne does need independent measurements of the luminosity distance to the black hole ($D_L$) and the size of the \nmaser-emitting disk ($D_{LS}$).  $D_L$ can be obtained from an assumed cosmology and the cosmological redshift, but \n$D_{LS}$ may be more difficult to measure.  The precision of black hole masses obtained from this method may be limited\nby our ability to measure, model, or estimate the radius of the maser-emitting part of the disk.  \n\nEinstein rings require linear alignment between the source, the lens, and the observer, which is not germane to the inclined\nmaser disk geometry (for unwarped disks). \nInstead, one would na\\\"{i}vely expect to see multiple images of the same maser, which can also be related to the black hole mass.  This\nexpectation, which is correct for isotropic emitters, is probably incorrect for masers.\n\nAn important difference between maser emission and the standard treatment of gravitational lensing is maser \nbeaming:  while there may be sightlines in a gravitational lens geometry that land on the emitter, emission \nmay not be seen if light is not beamed along that sightline.  For a general pointlike isotropic light source offset from \nthe lens by angle $\\theta_S$, there are two solutions to the lens equation:\n\\begin{equation}\n  \\theta_\\pm = {1\\over2} \\left( \\theta_S \\pm \\sqrt{\\theta_S^2 + 4 \\theta_E^2}\\right).\n\\end{equation}\n$\\theta_+$ represents the angle between the lens and the source image appearing outside $\\theta_E$, and \n$\\theta_-$ represents the angle of the image appearing inside $\\theta_E$\n\\citep[][Equation 24, Figures 5 and 7]{narayan1995}.\nFor an inclined maser disk configuration, no emission is directed \nsignificantly out of the disk plane, so no maser emission would be seen in the $\\theta_+$ direction.\nOn the other hand, $\\theta_-$ may be small enough that this sightline is included in the maser beam \npassing very close to the central black hole.  In this case, the source-observer deflection angle $\\alpha_-$ nearly matches the \ncomplement of the disk inclination:  $\\alpha_- \\simeq \\pi\/2 - i$ (Figure \\ref{fig:schematic}, right).  \n\nIn contrast to canonical lensing, \nwe expect that the maser beaming will produce only one maser spot image, where light is deflected in the \nmanner shown schematically in Figure \\ref{fig:schematic} (right).  This maser image will lie inside the Einstein radius.\nFor isotropic extended emitters, this image would also be demagnified, but masers are not isotropic emitters and \ncan be exceptionally compact (equivalently, they demonstrate high brightness temperatures).  The degree of demagnification\nis therefore unclear and requires numerical ray-tracing to assess.\n\nFor the fiducial parameters above, and assuming an inclination of $80^\\circ$,\n$\\theta_S = (R_{\\rm maser}\/D_S) \\sin\\alpha_- = 1.8$ mas, $\\theta_E = 0.021$ mas, and therefore $|\\theta_-| = 0.2$ $\\mu$as, \nwhich is equivalent to 11 $R_s$.  The maser beam spans this distance from \nthe black hole, so the maser emission can be lensed toward the observer in this case. \nFor the $10^9\\ M_\\odot$ black hole with a larger maser disk described above, \n$\\theta_S = (R_{\\rm maser}\/D_S) \\sin\\alpha_- = 7.2$ mas,\n$\\theta_E = 0.41$ mas, and therefore $|\\theta_-| = 24$ $\\mu$as, \nwhich is again equivalent to 11 $R_s$ (the impact parameter determines the \ndeflection angle, which is determined by the inclination).  \n\nOne would therefore expect lensed masers from inclined disks to be faint and appear to arise from \nthe black hole location ($\\theta_- \\ll 1$ mas).   \nThis treatment assumes a single pointlike maser rather than an extended continuous disk of masers or a set of distributed maser \nspots.  In this more realistic scenario, numerical ray-tracing is required to connect the observable lensed maser image to the black \nhole mass and maser disk configuration.  \n\nLensed masers from inclined disks may appear to be pointlike or they may describe arcs.\nIn-going masers from the far side of an edge-on disk or in-going masers from an inclined but warped disk may produce Einstein\nrings.  \nSingle-epoch VLBI maps can therefore provide black hole masses, but space-based VLBI may be required\nfor the typical $\\sim10^7\\ M_\\odot$ black hole associated with water maser disks.   The mass measurement precision will be limited by \nuncertainty about the maser disk size rather than by the distance to the object\n (the maser will provide the systemic redshift, which can be converted into a distance, and\nthe distance uncertainty will be dominated by peculiar velocity departures from the Hubble flow).\n\n\n\n\\subsection{Back-Side Masers in Edge-on Disks}  \n\nIf there are in-going radial water masers in maser disks (and there is no compelling reason to think otherwise), \nthen there should be systemic water masers seen in edge-on disks from the back side of the disk, \nboth lensed and unlensed by the black hole.  \nAre such masers in extant data?  \n\nIf there are also front-side systemic masers in an edge-on disk, then they will be orders of magnitude brighter \nthan back-side masers, even in the presence of strong lensing, simply due to amplification considerations.  \nThe front-side masers can amplify AGN radio continuum, whereas the back-side masers will either be driven by \nstimulated emission (but with an amplification pathway equal to the front-side masers) or by host galaxy continuum, \nwhich is substantially weaker than AGN continuum at 22 GHz in many cases.   The back-side maser contribution may therefore\nbe confused by the front-side emission, particularly since both types of maser emission will occur at the systemic velocity.\n\nIn rare cases, a back-side maser might be distinguishable from the front-side emission either by a position or a velocity offset\n(there is some spread to systemic velocity masers, e.g. \\citet{gao2016}).  The observational signature of a back-side maser would be an acceleration \nin the opposite sense of the front-side systemic maser acceleration (i.e., negative acceleration under the convention that \npositive velocities are redshifted).   This is not seen in published systemic maser acceleration measurements \n\\citep[e.g.][]{greenhill1995,nakai1995,braatz2010,kuo2015,gao2016}, but such a signal could be lost amid the brighter and numerous front-side masers.\n\nIf a back-side maser is lensed into an arc or Einstein ring by the central black hole, then it might be extended in VLBI maps.  Resolved \nsystemic maser emission would therefore be an additional observable signature of back-side masers.  \n\n\n\\section{Candidate Selection}\\label{sec:candidates}\n\nIf inclined water maser disks can be detected via gravitational lensing or deflection of in-going masers by massive black holes, then \nthey have likely already been detected in surveys for maser disks.  But they were rejected as disk candidates because they \nlacked high-velocity emission.  Inclined maser disks will appear to have maser\nemission only at the systemic velocity of the galaxy or AGN.  We therefore use extant water maser surveys to select inclined disk \ncandidates.\n\nUsing extant water maser surveys, most of which favor Seyfert 2 AGN, we examined single-dish spectra \ncompiled by the Megamaser Cosmology Project\\footnote{\\label{footnote:maser_url}\\url{https:\/\/safe.nrao.edu\/wiki\/bin\/view\/Main\/PrivateWaterMaserList}}\nto select objects showing\na narrow systemic velocity maser or maser complex.  We also imposed a 30 mJy line flux limit and excluded objects south of $-20^\\circ$ declination.  \nThis process identified 16 inclined maser disk candidates (Table \\ref{tab:obs}), and most candidates (14) have only been observed\nwith a single dish.  Those that do have interferometric maps are NGC 3556, which was mapped using the VLA in CnB and DnA configurations \\citep{tarchi2011} with no 22 GHz continuum detected (1$\\sigma$ rms noise of $\\sim$0.5 mJy beam$^{-1}$),  \nand NGC 3735,which was mapped using A-array \\citep{greenhill1997}, but no 22 GHz continuum was detected.  \n\n\nSince the inclinations of known maser disks are nearly edge-on, the number of inclined maser disks must be large, \nroughly an order of magnitude larger than the number of detected maser disks.\nHowever, the gravitational lensing or deflection of detectable in-going radial\nmasers adds substantial uncertainty to the detection expectations.   We do not know how common observable lensed inclined maser\ndisks are in the universe because we do not know the opening angle of the masers, the \nsize of the maser spots with respect to the black hole's Schwarzschild radius, the brightness of in-going masers, which will \ndepend on the 22 GHz seed photons from the host galaxy, or the degree of gravitational lensing demagnification.  \n\n\n\n\\section{Observations and Data Reduction}\\label{sec:obs}\n\nWe observed the 22.23508 GHz $6_{16}-5_{23}$ ortho water maser line and 20 GHz radio continuum \ntoward 16 candidate inclined maser disks \n(see Section \\ref{sec:candidates} and Table \\ref{tab:obs}) using the \nVLA in A configuration (the highest angular resolution configuration).  Observations \nof program 15A-297 spanned June 19 2015 through September 26 2015 in five sessions.  The \nfifth session occurred during the A-array to D-array reconfiguration.   Each session included visits to \nflux and bandpass calibrators, and observations of each target object were interleaved with nearby \ncomplex gain calibrators with a $\\sim$4 minute switching cadence.  \n\nSpectral line and continuum observations were simultaneous.  The spectral line observations were centered on the redshift of the\nhost galaxy, had 1.1--1.9 km s$^{-1}$ spectral resolution, used 1536 channels to span 128 MHz (1700--2900 km s$^{-1}$), \nand used dual circular polarization\nand 8-bit sampling.  Continuum observations spanned 4 GHz, 18--22 GHz, using 32 spectral windows spanning 128 MHz each \nusing 128 channels in dual circular polarization and 3-bit sampling.  \n\nTable \\ref{tab:obs} lists the details of the observations and the rms noise in the line cubes and continuum maps.  Typical beam sizes were\n80--100 milliarcseconds. Noise was about 3 mJy beam$^{-1}$ in 1.2 km~s$^{-1}$ channels in the spectral line cubes and \nabout 15--20 $\\mu$Jy beam$^{-1}$ in the continuum.  The exception was the $z\\simeq0.66$ water maser J0804+3607 \n\\citep{barvainis2005}\nthat was redshifted to 13.4 GHz, in Ku band, which necessarily had lower angular and spectral resolution and lower rms noise\nin the line but higher rms continuum noise.  In this case, the continuum was centered on 14 GHz and spanned 12--16 GHz.\n\nThe observing session on September 15 2015 had poor 22 GHz weather for A configuration and could not be calibrated or imaged.  \nNGC 3359, NGC 3556, and NGC 3735\nare therefore only listed in Table \\ref{tab:obs} and are not discussed further or included in any analysis of the remaining 13 objects.\n\nAll data reduction and analysis was performed using the Common Astronomy Software Applications package \\citep[CASA;][]{mcmullin2007}.  Calibration and flagging \nused a modified CASA pipeline plus additional manual flagging.  Imaging used Briggs weighting with robustness 0.5.  \nSpectra were extracted from spectral line cubes using a maser-centered beam, and integrated line maps were restricted to \nline-emitting channels.  All spectra use the optical velocity definition in the Barycentric reference frame.  \n\n\n\\floattable\n\\begin{deluxetable}{cccCrCccc}\n\\tablecaption{Journal of Observations \\label{tab:obs}}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{UT Date} &\n\\colhead{Integration} & \n\\multicolumn{3}{c}{Beam\\tablenotemark{a}} & \n\\multicolumn{2}{c}{Line} &\n\\colhead{Continuum} \\\\\n\\cline{4-6}\n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{Angular} & \n\\colhead{PA} & \n\\colhead{Physical} & \n\\colhead{rms} & \n\\colhead{$\\Delta v$} & \n\\colhead{rms} \\\\\n\\colhead{} & \n\\colhead{} & \n\\colhead{(m)} & \n\\colhead{(milliarcsec)} & \n\\colhead{($^\\circ$)} & \n\\colhead{(pc)} & \n\\colhead{(mJy bm$^{-1}$)} &\n\\colhead{(km s$^{-1}$)} &\n\\colhead{($\\mu$Jy bm$^{-1}$)}  \n}\n\\startdata\nNGC 291          & 2015-06-21 & 23.3  &  128\\times84 & 20.4 & 50\\times33 & 2.6 & 1.2 & 17  \\\\\nNGC 520b        & 2015-06-21 & 23.3  &  102\\times89 & 27.8 & 14\\times12 & 3.1 & 1.1 & 20 \\\\\nJ0350$-$0127 & 2015-06-21 & 23.3  &  100\\times87 & 20.5 & 81\\times70 & 3.1 & 1.2 & 15  \\\\\nIC 485              & 2015-09-26 & 23.4  &   84\\times82 & 19.6 & 47\\times46 & 2.2 & 1.2 & 18  \\\\\nJ0804+3607 & 2015-09-26 & 23.3 & 151\\times130 & $-$81.3 & 1065\\times917 & 0.9 & 1.9 & 47   \\\\\nCGCG 120$-$039 & 2015-09-26 & 23.3  & 85\\times81 & 33.7 & 44\\times42 & 2.1 & 1.2 & 20  \\\\\nJ0912+2304    & 2015-09-06 & 23.3  & 92\\times81 & 65.6 & 67\\times59 & 2.9 & 1.2 & 16  \\\\\nJ1011$-$1926 & 2015-09-06 & 23.4  & 173\\times86 & $-$20.8 & 98\\times49 & 5.3 & 1.2 & 19  \\\\\nNGC 3359\\tablenotemark{b}       & 2015-09-15 & 23.3  & 112\\times72 & 48.6 & 9\\times6 & \\nodata & 1.1 & \\nodata \\\\\nNGC 3556\\tablenotemark{b}       & 2015-09-15 & 23.3  & 104\\times72 & 45.0 & 6\\times4 & \\nodata & 1.1 & \\nodata  \\\\\nNGC 3735\\tablenotemark{b}       & 2015-09-15 & 23.3  & 112\\times68 & 36.8 & 21\\times13 & \\nodata & 1.2 & \\nodata  \\\\\nUGC 7016       & 2015-09-06 & 23.3  & 114\\times91 & $-$74.8 & 54\\times43 & 3.8 & 1.2 & 16  \\\\\nNGC 5256       & 2015-06-19 & 23.3  & 96\\times84 & $-$73.8 & 55\\times48 & 3.0 & 1.2 & 17 \\\\\nNGC 5691       & 2015-06-19 & 23.4  & 131\\times85 & 39.3 & 15\\times9 & 3.7 & 1.1 & 17 \\\\\nCGCG 168$-$018 & 2015-06-19 & 23.3 & 86\\times82 & 50.0 & 63\\times60 & 2.7 & 1.2 & 15  \\\\\nJ1939$-$0124 & 2015-06-19 & 25.0 & 96\\times87 & 37.7 & 41\\times38 & 3.0 & 1.2 & 16 \\\\\n\\enddata\n\\tablenotetext{a}{Synthesized beam properties for the spectral line observations.  The continuum maps are slightly different.}\n\\tablenotetext{b}{Objects observed on 2015 September 15 could not be calibrated or imaged due to poor weather.  }\n\\end{deluxetable}\n\n\n\n\\section{Results}\\label{sec:results}\n\nWater masers were detected in 9 out of 13 objects, and 20 GHz continuum emission was detected in 7 out of 13 objects.  \nOnly five objects show both maser and continuum emission, and two objects were detected in neither line nor continuum.  \nFigures \\ref{fig:NGC291}--\\ref{fig:2MASX1939} show a 1\\arcsec\\ square field of view of the first moment maser maps and\ncontinuum contours and they show spectra of the detected objects (in line, continuum, or both).  Table \\ref{tab:positions} lists the \nmaser and continuum centroids based on two-dimensional Gaussian fits.  For the spectral lines, these fits were made\nto the integrated line maps.  The maser emission was universally unresolved, but the continuum emission was formally resolved \nwhen deconvolved from the beam in all but two objects, IC 485 and CGCG 168$-$018.  \n\nTable \\ref{tab:positions} also lists the maser-continuum offsets in angular and physical units. Offsets for four of the five objects \ndetected in both maser and continuum are non-significant with $1\\sigma$ uncertainties ranging from about 1 to 40 pc.  The only object\nshowing a significant offset between the maser and continuum centroids is CGCG 168$-$018, with a $21.6\\pm2.7$ pc offset (Figure \\ref{fig:CGCG168-018}).  \n\nTable \\ref{tab:masers} lists the measured and derived water maser properties:  peak and integrated flux densities, luminosity distance, \nisotropic line luminosity, the range of velocities spanned by the line emission 3$\\sigma$ above the noise, the velocity of peak emission, and \nthe adopted systemic velocity.  For J0804+3607, we list redshifts rather than velocities.  The detected maser velocities are consistent with previous \nobservations, although the masers can be substantially offset from the systemic velocities, which are obtained from optical and HI 21 cm \nlines.  It is unclear whether these velocity offsets are physical (i.e., due to different line-emitting regions genuinely having different velocities, as\nis seen in shock-induced maser emission), due to obscuration (optical vs. radio lines), or due to measurement error, particularly \nin optical redshifts.  We therefore do not rely on the velocity offset between the maser emission and the adopted systemic  \nvelocity as a criterion for assessing the likelihood of a maser arising from an inclined disk.  \nIsotropic maser luminosities range from kilomaser values ($3.05 \\pm 0.26\\ L_\\odot$ in NGC 520b) to the exceptionally luminous, \n$L_{iso} = (1.8 \\pm 0.1) \\times 10^4\\ L_\\odot$ in J0804+3607 (Section \\ref{sec:discussion}).  \n\nTable \\ref{tab:continuum} shows the 20 GHz radio continuum properties of the seven detected objects.  We include the peak flux density, \nthe integrated flux density, the spectral index derived solely from the 18--22 GHz bandpass, and the deconvolved angular size.  \nAmong the six continuum sources with enough signal-to-noise to derive a significant spectral index, four are steep spectrum \n($\\alpha = -1$ to $-2$) and two are flat ($\\alpha \\simeq -0.2 \\pm 0.2$).  One of the latter, CGCG 120$-$039, is only marginally resolved.\\\nIC 485, which does not have a spectral index measurement, but which shows both maser and continuum emission, does not \nhave a resolved continuum.\n\n\n\\floattable\n\\begin{deluxetable}{cllllcc}\n\\tablecaption{Maser and Radio Continuum Positions \\label{tab:positions}}\n\\tablecolumns{7}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\multicolumn{2}{c}{Maser Centroid\\tablenotemark{a}} &\n\\multicolumn{2}{c}{Continuum Centroid\\tablenotemark{b}} &\n\\multicolumn{2}{c}{Offset} \\\\\n\\colhead{} & \n\\colhead{RA} & \n\\colhead{Dec} & \n\\colhead{RA} & \n\\colhead{Dec} & \n\\colhead{Angular} & \n\\colhead{Physical}\\\\\n\\colhead{} & \n\\colhead{(hms)} & \n\\colhead{(dms)} & \n\\colhead{(hms)} & \n\\colhead{(dms)} & \n\\colhead{(milliarcsec)} & \n\\colhead{(pc)} \n}\n\\startdata\nNGC 291          & \\nodata & \\nodata & 00:53:29.9101(11) & $-$08.46.03.740(14) & \\nodata & \\nodata \\\\\nNGC 520b        & 01:24:34.91412(14) & +03.47.29.7864(22) & 01:24:34.9099(28) & +03.47.29.7783(92) & 64(42) & 8.6(5.6) \\\\\nJ0350$-$0127 & 03:50:00.352168(29) & $-$01.27.57.39574(53) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nIC 485              & 08:00:19.752486(74) & +26.42.05.0526(10) & 08:00:19.7515(14) & +26.42.05.050(14) & 13(19) & 7.6(10.5)\\\\\nJ0804+3607 &  08:04:31.01144(40) & +36.07.18.1937(52) & 08:04:31.01138(12) & +36.07.18.19927(91) & 5.6(5.3) & 39.6(37.2) \\\\\nCGCG 120$-$039 & 08:49:14.07078(16) & +23.22.48.9408(20) &  08:49:14.07097(13) & +23.22.48.9346(17) & 6.7(2.7) & 3.5(1.4) \\\\\nJ0912+2304    & 09:12:46.36659(33) & +23.04.27.2421(28) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nJ1011$-$1926 & 10:11:50.56731(17) & $-$19.26.43.9645(59) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nUGC 7016       & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nNGC 5256\\tablenotemark{c} & \\nodata & \\nodata & 13:38:17.79219(15) & +48.16.41.1389(19) & \\nodata & \\nodata \\\\\n                       & \\nodata & \\nodata & 13:38:17.24843(76) & +48.16.32.2095(92) & \\nodata & \\nodata \\\\\nNGC 5691       & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nCGCG 168$-$018 &  16:30:40.90329(19) & +30.29.19.7066(29) &  16:30:40.90134(21) & +30.29.19.7216(20) & 29.3(3.6) & 21.6(2.7) \\\\\nJ1939$-$0124 & 19:39:38.91545(38) & $-$01.24.33.2553(39) & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n\\enddata\n\\tablenotetext{a}{All detected maser emission was unresolved.  See Table \\ref{tab:obs} for beam sizes.}\n\\tablenotetext{b}{All detected continuum emission was resolved except for IC 485 and CGCG 168$-$018.  \n  See Table \\ref{tab:continuum} for continuum measurements.}\n\\tablenotetext{c}{NGC 5256 shows two widely-separated continuum components (both are listed).}\n\\tablecomments{Coordinates are epoch J2000, and parenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n \n\n\n\\floattable\n\\begin{deluxetable}{crrrrcrrc}\n\\tablecaption{Water Maser Properties and Redshifts \\label{tab:masers}}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{$S_{\\rm peak}$\\tablenotemark{a}} &\n\\colhead{$S_{\\rm int}$\\tablenotemark{b}} &\n\\colhead{$D_L$} & \n\\colhead{$L_{\\rm iso}$\\tablenotemark{c}} & \n\\colhead{$v_{\\rm range}$\\tablenotemark{d}} & \n\\colhead{$v_{\\rm peak}$} & \n\\colhead{$v_{\\rm sys}$} & \n\\colhead{Ref.\\tablenotemark{e}} \\\\\n\\colhead{} & \n\\colhead{(mJy)} & \n\\colhead{(mJy km s$^{-1}$)} & \n\\colhead{(Mpc)} & \n\\colhead{($L_\\odot$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{(km s$^{-1}$)} & \n\\colhead{} \n}\n\\startdata\nNGC 520b        & 35(3) & 166(14) & 28 & 3.05(26) & 2270--2273 & 2272(1) & 2288(8) & 1 \\\\\nJ0350$-$0127 &  347(4) & 7200(150)  & 180 & 5181(108) & 12336--12393\\tablenotemark{f} & 12369(1) & 12322(18) & 2 \\\\\nIC 485              & 78(2) & 2470(130) & 125 & 868(46) & 8307--8387 & 8356(1) & 8338(10) & 3 \\\\ \nJ0804+3607\\tablenotemark{g} & 9(1) & 80(6) & 3997 & $1.8(1)\\times10^4$ & 0.66038--0.66051 & 0.66045(1) & 0.65654(37)  & 4 \\\\\nCGCG 120$-$039 & 82(2) & 438(46) & 116 &  133(14) & 7559--7565 & 7565(1) & 7684(26) & 5\\\\ \nJ0912+2304    & 16(3) & 252(44) & 164 & 151(26) & 10855--10878 & 10855(1) & 10861(26) &  6\\\\\nJ1011$-$1926 & 44(4) & 624(82) & 123 & 213(28) & 8043--8065 & 8048(1) & 8065(31) & 7 \\\\\nCGCG 168$-$018 & 25(2) & 279(36) & 162 & 163(21) & 11134--11164 & 11140(1) & 11015(29) & 5 \\\\ \nJ1939$-$0124 & 30(2) & 519(80) & 93 & 102(16) & 6170--6206 & 6198(1) & 6226(20)  & 8 \\\\\n\\enddata\n\\tablenotetext{a}{The peak flux density was obtained from a spectrum formed from a single beam centered \non the peak (unresolved) maser emission.}\n\\tablenotetext{b}{The integrated flux density of the water maser complex was \nobtained from fitting a single two-dimensional Gaussian to the velocity-integrated spectral line cube.}\n\\tablenotetext{c}{The isotropic luminosity is computed from the integrated line flux density $S_{int}$ via \n  $L_{iso} = 23.1\\ L_\\odot \\times S_{\\rm int} ({\\rm mJy\\ km\\ s}^{-1}) \\times D_L ({\\rm Gpc})^2\/(1+z)$, where $D_L$ is the luminosity distance and \n  $z$ is the cosmological redshift.}\n\\tablenotetext{d}{$v_{\\rm range}$ is the velocity range over which the water maser exceeds 3$\\sigma$ significance.}\n\\tablenotetext{e}{References for the systemic velocities:\n1 -- \\citet{springob2005}; \n2 -- \\citet{huchra2012};\n3 -- \\citet{RC3};\n4 -- \\citet{hewett2010};\n5 -- \\citet{SDSSDR5};\n6 -- \\citet{SDSSDR6}; \n7 -- \\citet{6dFDR3};\n8 -- \\citet{theureau2007}.}\n\\tablenotetext{f}{There appears to be an additional narrow maser line at 12422 km s$^{-1}$ (Figure \\ref{fig:2MASX0350}).}\n\\tablenotetext{g}{We list redshifts instead of velocities for $v_{\\rm range}$, $v_{\\rm peak}$, and $v_{\\rm sys}$ for J0804+3607.}\n\\tablecomments{Only objects with detected water maser emission are listed (see Table \\ref{tab:obs} for rms noise values for nondetections).\nParenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n\n\\floattable\n\\begin{deluxetable}{ccccCcC}\n\\tablecaption{20 GHz Radio Continuum Properties \\label{tab:continuum}}\n\\tablecolumns{7}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Galaxy} & \n\\colhead{$S_{\\rm peak}$\\tablenotemark{a}} &\n\\colhead{$S_{\\rm int}$\\tablenotemark{a}} &\n\\colhead{$\\alpha$\\tablenotemark{b}} &\n\\multicolumn{3}{c}{Size\\tablenotemark{a}} \\\\\n\\cline{5-7}\n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{} & \n\\colhead{Angular} & \n\\colhead{PA} & \n\\colhead{Physical}\\\\\n\\colhead{} & \n\\colhead{($\\mu$Jy bm$^{-1}$)} & \n\\colhead{(mJy)} & \n\\colhead{} & \n\\colhead{(milliarcsec)} & \n\\colhead{($^\\circ$)} & \n\\colhead{(pc)} \n}\n\\startdata\nNGC 291          & 263(24) & 1.93(20) & $-$2.0(3) & 451(52)\\times146(19) & 51.8(3.4) & 177(20)\\times57(7) \\\\\nNGC 520b        & 272(31) & 4.31(52) & $-$0.23(19) & 775(100)\\times164(24) & 95.6(2.3) & 105(14)\\times22(3) \\\\\nIC 485              &  77(15)  &0.180(48)& \\nodata & < 93\\times88            & $-$69.5 &    < 52\\times49 \\\\\nJ0804+3607 & 4247(83) & 4.71(16) & $-$1.5(3) & 62(12)\\times23(13) & 71(17) & 437(85)\\times162(92) \\\\\nCGCG 120$-$039 & 482(20) & 0.555(39) & $-$0.27(16) & 44(8)\\times23(14) & 133(33) & 23(4)\\times12(7) \\\\\nNGC 5256\\tablenotemark{c} & 633(28) & 1.543(68) & $-$2.0(2) & 127(6)\\times96(8) & 148.7(9.6) & 72(3)\\times55(5)\\\\\n                      & 78(17) & 0.48(11) & \\nodata & 433(23)\\times68(58) & 163.1(3.1) & 247(13)\\times39(33) \\\\\nCGCG 168$-$018 &  267(16) & 0.308(30) & $-$0.95(17) & < 87\\times85 & 77.8 & < 64\\times63\\\\\n\\enddata\n\\tablenotetext{a}{The peak flux density, integrated flux density, and the radio source size and orientation were \nobtained from fitting a single two-dimensional Gaussian to the source image and \ndeconvolving the source from the beam. Upper limits indicate unresolved detections and list the continuum beam parameters.}\n\\tablenotetext{b}{The spectral index $\\alpha$ was measured at the peak of the continuum emission solely from within the 18--22 GHz bandpass\nand follows the convention $S_\\nu \\propto\\nu^\\alpha$. It is not listed for objects with inadequate signal-to-noise in the 20 GHz continuum. }\n\\tablenotetext{c}{NGC 5256 shows two widely-separated continuum components (both are listed).}\n\\tablecomments{Only objects with detected 20 GHz continuum emission are listed (see Table \\ref{tab:obs} for rms noise values for nondetections).  \nParenthetical values indicate uncertainties in the ultimate digits.}\n\\end{deluxetable}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.535\\textwidth]{NGC291_overlay.eps}\n\\includegraphics[width=0.46\\textwidth]{NGC291_spec.eps}\n\\caption{Left:  NGC 291 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 393 pc.  \nRight:  water maser nondetection spectrum at the continuum peak with the systemic velocity and its uncertainty indicated by the vertical bars\n\\citep[$5705\\pm4$ km s$^{-1}$;][]{SDSSDR2}.\nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:NGC291}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{NGC520b_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{NGC520b_spec.eps}\n\\includegraphics[width=1.0\\textwidth,trim={0 0 0 0},clip]{NGC520b_overlay_m0_wide.eps}\n\\caption{Top Left:  NGC 520b integrated water maser (image) and 20 GHz radio continuum (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 135 pc.  \nTop Right:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars  (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered\non the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \nBottom: a 5\\arcsec$\\times$2\\arcsec\\ (675 pc $\\times$ 170 pc) field of view to show the full extent of the 20 GHz continuum.\n\\label{fig:NGC520b}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.52\\textwidth]{2MASX0350_overlay.eps}\n\\includegraphics[width=0.475\\textwidth]{2MASX0350_spec.eps}\n\\caption{Left:  J0350$-$0127  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 805 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX0350}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.525\\textwidth]{IC485_overlay.eps}\n\\includegraphics[width=0.47\\textwidth]{IC485_spec.eps}\n\\caption{Left: IC 485 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 3 and 4 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 562 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}). \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:IC485}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{SDSSJ0804_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{SDSSJ0804_spec.eps}\n\\caption{Left:  J0804+3607 integrated water maser (image) and 14 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 7.055 kpc.  \nRight:  water maser spectrum centered on the peak maser emission at $z=0.66045$, plotted in the object rest frame (Table \\ref{tab:masers}).  \n\\label{fig:SDSSJ0804}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{CGCG120-039_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{CGCG120_039_spec.eps}\n\\caption{Left:  CGCG 120$-$039 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 519 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:CGCG120-039}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.52\\textwidth]{2MASX0912_overlay.eps}\n\\includegraphics[width=0.475\\textwidth]{2MASX0912_spec.eps}\n\\caption{Left:  J0912+2304  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 723 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars  (Table \\ref{tab:masers}).  \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX0912}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{2MASX1011_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{2MASX1011_spec.eps}\n\\caption{Left: J1011$-$1926  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 564 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX1011}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.51\\textwidth]{NGC5256_overlay_c1.eps}\n\\includegraphics[width=0.485\\textwidth]{NGC5256_c1_spec.eps}\n\\includegraphics[width=0.51\\textwidth]{NGC5256_overlay_c2.eps}\n\\includegraphics[width=0.485\\textwidth]{NGC5256_c2_spec.eps}\n\\caption{Left:  NGC 5256 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, 16, and 32 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 570 pc.  \nRight:  water maser nondetection spectra at the continuum peaks with the systemic velocity and its uncertainty indicated by the vertical bars\n\\citep[$8353\\pm13$ km s$^{-1}$;][]{RC3}.\nThe spectra are roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \nTwo continuum sources were detected, but no water maser emission was detected toward either continuum source or in the \nlarger field of view.\n\\label{fig:NGC5256}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{CGCG168-018_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{CGCG168_018_spec.eps}\n\\caption{Left:  CGCG 168$-$018 integrated water maser (image) and 20 GHz radio continuum  (contours) maps.  Continuum contours \nindicate 4, 8, and 16 times the rms noise listed in Table \\ref{tab:obs}.   The spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 737 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:CGCG168-018}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.53\\textwidth]{2MASX1939_overlay.eps}\n\\includegraphics[width=0.465\\textwidth]{2MASX1939_spec.eps}\n\\caption{Left:  J1939$-$0124  integrated water maser map.  The 20 GHz continuum was not significantly detected.\nThe spectral line beam is shown in the lower left\n(properties listed in Table \\ref{tab:obs}). The 1\\arcsec\\ field of view is equivalent to 432 pc.  \nRight:  water maser spectrum with the systemic velocity and its uncertainty indicated by the vertical bars (Table \\ref{tab:masers}).  \nThe spectrum is roughly centered on the previous single-dish water maser detection (see Section \\ref{sec:discussion}).  \n\\label{fig:2MASX1939}}\n\\end{figure*}\n\n\n\\section{Analysis} \\label{sec:analysis}\n\nFigure \\ref{fig:offsets} shows the projected physical offset between the water maser line and continuum centroids for the \nfive objects detected in both line and continuum (Table \\ref{tab:positions}).  Only CGCG 168$-$018 shows a significant offset\nof $29.3\\pm3.6$ milliarcseconds or $21.6\\pm2.7$ pc, but based \non its 18--22 GHz  continuum spectral index of $-0.95\\pm0.17$ (Table \\ref{tab:continuum}), \nthe continuum could be jet emission and not the core of the AGN.  \nAll other maser-continuum\noffsets are not significant and are therefore consistent with\ninclined maser disk expectations. \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{offsets.eps}\n\\caption{Maser-continuum offsets for those objects detected in both.  Only CGCG 168$-$018 shows a significant offset.  Error bars are \nplotted symmetrically.\n\\label{fig:offsets}}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nAmong the 13 water maser hosts, we did not detect water masers in four objects:\nNGC 291, UGC 7016, NGC 5256, and NGC 5691.\nThe water maser in NGC 520b is likely associated with star formation and is rejected \nas an inclined maser disk candidate (see \\ref{subsubsec:NGC520b}).  \nObjects with unresolved masers but not detected in the 20 GHz continuum remain ambiguous \n(these are J0350$-$0127, J0912+2304, J1011$-$1926, and J1939$-$0124; note that none of these\nhave other sub-arcsecond radio continuum observations), as does \nCGCG 168$-$018, the only object to show a significant maser-continuum offset (see \\ref{subsubsec:CGCG168-018}).\nIC 485, J0804+3607, and CGCG 120$-$039 remain inclined maser disk candidates:  they show no \nsignificant maser-continuum offset and no high velocity lines (by selection).  While the maser\nemission from IC 485 is broad and multi-component, this type of structure spanning $\\sim$100 km s$^{-1}$ is seen in the systemic masers\nin many disk systems \\citep[e.g.,][]{kuo2011}.  J0808+3607 and CGCG 120$-$039 have narrow maser lines and are particularly \ngood inclined disk candidates.  \n\n\\subsection{Individual Objects}\n \nWe discuss the previous water maser observations, general characteristics, and our results for each \nindividual object below. \n\n\n\\subsubsection{NGC 291}   \nNGC 291 is a barred spiral galaxy with a Seyfert 2 nucleus \\citep{kewley2001,nair2010}.\nThe maser was detected as a narrow $\\sim$60 mJy line in 2006 by GBT program \nGBT06A-009,\\textsuperscript{\\ref{footnote:maser_url}}\nbut we did not detect it (2.6 mJy beam$^{-1}$ rms per 1.2 km s$^{-1}$ \nchannel; Table \\ref{tab:obs} and Figure \\ref{fig:NGC291}) nor did \\citet{kondratko2006} in 2002 (15 mJy rms per 1.3 km s$^{-1}$ \nchannel).  \nWe detect extended 20 GHz continuum with 18--22 GHz spectral index $-2.0\\pm0.3$\n(Table \\ref{tab:continuum}). \n\n\n\\subsubsection{NGC 520b}    \\label{subsubsec:NGC520b}\nNGC 520b is one galaxy in the colliding pair in NGC 520 \\citep[e.g.,][]{stanford1990}.  \n\\citet{castangia2008} detected a maser in 2005, measuring a $\\sim40$ mJy peak with the VLA \nand $L_{iso} \\sim 1\\  L_\\odot$.  Our VLA detection shows a $35\\pm3$ mJy peak and\n$L_{iso} = 3.05\\pm0.26\\ L_\\odot$ (Figure \\ref{fig:NGC520b} and Table \\ref{tab:masers}).  \nOur VLA map shows extended 20 GHz radio continuum emission (Figure \\ref{fig:NGC520b}, bottom) with a \nflat 18--22 GHz spectral index of $-0.23\\pm0.19$ (Table \\ref{tab:continuum}) and  a similar east-west \nmorphology to the \\citet{castangia2008} 14.9 GHz map.  In our map the maser is slightly north of the \\citet{castangia2008}\nmaser position.  There is a significant maser-continuum peak offset, although the single-component centroid \nfits listed in Table \\ref{tab:positions} are consistent because the radio emission is extended.  \\citet{castangia2008}\nfavor a star formation origin for the water maser in NGC 520b, but cannot rule out a low luminosity AGN.  \nThe preponderance of evidence suggests that NGC 520b is a poor candidate for an inclined maser disk.  \n\n\n\\subsubsection{J0350$-$0127}  \nJ0350$-$0127 is an almost otherwise unknown spiral or possibly irregular galaxy included in the 2MASS Redshift Survey \\citep{huchra2012}\nand detected in water maser emission by the GBT programs GBT09-051\\textsuperscript{\\ref{footnote:maser_url}} in 2010 and \nGBT10C-019\\textsuperscript{\\ref{footnote:maser_url}} in 2011.  \nThe peak flux density was $\\sim$350 mJy in 2011, in good agreement with our VLA measurement of $347\\pm4$ mJy (Table \\ref{tab:masers} and \nFigure \\ref{fig:2MASX0350}).  \nThis is a broad and luminous maser ($L_{\\rm iso} = 5181\\pm108\\ L_\\odot$) with no associated 20 GHz continuum, down to an rms noise of 15 $\\mu$Jy beam$^{-1}$\n(Table \\ref{tab:obs}).  \nWithout a continuum detection, the nature of this maser remains ambiguous, although it is almost certainly associated with an AGN.\n\n\n\\subsubsection{IC 485}  \nIC 485 is a spiral galaxy detected in the water maser line in 2006 in GBT program\nGBT06C-035.\\textsuperscript{\\ref{footnote:maser_url}}  The maser has a \n$\\sim$80 mJy peak and shows a broad profile.  \\citet{zhu2011} lists this object as \na maser nondetection.  Our VLA observations show a \nbroad, multi-component maser with a similar peak ($78\\pm2$ mJy) \nand a high luminosity, $L_{iso} = 868\\pm46\\ L_\\odot$ (Figure \\ref{fig:IC485} and Table \\ref{tab:masers}).  The 20 GHz continuum \nis detected but unresolved and faint  ($77\\pm15$ $\\mu$Jy beam$^{-1}$ peak flux density; Table \\ref{tab:continuum}).\nThis galaxy was also detected at 1.4 GHz (4.4 mJy), and its dominant radio energy source was classified as star formation \nby \\citet{condon2002}.  This classification does not exclude the presence of an AGN: \\citet{liu2011} classify the optical nucleus \nof IC 485 as a LINER.  \nThe maser-continuum offset is not significant --- less than 10.5 pc (1$\\sigma$; Table \\ref{tab:positions}) --- so this maser remains an inclined disk candidate.\n\n\n\\subsubsection{J0804+3607}   \nThis is a type 2 quasar showing luminous water maser emission at $z\\simeq0.66$ \\citep{zakamska2003,barvainis2005}.  \nThe isotropic maser luminosity was $2.31\\pm0.46\\times10^4\\ L_\\odot$ in 2005 \n\\citep[][error from the quoted 20\\% calibration uncertainty]{barvainis2005} and $1.8\\pm0.1\\times10^4\\ L_\\odot$ in 2015 (this work),\nconsistent with no variation.   The maser and 20 GHz continuum \nemission are coincident to within 40 pc (1$\\sigma$; Table \\ref{tab:positions} and Figure \\ref{fig:SDSSJ0804}), \nwhich is less precise than the other objects due to the much larger distance and lower observing frequency.  The 12--16 GHz spectral \nindex is $-1.5\\pm0.3$ (Table \\ref{tab:continuum}).  This object remains an inclined maser disk candidate. \n\n\n\\subsubsection{CGCG 120$-$039}\nThe water maser in this little-studied galaxy was detected in 2013 in GBT program\n GBT13A-236.\\textsuperscript{\\ref{footnote:maser_url}}\nIt had a peak flux density of $\\sim$210 mJy, was blueshifted from the systemic velocity, and \nshowed several components.  \nWe detect the maser --- now at $82\\pm2$ mJy peak, but still showing multiple components (Figure \\ref{fig:CGCG120-039})\n--- and the 20 GHz continuum at $0.56\\pm0.04$ mJy, which are spatially coincident to within a remarkably small $3.5\\pm1.4$ pc \n(Table \\ref{tab:positions}).  \nThe 18--22 GHz spectral index is flat:  $\\alpha = -0.27\\pm0.16$ (Table \\ref{tab:continuum}).\nThis object is a good inclined maser disk candidate.  \n\n\\subsubsection{J0912+2304}  \nThe water maser in the galaxy J0912+2304 was detected in 2008 by GBT program GBT07A-034,\\textsuperscript{\\ref{footnote:maser_url}}\nshowing a $\\sim$30 mJy peak.  \\citet{zhu2011} list this as a water maser nondetection.  The VLA observations show a $16\\pm3$ mJy \npeak, but no 20 GHz continuum down to an rms noise of 16 $\\mu$Jy beam$^{-1}$ (Figure \\ref{fig:2MASX0912} and Tables \\ref{tab:obs} and\n\\ref{tab:masers}).  The provenance of the maser remains ambiguous.\n\n\\subsubsection{J1011$-$1926}  \nThe maser in this almost unknown galaxy was detected in GBT program GBT07A-066\\textsuperscript{\\ref{footnote:maser_url}}\nin 2008.  \\citep{zhu2011} list this object as a water maser nondetection.  \nThe GBT detection shows a broad line with a $\\sim$80 mJy peak.  The VLA detection shows a broad multi-component maser with a \n$44\\pm4$ mJy peak in good agreement with the systemic velocity (Figure \\ref{fig:2MASX1011} and Table \\ref{tab:masers}).  \nWe do not detect the 20 GHz continuum.  The nature of this maser therefore remains ambiguous.\n\n\\subsubsection{UGC 7016}   \nThe water maser in this barred spiral \\citep{RC3} was \ndetected in 2007 by GBT program GBT07A-034,\\textsuperscript{\\ref{footnote:maser_url}} but \\citet{zhu2011} list this as a nondetection.\nThe GBT detection spectrum had a peak flux density of $\\sim$55 mJy, but we did not detect the maser or the 20 GHz continuum emission.\nVLA rms noise values were 3.8 mJy beam$^{-1}$ per 1.2 km s$^{-1}$ channel in the spectral line cube and \n16 $\\mu$Jy beam$^{-1}$ in the continuum map (Table \\ref{tab:obs}).\n\n\n\\subsubsection{NGC 5256}    \n\\citet{braatz2004} discovered the water maser in this merging luminous infrared galaxy in 2003.  \nIt had a peak flux density of 99 mJy ($L_{iso} = 30\\ L_\\odot$)  and was redshifted from the systemic\nvelocity by $\\sim$300 km s$^{-1}$.  \\citet{braatz2004} claim that all maser emission originates \nfrom the southern nucleus \\citep[a Sey 2 nucleus according to, e.g.,][]{mazzarella1993}.\nWe detect two 20 GHz continuum sources that are consistent with the positions of the two nuclei \n\\citep[Tables \\ref{tab:positions} and \\ref{tab:continuum};][]{brown2014},\nbut neither location (nor the larger region that includes the overlap region between the two nuclei)\nshows maser emission down to an rms noise of 3.0 mJy beam$^{-1}$ in 1.2 km s$^{-1}$ channels\n(Table \\ref{tab:obs} and Figure \\ref{fig:NGC5256}).\nThe spectral index of the northern nuclear continuum is $-2.0\\pm0.2$ in the 18--22 GHz band  (Table \\ref{tab:continuum}).  \n\n\n\\subsubsection{NGC 5691}   \nThe water maser in NGC 5691, a barred non-Seyfert spiral galaxy \\citep{mulchaey1997}, was detected in \n2009 by GBT program AGBT08C-035\\textsuperscript{\\ref{footnote:maser_url}}\nwith a $\\sim$45 mJy peak flux density.  It was listed by \\citet{zhu2011} as a maser nondetection.  \nWe did not detect the maser or any 20 GHz continuum in this galaxy, with rms noise levels of 3.7 mJy beam$^{-1}$ per 1.1 km s$^{-1}$ channel\nand 17 $\\mu$Jy beam$^{-1}$, respectively (Table \\ref{tab:obs}).\n\n\\subsubsection{CGCG 168$-$018}  \\label{subsubsec:CGCG168-018}\nCGCG 168$-$018 is a little-studied galaxy classified as an AGN by \\citet{schawinski2010} and\nlisted as a water maser nondetection by \\citet{zhu2011}.\nWater maser emission was detected by GBT program GBT07A-066\\textsuperscript{\\ref{footnote:maser_url}}\nin 2008.  The GBT water maser spectrum shows \na $\\sim$50 mJy peak, detected by this work at $25\\pm2$ mJy (Table \\ref{tab:masers}).  \nThe $0.31\\pm0.03$ mJy continuum shows a 18-22 GHz spectral index of $-0.95\\pm0.17$ (Table \\ref{tab:continuum}).   \nThe maser emission and 20 GHz continuum\nare unresolved but show a significant relative offset of $21.6\\pm2.7$ pc (Table \\ref{tab:positions} and Figure \\ref{fig:CGCG168-018}).  \nThis is the only object in the sample\nthat shows a significant offset between the maser and continuum emission (Figure \\ref{fig:offsets}).\nWhile the offset suggests that the maser emission is not deflected from an inclined maser disk, the continuum \nspectral index suggests that the continuum may arise from a jet, and the radio core is not detected.  The provenance \nof the maser therefore remains ambiguous.\n\n\\subsubsection{J1939$-$0124}   \n\\citet{greenhill2003} detected the water maser in this spiral galaxy hosting a Sey 2 nucleus in 2002 using the Tidbinbilla antenna.  \nThe maser showed a peak flux density of $\\sim$28 mJy, and \\citet{henkel2005} report $L_{iso} \\simeq 160\\ L_\\odot$. \nNo 22 GHz continuum was detected by \\citet{greenhill2003} using the VLA ($< 2.8$ mJy).  \nWe likewise detect no continuum, with rms noise of 16 $\\mu$Jy beam$^{-1}$ (Table \\ref{tab:obs}), but we do detect the \nmaser emission with peak $30\\pm2$ mJy ($L_{iso} = 102\\pm16 \\ L_\\odot$) although with a substantially different maser profile \n(Figure \\ref{fig:2MASX1939} and Table \\ref{tab:masers}).  \nThe 6 and 20 cm continua were detected at $5.4\\pm0.4$ mJy  and $15.5\\pm1.0$ mJy, respectively, by \\citet{vader1993}, and the \n6 cm continuum position agrees with the maser position to within $\\sim$1\\arcsec ($\\sim$430 pc).  \nThe nature of this maser remains ambiguous.  \n\n\n\n\\section{Conclusions}\n\nThis paper presents a physical mechanism that may enable detection of inclined water maser disks orbiting massive black holes\nvia the lensing\/deflection of in-going systemic masers.  The observational signature of an inclined disk \nis a maser line or line complex with limited Doppler extent that appears to arise at the location of the black \nhole, as identified by its radio continuum core.  With enough angular resolution, it may be possible to \nmeasure the black hole mass if the maser emission forms a lensing arc or Einstein ring, but the mass\nprecision will be limited by one's ability to measure or estimate the size of the maser-emitting portion of the disk.  \n\nWe suggest that if inclined maser disks can be detected at all, then they have probably already been detected\nin single-dish surveys but discarded for interferometric follow-up because they did not show high-velocity lines.\nWe present original 0.07--0.17\\arcsec\\ (4--100 pc) resolution VLA observations of inclined maser disk candidates with the goal\nof identifying systems where the maser emission is unresolved and is coincident with the 20 GHz continuum \nemission.  \n\nOf the 16 masers observed with the VLA, we obtained useful data for 13, and among these, five were \ndetected in both 22 GHz maser line emission and in 20 GHz continuum.  Of these five, one maser is most likely\nassociated with star formation (NGC 520b), and one shows a significant spatial offset between the maser emission \nand the continuum (CGCG 168$-$018, but it could still host an inclined maser disk --- this case is ambiguous).  \nThree objects are good inclined maser disk candidates that merit further study with VLBI:  \nIC 485, J0804+3607, and CGCG 120$-$039.\nFive maser hosts remain ambiguous, based either on non-detected or offset continua:\nJ0350$-$0127,  J0912+2304, J1011$-$1926, CGCG 168$-$018, and J1939$-$0124.  \n\nMore straightforward methods for measuring black hole masses from molecular lines may be in the offing.  For example,\n\\citet{davis2013} and \\citet{barth2016a,barth2016b} have used carbon monoxide kinematics in thin disks that approach or are\nwithin the black hole gravitational sphere of influence to obtain black hole mass measurements.  \\citet{barth2016a}, in particular, \ndemonstrate the ability of ALMA to measure black hole masses with $\\sim$10\\% uncertainty.   Although they lack the \nintrinsic brightness of masers that enables VLBI mapping, \nthermal molecular lines have the advantage of being observable at any disk inclination.\n\n\n\\acknowledgments\n\nWe thank A. Hamilton and M. Eracleous for helpful discussion and B. Butler for assistance with \nmetadata repair of bespoke observing configurations.  \nWe also thank the anonymous referee for important feedback.\nThis research has made use of NASA's Astrophysics Data System Bibliographic Services and\nthe NASA\/IPAC Extragalactic Database (NED),\nwhich is operated by the Jet Propulsion Laboratory, California Institute of Technology,\nunder contract with the National Aeronautics and Space Administration.\n\n \\facility{VLA}\n\n\\software{CASA}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Compact representation of cuts}\n\\label{sec:compact}\n\\newcommand{{\\mathcal F}}{{\\mathcal F}}\n\nThis part serves as a ``communication language'' for various components in our data structure.\nSince a cut can have up to $\\Omega(m)$ edges, the data structure cannot afford to describe it explicitly. We will use a compact representation of cuts~\\cite{ChekuriQ17}, which allows us to describe any $(1+\\epsilon)$-approximate solution in a given weighted graph using $\\tilde{O}(1)$ bits; notice that, in the MWU framework, we only care about (punishing) near-optimal solutions, so it is sufficient for us that we are able to concisely describe such cuts.\n\n\n\nFormally, we say that a family ${\\mathcal F}$ of  subsets of edges is $\\epsilon$-canonical for $(G,\\textbf{w})$ if (i) $|{\\mathcal F}| \\leq \\widetilde{O}(|E|)$, (ii) any $(1+\\epsilon)$-approximate minimum cut of $(G,\\textbf{w})$ is a disjoint union of at most $\\tilde{O}(1)$ sets in ${\\mathcal F}$, (iii) any set $S \\in {\\mathcal F}$ can be described concisely by $\\tilde{O}(1)$ bits, and (iv) every edge in the graph belongs to $\\tilde{O}(1)$ sets in ${\\mathcal F}$.\nIt follows that any $(1+\\epsilon)$-approximate cut admits a short description.\nDenote by $[[S]]$ a short description of cut $S \\in {\\mathcal F}$, and for each $(1+\\epsilon)$ approximate cut $C$, $[[C]]$ a short description of $C$.\n\n\n\n\n\\begin{lemma}[implicit in \\cite{ChekuriQ17}] \\label{lem:canonical cuts}\nThere exists a randomized data structure that, on input $(G,\\textbf{w})$, can be initialized in near-linear time,  (w.h.p) constructs an $\\epsilon$-canonical family ${\\mathcal F} \\subseteq 2^{E(G)}$, and handles the following queries:\n\\begin{itemize}\n\n    \\item Given a description $[[C]]$ of a $(1+\\epsilon)$-approximate cut, output a list of $\\tilde{O}(1)$ subsets in ${\\mathcal F}$ such that $C$ is a disjoint union of those subsets in $\\tilde{O}(1)$ time.\n\n    %\n\n    \\item Given a description of $[[S]]$, $S \\in {\\mathcal F}$, output a list of edges in $S$ in $\\widetilde{O}(|S|)$ time.\n\n    %\n     %\n   \n\\end{itemize}\n\\end{lemma}\n\n\n\n\\subsection{Range Cut-listing Data Structure}\n\\label{sec:cut listing}\n\n\n\nThe cut listing data structure is encapsulated in the following theorem.\n\n\\begin{theorem} [Range Cut-listing Data Structure \\cite{ChekuriQ17}] \\label{thm:cut listing}\nThe cut-listing data structure, denoted by $\\mathcal{D}$, maintains dynamically changing weighted graph $(G,\\widehat{\\textbf{w}})$ and supports the following operations.\n\\begin{itemize}\n\\item $\\mathcal{D}$.\\textsc{Init}$(G,\\vecw^{\\operatorname{init}},\\lambda, \\epsilon)$ where $G$ is a graph, $\\widehat{\\textbf{w}}$ is an initial weight function, and ${\\sf mincut}_{\\widehat{\\textbf{w}}} \\geq \\lambda$: initialize the data structure and  the weight $\\widehat{\\textbf{w}} \\leftarrow \\vecw^{\\operatorname{init}}$ in $\\tilde{O}(m)$ time.\n\n\n     \\item $\\mathcal{D}$.\\textsc{FindCut}$():$ output either a short description of a $(1+O(\\epsilon))$-approximate mincut $[[C]]$ or $\\emptyset$ (when ${\\sf mincut}_{\\widehat{\\textbf{w}}} > (1+\\epsilon) \\lambda$). The operation takes amortized $\\tilde{O}(1)$ time.\n\n    \\item $\\mathcal{D}$.\\textsc{Increment$(\\Delta)$} where $\\Delta = \\{(e,\\delta_e)\\}$ is the set of increments (defined by a pair of an edge $e \\in E$ and a value $\\delta_e \\in \\mathbb{R}_{\\geq 0}$): For each $(e,\\delta_e) \\in \\Delta$, $\\widehat{\\textbf{w}}(e) \\gets \\widehat{\\textbf{w}}(e) + \\delta_e$.   The operation takes $\\tilde{O}(|\\Delta|)$ time (note that $|\\Delta|$ corresponds to the number of increments).\n\\end{itemize}\n\\end{theorem}\n\n\nAs outlined earlier, the cut listing data structure will be invoked with $\\widehat{\\textbf{w}} = \\textbf{w}_{\\rho}$.\n\n\n\n\n\n\n\\subsection{Truncated Lazy MWU Increment}\n\\label{sec:lazy weight}\n\nThe data structure is formally summarized by the definition below.\n\n\n\n\n\n\n\n\n\\begin{definition}[Truncated Lazy MWU Increment]\n A truncated lazy MWU increment denoted by $\\mathcal{L}$ maintains the approximate weight function $\\textbf{w}$ explicitly, and exact weight $\\vecw^{\\operatorname{mwu}}$ implicitly and  supports the following operations:\\footnote{This is implicit in the sense that $w$ is divided into parts and they are internally stored in different memory segments. Whenever needed, the real weight can be constructed from the memory content in near-linear time.}\n\n \\begin{itemize}\n     \\item   $\\mathcal{L}.\\textsc{Init}(G,\\vecw^{\\operatorname{init}},\\rho)$ where $G$ is a graph, $\\vecw^{\\operatorname{init}}$ is the initial weight function, $\\rho \\in \\mathbb{R}_{>0}$: Intialize the data structure, and set $\\textbf{w} \\leftarrow \\vecw^{\\operatorname{init}}$.\n\n     \\item $\\mathcal{L}.\\textsc{Punish}([[C]])$ where $C$ is a cut: Internally punish the free cut $(C,F)$ for some $F$ (to be made precise later) and output a list of increment $\\Delta = \\{(e,\\delta_e)\\}$ so that for each $e \\in E$, $\\vecw^{\\operatorname{init}}(e)$ plus the total increment over $e$ is $\\textbf{w}_{\\rho}(e)$.\n\n     \\item  $\\mathcal{L}.\\textsc{Flush}()$: Return the exact weight $\\vecw^{\\operatorname{mwu}}$.\n \\end{itemize}\n\\end{definition}\n\nRemark that the output list of increments returned by {\\sc Punish} is mainly for the purpose of syncing with the cut listing data structure (so it aims at maintaining $\\textbf{w}_{\\rho}$ instead of $\\textbf{w}$).  Also, in the {\\sc Punish} operation, the data structure must compute the set $F \\subseteq C$ of free edges efficiently (these are the edges whose weights would not be increased).\n This is one of the reasons for which we cannot use the lazy update data structure in~\\cite{ChekuriQ17} as a blackbox. Section~\\ref{sec:dataStructures} will be devoted to proving the following theorem.\n\n\\begin{theorem} \\label{thm:tlmi}\nThere exists a lazy MWU increment with the following time complexity: (i) init operation takes $\\tilde{O}(m)$ time, (ii)  \\textsc{Punish} takes $\\tilde{O}(K) +\\widetilde{O}\\left(\\sum_e \\log \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)$ time in total where $K$ is the number of calls to $\\textsc{Punish}$ and outputs at most $\\widetilde{O}\\left(\\sum_e \\log \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)$ increments, and (iii) flush takes $\\tilde{O}(m)$ time.   Moreover, the Invariant~\\ref{inv:new approx weight} is maintained throughout the execution.\n\\end{theorem}\n\n\n\n %\n\n  %\n\n\n\\subsection{A Fast Range Punisher for Normalized Free Cut Problem}\n\n\n\n\\label{sec:implementationPunisher}\n\n\\newcommand{\\mathcal{T}}{\\mathcal{T}}\n\\newcommand{\\mathcal{D}}{\\mathcal{D}}\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\n\n\n\nNow we have all necessary ingredients to prove~\\Cref{lem:fast range punisher}. The algorithm is very simple and described in~\\Cref{alg:fast range punisher}. We initialize the cut-listing data structure ${\\mathcal D}$ so that it maintains the truncated weight $\\textbf{w}_{\\rho}$ and the lazy weight data structure $\\mathcal{L}$.  \nWe iteratively use ${\\mathcal D}$ to find a cheap cut in $(G,\\textbf{w}_{\\rho})$ until no such cut exists. Due to our mapping theorem, such a cut found can be used for our problem, and the data structure $\\mathcal{L}$ is responsible for punishing the weights (Line 8) and returns the list of edges to be updated (this is for the cut-listing ${\\mathcal D}$ to maintain its weight function $\\textbf{w}_{\\rho}$).\n\n\n\n\\subsubsection*{Algorithm}\n\\begin{algorithm}[H]\n\\KwIn{$G,\\vecw^{\\operatorname{init}},\\lambda , \\epsilon$ such that ${\\sf OPT}_{\\vecw^{\\operatorname{init}}}\\geq \\lambda$.}\n\\KwOut{a correct weight function $\\textbf{w} = \\vecw^{\\operatorname{mwu}}$ such that ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$.}\n\\BlankLine\n$\\textbf{w} \\gets \\vecw^{\\operatorname{init}}$ and $\\rho \\gets (1+\\epsilon)\\lambda$\\;\nLet $\\textbf{w}_{\\rho}$ be the truncated weight function of $\\textbf{w}$.\\;\n\\lIf{ ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$}{ \\Return{$\\textbf{w}$.}} \\label{line:early return}\n  Let $\\mathcal{D}$ and $\\mathcal{L}$ be cut listing data structure, and truncated lazy MWU increment. \\;\n  $\\mathcal{D}.\\textsc{Init}(G,\\textbf{w}_{\\rho}, k\\rho\/(1+\\epsilon), \\epsilon)$\\;\n  $\\mathcal{L}.\\textsc{Init}(G,\\textbf{w},\\rho,\\epsilon)$\\;\n  \\While{$\\mathcal{D}.\\textsc{FindCut}()$ \\normalfont{returns} $[[C]]$}\n  {\n    $\\Delta \\gets \\mathcal{L}.\\textsc{Punish}([[C]])$\\;\n    $\\mathcal{D}.\\textsc{Increment}(\\Delta)$\\;\n  }\n  $\\textbf{w} \\gets \\mathcal{L}.\\textsc{Flush}()$\\;\n\n  \\Return{$\\textbf{w}$.} \\label{line:late return}\n\\caption{\\textsc{FastRangePunisher}($G,\\textbf{w},\\lambda$)}\n\\label{alg:fast range punisher}\n\\end{algorithm}\n\n\\subsubsection*{Analysis}\nBy input assumption, we have ${\\sf OPT}_w \\geq \\lambda$. If $\\textbf{w}$ is returned at line \\ref{line:early return}, then ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$. By \\Cref{thm:approx mapping}(\\ref{item:approx mapping1}), ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq {\\sf OPT}_{\\textbf{w}} \\geq \\rho= (1+\\epsilon)\\lambda$, and we are done (since minimum cut can be computed in near-linear time). Now, we assume that $\\textbf{w}$ is returned at the last line. The following three claims imply \\Cref{lem:fast range punisher}.\n\n\\begin{claim} For every cut $[[C]]$ returned by the range cut listing data structure during the execution of \\Cref{alg:fast range punisher}, we have that $(C,H_{\\textbf{w},\\rho} \\cap C)$ is a $(1+O(\\epsilon))$-approximation to ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}}$ at the time $[[C]]$ is returned.\n\\end{claim}\n\nWe remark that it is important that our cut punished must be approximately optimal w.r.t. the actual MWU weight.\n\n\\begin{proof}\n  By definition of $\\mathcal{L}.\\textsc{Flush}()$ operation, we always have that the exact weight function and approximate weight function are identical at the beginning of the loop.  By definition of $\\mathcal{L}.\\textsc{Punish}([[C]])$, the total increment plus the initial weight at the beginning of the loop for every edge $e$ is $\\textbf{w}_{\\rho}(e)$ and \\Cref{inv:new approx weight} holds. Therefore, by definition of $\\mathcal{D}.\\textsc{Increment}(\\Delta)$, the range cut-listing data structure maintains the weight function $\\textbf{w}_{\\rho}$ internally. We now bound the approximation of each cut $[[C]]$ that $\\mathcal{D}.\\textsc{FindCut}()$ returned. Let $F =  H_{\\textbf{w},\\rho}\\cap C$. By definition of $\\textsc{FindCut}()$, we have that $\\textbf{w}_{\\rho}(C) < k\\rho$. By \\Cref{thm:approx mapping}(\\ref{item:approx mapping2}), ${\\sf val}_{\\textbf{w}}(C, F) < (1+\\epsilon)\\lambda$. By \\Cref{inv:new approx weight}, we have that ${\\sf val}_{\\vecw^{\\operatorname{mwu}}}(C, \\tilde F) < (1+O(\\epsilon))\\lambda$.  Since ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq {\\sf OPT}_{w^{\\operatorname{init}}} \\geq \\lambda$, we have $(C, F)$ is a $(1+O(\\epsilon))$-approximation to ${\\sf OPT}_{w}$.\n\\end{proof}\n\n\\begin{claim}\nAt the end of \\Cref{alg:fast range punisher},  we have ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq (1+\\epsilon)\\lambda.$\n\\end{claim}\n\\begin{proof}\n  Consider the time when $\\mathcal{D}.\\textsc{FindCut}()$ outputs $\\emptyset$. The fact that this procedure terminates means that ${\\sf mincut}_{\\textbf{w}_{\\rho}} \\geq k\\rho$. Therefore,  \\Cref{thm:approx mapping}(\\ref{item:approx mapping1})  implies that  ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$. Let $(C^*, F^*)$ be an optimal normalized free cut with respect to $\\vecw^{\\operatorname{mwu}}$. We have\n\n  \\begin{align*}\n    {\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} &= {\\sf val}_{\\vecw^{\\operatorname{mwu}}}(C^*,F^*) \\\\\n    &\\geq {\\sf val}_{\\textbf{w}}(C^*,F^*) \\\\\n    & \\geq {\\sf OPT}_{\\textbf{w}} \\\\\n    &\\geq (1+\\epsilon)\\lambda\n  \\end{align*}\n\n  \\noindent where the first inequality follows from \\Cref{inv:new approx weight}.\n  %\n\\end{proof}\n\n\n\\begin{claim}\n  \\Cref{alg:fast range punisher} terminates in $\\tilde{O}(m+K+\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)}))$ time where $K$ is the number of \\textsc{Punish} operations.\n\\end{claim}\n\n\\begin{proof}\nWe first bound the running time due to truncated lazy MWU increment. By \\Cref{thm:tlmi}, the total running time due to $\\mathcal{L}$ (i.e.,  $\\mathcal{L}.\\textsc{Init}, \\mathcal{L}.\\textsc{Punish}, \\mathcal{L}.\\textsc{Flush}$)  is  $\\tilde{O}(m+K+\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)}))$ time where $K$ is the number of \\textsc{Punish} operations.  We bound the running time due to cut-listing data structure. Observe that the number of cuts listed equals the number of calls of \\textsc{Punish} operations, and the total number of edge increments in $\\mathcal{D}$ is $\\widetilde{O}\\left(\\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)})\\right)$. By \\Cref{thm:cut listing}, the total running time due to $\\mathcal{D}$ (i.e, $\\mathcal{D}.\\textsc{Init}, \\mathcal{D}.\\textsc{FindCut}(), \\mathcal{D}.\\textsc{Increment}(\\Delta)$) is as desired.\n\\end{proof}\n\n\n\\subsection{Additive Accuracy}\n\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\n\nNotice that, at any time, we always have $\\vecw^{\\operatorname{mwu}}(e)= \\frac{1}{c(e)} \\cdot\n\\exp \\left( \\vecv^{\\operatorname{mwu}}(e) s(e)\\right)$ for some positive real numbers $\\vecv^{\\operatorname{mwu}}(e)$\nand $s(e) = \\frac{\\epsilon}{c(e)}$.\nFor the true vector $\\vecv^{\\operatorname{mwu}}$, the update rule for $(C, F)$ becomes the\nfollowing: $\\vecv^{\\operatorname{mwu}}(e) \\leftarrow \\vecv^{\\operatorname{mwu}}(e) + c_{\\min}$ for all $e \\in C\n\\setminus F$. This update causes all\nedges in $C \\setminus F$ to increase their $\\vecv^{\\operatorname{mwu}}(e)$ by the same amount\nof $c_{\\min}$. By \\Cref{thm:approx mapping}(2), it is enough to use\n$F$ to be always $H_{\\rho, \\textbf{w}} \\cap C$, i.e., the set of heavy edges with respect to $\\textbf{w}$ inside\n$C$.  From now on, we always use $H_{\\textbf{w},\\rho} \\cap C$ as a free\nedge set whenever we punish $C$.  \n\nInstead of maintaining the approximate vector $\\textbf{w}$ for the real\nvector $\\vecw^{\\operatorname{mwu}}$, we instead work\nwith the additive form of the approximate vector $\\textbf{v}$ for the real vector $\\vecv^{\\operatorname{mwu}}$,\nand we bound the additive error:\n\\begin{align} \\label{eq:additive error small}\n  \\forall e \\in E, \\vecv^{\\operatorname{mwu}}(e) - \\eta\/s(e) \\leq \\textbf{v}(e) \\leq \\vecv^{\\operatorname{mwu}}(e) \n\\end{align}\nNext, we show that it is enough to work on $\\vecv^{\\operatorname{mwu}}$ with additive errors. \n\\begin{proposition}\n  If \\Cref{eq:additive error small} holds, then $\\forall e \\in E, \\vecw^{\\operatorname{mwu}}(e)\n  (1-\\eta) \\leq \\textbf{w}(e) \\leq \\vecw^{\\operatorname{mwu}}(e)$. \n\\end{proposition}\n\\begin{proof}\n  Fix an arbitrary edge $e \\in E$, we have $\\textbf{w}(e) \\leq\n  \\vecw^{\\operatorname{mwu}}(e)$. Moreover,\n$$ \\textbf{w}(e) \\geq \\frac{1}{c(e)} \\exp( (\\vecv^{\\operatorname{mwu}}(e) - \\eta\/s(e)) s(e)) =\n\\frac{1}{c(e)} \\cdot \\frac{\\exp(\\vecv^{\\operatorname{mwu}}(e)s(e))}{\\exp(\\eta)} \\geq (1-\\eta) \\vecw^{\\operatorname{mwu}}(e).$$\n\\end{proof}\n\\subsection{Local Bookkeeping} \\label{sec:local book}\n\n\\newcommand{\\operatorname{q}}{\\operatorname{q}}\n\\newcommand{\\operatorname{ref}}{\\operatorname{ref}}\n\\newcommand{\\operatorname{last}}{\\operatorname{last}}\n\\newcommand{\\operatorname{diff}}{\\operatorname{diff}}\n\\newcommand{\\operatorname{priority}}{\\operatorname{priority}}\n\nWe describe the set of variables to maintain in order to support\n$\\textsc{Punish}$ operation efficiently. Let $\\mathcal{F}$ be a\n$\\epsilon$-canonical family of subsets of edges (as defined in\n\\Cref{lem:canonical cuts}). We call each subset of edges in $\\mathcal{F}$ as a canonical cut.  Let $\\bar E = E \\setminus H_{\\textbf{w}, \\rho}\n$ be the set of non-heavy edges where $H_{\\rho, \\textbf{w}} = \\{e \\in E\\colon \\textbf{w}(e) \\geq \\rho\\}$ is the set of heavy edges. We define a bipartite graph $\\mathcal{B} =\n(\\mathcal{F}, \\bar E, E_{\\mathcal{B}})$ where the first vertex partition  is the set\nof canonical cuts $\\mathcal{F}$ , the\nsecond vertex partition is $\\bar E$, and for each $S \\in \\mathcal{F}$ and for\neach $e \\in \\bar E$, we add an edge  $(S,e)$ to $E_{\\mathcal{B}}$ if and\nonly if $e \\in S$.  Let $\\operatorname{q}(\\mathcal{B}) =$ the maximum degree of\nnodes in $\\bar E$ in graph $\\mathcal{B}$.  Since $\\mathcal{F}$ is\n$\\epsilon$-canonical, $\\operatorname{q}(\\mathcal{B}) = \\tilde{O}(1)$. By \\Cref{lem:canonical\n  cuts}, given a description\n$[[C]]$ of 1 or 2$-$repsecting cut, we can compute a list of at most\n$\\tilde{O}(1)$ canonical cuts in $\\mathcal{F}$ in $\\tilde{O}(1)$ time.\n\nWe maintain the following variables:\n\\begin{enumerate}\n\\item For each canonical cut $S \\in \\mathcal{F}$,\n  \\begin{enumerate} \n     \\item we have a non-negative\n  real number $\\operatorname{ref}(S)$ representing the reference point for\n  the total increase in $S$ so far.  %\n  \\item Also, we create a min priority queue\n  $Q_{S}$ containing the set of neighbors $N_{\\mathcal{B}}(S)$ (which is the\n  set of edges in $\\bar E$ that $S$ contains). %\n %\n  \\item Also, we define $c_{\\mathcal{B}}(S) = \\min_{e \\in N_{\\mathcal{B}}(S)}c(e)$ for the purpose of\n  computing $c_{\\min}$  which is the minimum capacity $c(e)$ for all\n  edge $e$ in the cut (excluding  heavy edges) that we want to punish. %\n  \\end{enumerate}\n\\item For each edge $(S,e) \\in E_{\\mathcal{B}}$, we  have a number\n  $\\operatorname{last}(S,e)$ representing the last update point for $e$ in $S$.\n \\item For each edge $e \\in E$, we maintain $\\textbf{v}(e)$.\n\\end{enumerate}\n\nFor each edge $(S,e) \\in E_{\\mathcal{B}}$, we define $\\operatorname{diff}(S,e) =\n\\operatorname{ref}(S) - \\operatorname{last}(S,e) \\geq 0$.  This difference represents the total\nslack from the exact weight of $e$ on $S$ (we will ensure that the\nslack is non-negative by being ``lazy''). When summing over all canonical cuts that contains $e$, we\nensure that $\\sum_{S \\ni  e}\\operatorname{diff}(S,e) =\n\\vecv^{\\operatorname{mwu}}(e) - \\textbf{v}(e)$.  More formally, we maintain the following\ninvariants throughout the execution of the truncated lazy increment.\n\n\\begin{invariant}  \\label{inv:lazy update}Let $\\eta' = \\eta\/\\operatorname{q}({\\mathcal B})$. %\n  \\begin{enumerate}[(a)]\n   \\item \\label{item:total diff not far} for all $e \\in \\bar E$, $ \\frac{\\eta}{s(e)}  \\geq \\sum_{S:\n       (S,e) \\in E_{\\mathcal{B}}}\\operatorname{diff}(S,e) = \\vecv^{\\operatorname{mwu}}(e) - \\textbf{v}(e)$,  \n   \\item \\label{item:priority set correctly} for all $(S,e) \\in E_{\\mathcal{B}}$, $Q_S.\\operatorname{priority}(e) =\n     \\operatorname{last}(S,e) +  \\frac{\\eta'}{s(e)}$, and\n   \\item  \\label{item:heavy edges exact}for all $e \\in H_{\\textbf{w},\\rho}$, $\\vecv^{\\operatorname{mwu}}(e) = \\textbf{v}(e)$.\n %\n  %\n   \\end{enumerate}\n\\end{invariant}\n\nIntuitively, the first invariant means for each $e \\in \\bar E$, the total difference over all\n$S \\ni e$ is bounded. The second invariant ensures that $\\operatorname{ref}(S)\n\\leq Q_S.\\operatorname{priority}(e)$ if and only if \n$\\operatorname{diff}(S,e)$ is small, and we can apply extract min operations on $Q_S$ to detect all\nedges whose priority exceeds the reference point efficiently.  The\nthird invariant means we restore the exact value for all heavy\nedges.\n\n\\begin{proposition}\n\\Cref{inv:lazy update}\\ref{item:total diff not far} implies \\Cref{eq:additive error small}.\n\\end{proposition}\n\nAlso, this invariant allows us to ``reset'' $\\textbf{v}$ to be $\\vecv^{\\operatorname{mwu}}$\nefficiently. \n\n\\begin{algorithm}[H]\n  \\BlankLine\n  $\\textbf{v}(e) \\gets \\textbf{v}(e) + \\sum_{S: (S,e) \\in E_{\\mathcal{B}}} \\operatorname{diff}(S,e)$ \\; %\n  \\For{\\normalfont{each} $S: (S,e) \\in E_{\\mathcal{B}}$}{\n  $\\operatorname{last}(S,e) \\gets \\operatorname{ref}(S)$ \\;\n   $Q_S.\\operatorname{priority}(e) \\gets \\operatorname{last}(S,e) + \\frac{\\eta'}{s(e)}$ \\;\n }\n\\caption{\\textsc{Reset}($e$)}\n\\label{alg:reset}\n\\end{algorithm}\n\nSince priority queue supports the change of priority in $O(\\log m)$ time, we\nhave:\n\\begin{proposition}\n  The procedure \\textsc{Reset} can be implemented in time $O( \\operatorname{q}(\\mathcal{B}) \\cdot \\log m) = \\tilde{O}(1)$.\n\\end{proposition}\n\n\\subsection{Init}\nDefine $\\textbf{v} = v_0$ where $v_0$ is the additive form of $w_0$. We construct the bipartite graph $\\mathcal{B} = (\\mathcal{F},\n\\bar E, E_{\\mathcal{B}})$ as defined in \\Cref{sec:local book}. We use \\Cref{lem:canonical cuts} to construct\n$\\mathcal{B}$ in $\\tilde{O}(m)$ time. For each $S \\in \\mathcal{F}$, we create a min priority\nqueue $Q_S$ containing all the elements in $N_{\\mathcal{B}}(S)$ where for\neach $e \\in N_{\\mathcal{B}}(S)$, we set $Q_s.\\operatorname{priority}(e) =\n\\eta'\/s(e)$. We also define $\\operatorname{ref}(S) = 0$ for all $S \\in \\mathcal{F}$, and\n$\\operatorname{last}(S,e) = 0$ for all $(S,e) \\in E_{\\mathcal{B}}$. By design, the\ninvariants are satisfied. The total running time of this step is\n$\\tilde{O}(m)$.  \n\\subsection{Punish}\n\nGiven a short description of 1 or 2-respecting cut $[[C]]$, we apply\n\\Cref{lem:canonical cuts} to obtain a set $\\mathcal{S} \\subseteq \\mathcal{F}$ of $\\tilde{O}(1)$ canonical\ncuts whose disjoint union is $C$ in $\\tilde{O}(1)$ time.  Recall that the\nupdate increases $\\vecv^{\\operatorname{mwu}}(e)$ by $c_{\\min}$ for each $e \\in C -\nH_{\\textbf{w},\\rho}$ where $c_{\\min} = \\min_{e \\in C - H_{\\textbf{w},\\rho}} c(e)$.\n\\begin{claim}\nWe can compute $c_{\\min}$ in $\\tilde{O}(1)$ time.\n\\end{claim}\n\\begin{proof}\nBy definition of $c_{\\mathcal{B}}(S)$,   $\\min_{S \\in \\mathcal{S}}\nc_{\\mathcal{B}}(S) = \\min_{S \\in \\mathcal{S}} \\min_{e \\in N_{\\mathcal{B}}(S)}c(e) =  \\min_{e \\in C -\n  H_{\\textbf{w},\\rho}} c(e) = c_{\\min}$. The claim follows because there are $\\tilde{O}(1)$ canonical\ncuts in $\\mathcal{S}$ and we maintain the value $c_{\\mathcal{B}}(S)$ for every $S \\in \\mathcal{F}$.%\n\\end{proof}\n\nIn the first step, for each $S \\in \\mathcal{S}$, we set $\\operatorname{ref}(S) \\gets\n\\operatorname{ref}(S) + c_{\\min}$. This takes $\\tilde{O}(1)$ time because $|\\mathcal{S}| =\n\\tilde{O}(1)$ and potentially causes a violation to \\Cref{inv:lazy\n  update}\\ref{item:total diff not far}.\n\nIn the second step, we check and fix the invariant violation as follows.  For\neach $S \\in \\mathcal{S}$, let $W_S = \\{ e \\in S \\setminus H_{\\textbf{w},\\rho} \\colon \\operatorname{ref}(S) > Q_S.\\operatorname{priority}(e)\\}$ be the set of all edges\nin $S \\setminus H_{\\textbf{w},\\rho}$ whose priority in $Q_S$ is smaller\nthan the reference point $\\operatorname{ref}(S)$. For each $e \\in W_S$, we call\nthe procedure $\\textsc{Reset}(e)$. This take times $O(r \\cdot q(\\mathcal{B})\n\\log m) = \\tilde{O}(r)$ where $r$ is the number of calls to $\\textsc{Reset}$\nprocedure. There will be new heavy edges after this step, which means\nwe need to update $\\mathcal{B}$ to correct the set $\\bar E$. \n %\n\nIn the third step, we identify new heavy edges from the set of edges\nthat we called \\textsc{Reset} procedure in the second step, then we\nremove each edge in the set from the associated priority queues and\nfrom the graph $\\mathcal{B}$ as follows. Let $U = \\bigcup_{S \\in \\mathcal{S}}W_S$. Define $U_{H} = \\{ e \\in U \\colon \\textbf{w}(e) \\geq\n\\rho\\}$.  For each $e \\in U_H$, for all $D \\in N_{\\mathcal{B}}(e)$, remove $e$ from the priority queue\n$Q_D$ and update the value $c_{\\mathcal{B}}(D)$ (to get a new minimum after\nremoving $e$). Finally, delete all nodes in $U_H$ from $\\mathcal{B}$. The third step\ntakes $O( |U| + |U_H| \\operatorname{q}(\\mathcal{B}) \\log m + |U_H| \\operatorname{q}(\\mathcal{B}))\n= \\tilde{O}(r)$ time. The running time follows because the $|U| = r$ and $|U_H| \\leq |U|$.\n\nFinally, we output $\\Delta$ where $\\Delta$ is constructed as\nfollows. For each $e \\in U$, let $\\textbf{w}'(e)$ be the weight of $e$\nbefore \\textsc{Reset}$(e)$ is invoked. If $e \\not \\in U_H$, then we define $\\delta_e =\n\\textbf{w}(e) - \\textbf{w}'(e)$. Otherwise, we define $\\delta_e = \\rho -\n\\textbf{w}'(e)$. Then, we add $(e,\\delta_e)$ to $\\Delta$. \n\n\n\\begin{lemma}\n  If \\Cref{inv:lazy update} holds before calling $\\textsc{Punish}([[C]])$, then\n  \\Cref{inv:lazy update} holds afterwards.\n\\end{lemma}\n\\begin{proof}\n  In the first step, we have $ \\bigcup_{S \\ni \\mathcal{S}} N_{\\mathcal{B}}(S) = C\n  \\setminus H_{\\textbf{w},\\rho}$, and thus the violation to \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} can only happen due to some edge $e \\in C\n\\setminus H_{\\textbf{w},\\rho}$.  Because the unions are over disjoint sets, for each\nedge $e \\in C \\setminus H_{\\textbf{w},\\rho}$, there is a unique\ncanonical cut $S_e \\in \\mathcal{S}$ such that $N_{\\mathcal{B}}(S_e) \\ni e$.\n\n\\begin{claim} \n  If there is a violation to \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} due to an edge $e \\in C \\setminus H_{\\textbf{w},\\rho}$, then \\textsc{Reset}$(e)$ is invoked in the second step.\n\\end{claim}\n\\begin{proof}\n    Since  \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} is violated due to an edge $e$, we have $\\sum_{S':\n  (S',e) \\in E_{\\mathcal{B}}} \\operatorname{diff}(S',e) > \\eta\/s(e)$. By averaging argument,\nthere is a canonical cut $S^*$ such that $\\operatorname{diff}(S^*,e) >\n\\frac{\\eta}{s(e)} \\cdot q(\\mathcal{B})$. Since $\\operatorname{diff}(S_e,e)$ is the\nonly term in the summation that is increased, we have $S^* = S_e$. Therefore, we have %\n$$ \\frac{\\eta}{s(e)}\\cdot q(\\mathcal{B}) < \\operatorname{diff}(S_e,e) = \\operatorname{ref}(S_e)\n- \\operatorname{last}(S_e,e) \\overset{\\ref{item:priority set correctly}}{=} \\operatorname{ref}(S_e) - Q_{S_e}.\\operatorname{priority}(e) +\n\\frac{\\eta'}{s(e)}.$$  Therefore, $\\operatorname{ref}(S_e) >\nQ_{S_e}.\\operatorname{priority}(e)$, and so $e \\in W_S$ as defined in the\nsecond step. Hence, \\textsc{Reset}$(e)$ is invoked. \n\\end{proof}\n\nSince \\textsc{Reset}$(e)$ is invoked for every violation, we have that \\Cref{inv:lazy update}\\ref{item:total\n  diff not far} is maintained. By design, the second invariant is\ntrivially maintained whenever \\textsc{Reset} is invoked, and also the last\ninvariant is automatically maintained by the third step. This completes the proof.\n\\end{proof}\n\n\\subsection{Flush}\nFor each $e \\in \\bar E$, we call the procedure\n$\\textsc{Reset}(e)$. Then, we output $\\textbf{w}$ which is the same as\n$\\vecw^{\\operatorname{mwu}}$.  The total running time is $O(q(\\mathcal{B}) |\\bar{E}|)\n= \\tilde{O}( m )$.  \n \n\\subsection{Total Running Time}\n\nThe initialization takes $\\tilde{O}(m)$. Let $K$ be the number of calls to\n\\textsc{Punish}$([[C]])$ and let $I$ be the number of calls to\n\\textsc{Reset}$(e)$ before calling \\textsc{Flush}(). The total running time\ndue to the first step is $O(K\\log^2 n) = \\tilde{O}(K)$, and total running\ntime due to the second and third steps is $\\tilde{O}(I)$. It remains to\nbound $I$, the total number of calls to \\textsc{Reset}$(e)$. Since\neach \\textsc{Reset}$(e)$ increases of weight $\\vecw^{\\operatorname{mwu}}(e)$ by a factor of\n$1+\\eta'$, the total number of resets is\n\n\\begin{align*}\n  O(\\sum_{i \\in [n]}  \\log_{1+O(\\eta')} \\left(\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}\\right)) &= O\\left(\\frac{q({\\mathcal B})}{\\eta} \\cdot  \\sum_{i \\in [n]} \n                                                 \\log ( \\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})  \\right) \\\\\n  &=  \\tilde{O} \\left(\\frac{1}{\\eta} \\cdot  \\sum_{i \\in [n]} \n  \\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})  \\right).\n\\end{align*}\n\n\n\n\n\n\n\\section{Introduction}\n\n\nIn the $k$-Edge-Connected Spanning Subgraph problem ($k$ECSS), we are given an undirected $n$-node $m$-edge graph $G=(V,E)$ together with edge costs, and want to find a minimum-cost $k$-edge connected spanning subgraph.\\footnote{\\label{foot:intro:sub-multigraph}Note that this problem should not be confused with a variant that allows to pick the same edge multiple time, which is sometimes also called $k$ECSS (e.g.,~\\cite{ChekuriQ17}). We follow the convention in \\cite{CzumajL07} and call the latter variant minimum-cost $k$-edge connected spanning sub-multigraph ($k$ECSSM) problem. (See also the work by Pritchard~\\cite{Pritchard10}.)}\nFor $k=1$, this is simply the minimum spanning tree problem, and thus can be solved in $O(m)$ time~\\cite{karger1995randomized}.\nFor $k\\geq 2$, the problem is a classical NP-hard problem whose first approximation algorithm was given almost four decades ago, where Frederickson and Jaja \\cite{FredericksonJ81} gave a $3$-approximation algorithm that runs in $O(n^2)$ time for the case of $k=2$.\nThe approximation ratio was later improved to $2$ by an\n$\\tilde O(mnk)$-time\nalgorithm of Khuller and Vishkin~\\cite{KhullerV94}.\\footnote{$\\tilde O$ hides $\\operatorname{polylog}(n)$ factor.}\nThis approximation factor of $2$  has remained the best for more than 30 years, even for a special case of $2$ECSS called the weighted tree augmentation problem.\nWhen the running time is of the main concern, the fastest known algorithm takes  $O(n^2)$  time at the cost of a significantly higher $(2k-1)$-approximation guarantee, due to Gabow, Goemans, and Williamson~\\cite{GabowGW98}.\n\nThis above state-of-the-arts leave a big gap between\nalgorithms achieving the best approximation ratio and the best time complexity. This gap exists even for $k=2$.\nIn this paper, we improve the running time of both aforementioned algorithms of~\\cite{KhullerV94,GabowGW98} while keeping the approximation ratio arbitrarily close to two.\nOur main contribution is a near-linear time algorithm that $(1+\\epsilon)$-approximates the optimal {\\em fractional} solution.\n\n\\begin{theorem}\\label{thm:intro:fractional $k$ECSS}\nFor any $\\epsilon > 0$, there is a randomized $\\widetilde{O}(m\/\\epsilon^2)$-time algorithm that outputs a $(1+\\epsilon)$-approximate {\\em fractional} solution for $k$ECSS.\n\\end{theorem}\n\nFollowing, in the high-level, the arguments of Chekuri and Quanrud \\cite{ChekuriQuanrud18} (i.e. solving the minimum-weight $k$ disjoint arborescences in the style of \\cite{KhullerV94} on the support of the sparsified fractional solution), the above fractional solution can be turned into a fast $(2+\\epsilon)$-approximation algorithm for the integral version of $k$ECSS.\n\n\\begin{corollary}\\label{thm:intro:integral $k$ECSS}\nFor any $\\epsilon>0$, there exist\n\\begin{itemize}\n    \\item a randomized $\\widetilde{O}(m\/\\epsilon^2)$-time algorithm that estimates the {\\em value} of the optimal solution for $k$ECSS to within a factor $(2+\\epsilon)$, and\n\n    \\item a randomized $\\tilde O\\left(\\frac{m}{\\epsilon^2} + \\frac{k^2n^{1.5}}{\\epsilon^2}\\right)$-time algorithm\nthat produces a feasible $k$ECSS solution of cost at most $(2+\\epsilon)$ times the optimal value.\n\\end{itemize}\n\\end{corollary}\n    %\n    %\n    %\n    %\n    %\n\nWe remark that the term $\\tilde{O}(k^2 n^{1.5})$ is in fact ``tight'' up to the state-of-the-art algorithm for finding minimum-weight $k$ disjoint arborescences.\\footnote{More formally, if a minimum-weight union of $k$ edge-disjoint arborescences can be found in time $T(k,m,n)$, then our algorithm would run in time $T(k, kn, n)$. The term $O(k^2 n^{1.5})$ came from Gabow's algorithm~\\cite{gabow1995matroid} that runs in time $O(km \\sqrt{n} \\log (n c_{\\max}))$.}\n\nPrior to our results, a sub-quadratic time algorithm was not known even for special cases of $k$ECSS, called\n{\\em $k$-Edge-Connected Augmentation} ($k$ECA). In this problem, we are given a $(k-1)$-edge-connected subgraph $H$ of a graph $G$, and we want to minimize the total cost of adding edges in $G$ to $H$ so that $H$ becomes $k$-edge connected. It is not hard to see that if we can $\\alpha$-approximates $k$ECSS, then we can $\\alpha$-approximates $k$ECA by assigning cost 0 to all edges in $H$. This problem previously admits a $O(kn^2)$-time $2$-approximation algorithm for any even integer $k$~\\cite{khuller1993approximation}\\footnote{In Khuller and Vishkin~\\cite{khuller1993approximation}, the $k$ECA problem aims at augmenting the connectivity from $k$ to $(k+1)$ (but for us it is from $(k-1)$ to $k$.)}. The approximation ratio of 2 remains the best even for 2ECA. Our result in \\Cref{thm:intro:integral $k$ECSS} improves the previously best time complexity by a $\\tilde{\\Theta}(\\sqrt{n})$ factor.\n\n\\textbf{Perspective.} The gap between algorithms with best approximation ratio and best time complexity in fact reflects a general lack of understanding on {\\em fast approximation algorithms}. While polynomial-time algorithms were perceived by many as efficient, it is not a reality in the current era of large data, where it is nearly impossible to take $O(n^3)$ time to process a graph with millions or billions of nodes. Research along this line includes algorithms for sparsest cut \\cite{KhandekarRV09,KhandekarKOV2007cut,Sherman09,Madry10-jtree}, multi-commodity flow \\cite{GargK07,Fleischer00,Madry10}, and travelling salesman problem \\cite{ChekuriQ17,ChekuriQuanrud18}. Some of these algorithms have led to exciting applications such as fast algorithms for max-flow \\cite{Sherman13}, dynamic connectivity \\cite{NanongkaiSW17,ChuzhoyGLNPS19,SaranurakW19,Wulff-Nilsen17,NanongkaiS17dynamic}, vertex connectivity \\cite{LiNPSY21} and maximum matching \\cite{BrandLNPSSSW20}.\n\nThe $k$ECSS problem belongs to the class of survivable network design problems (SNDPs), where the goal is to find a subgraph ensuring that every pair of nodes $(u,v)$ are $\\kappa(u,v)$-edge-connected for a given function $\\kappa$. ($k$ECSS is the {\\em uniform} version of SNDP where $\\kappa(u,v)=k$ for every pair $(u,v)$.)\nThese problems typically focus on building a network that is resilient against device failures (e.g. links or nodes), and are arguably among the most fundamental problems in combinatorial optimization.\nResearch in this area has generated a large number of beautiful algorithmic techniques during the 1990s, culminating in the result of Jain~\\cite{Jain01} which gives a $2$-approximation algorithm for the whole class of SNDPs. Thus, achieving a {\\em fast} $2$-approximation algorithm for SNDPs is a very natural goal.\n\n\nTowards this goal and towards developing fast approximation algorithms in general, there are two common difficulties:\n\\begin{enumerate}\n    \\item Many approximation algorithms inherently rely on solving a {\\em linear program (LP)} to find a fractional solution, before performing  rounding steps. However, the state-of-the-art general-purpose linear program solvers are still quite slow, especially for $k$ECSS and SNDP where the corresponding LPs are {\\em implicit}.%\n\n    In the context of SNDP, the state-of-the-art (approximate) LP solvers still require at least quadratic time: Fleischer~\\cite{fleischer2004fast} designs an $\\tilde{O}(m n k)$ for solving $k$ECSS LP, and more generally for SNDP and its generalization~\\cite{fleischer2004fast,FeldmannKPS16} with at least $\\Theta(m \\min\\{n,k_{\\max}\\})$ iterations  of minimum cost flow's computation  are the best known running time where $k_{\\max}$ is the maximum connectivity requirements.\n\n    \\item Most existing techniques that round fractional solutions to integral ones are not ``friendly'' for the design of fast algorithms. For instance, Jain's celebrated iterative rounding~\\cite{Jain01} requires solving the LP  $\\Omega(m)$ times.  Moreover, most LP-based network design algorithms are fine-tuned to optimize approximation factors, while designing near-linear time LP rounding algorithms requires limiting ourselves to a relatively small set of tools, about which we currently have very limited understanding.\n\n\\end{enumerate}\n\n\n\n\nThis paper completely resolves the first challenge for $k$ECSS  and manages to identify a fundamental bottleneck of the second challenge.\n\n\n\\textbf{Challenges for LP Solvers.} Our main challenge is handling the so-called {\\em box constraints} in the LPs. To be concrete, below is the LP relaxation of $k$ECSS on graph $G=(V,E)$.\n\\begin{align}\n\\min \\{\\sum_{e \\in E} c_e x_e: \\sum_{e \\in \\delta_G(S)} x_e \\geq k\\ (\\forall S \\subseteq V), x \\in [0,1]^E \\} \\label{eq:intro:$k$ECSS LP}\n\\end{align}\nwhere $\\delta_G(S)$ is the set of edges between nodes in $S$ and $V\\setminus S$.\nThe box constraints refer to the constraints  $x \\in [0,1]^E$.\nWithout these constraints, we can select the same edge multiple times in the solution; this problem is called $k$ECSSM in \\cite{CzumajL07} (see \\Cref{foot:intro:sub-multigraph}). Removing the box constraints often make the problem significantly easier. For example, the min-cost $st$-flow problem without the box constraints become computing the shortest $st$-path, which admits a much faster algorithm.\n\nFor $k$ECSS, it can be shown that solving \\eqref{eq:intro:$k$ECSS LP} without the box constraints can be reduced to solving \\eqref{eq:intro:$k$ECSS LP} {\\em with $k=1$} and multiplying all $x_e$ with $k$. In other words, without the box constraints, fractional $k$ECSS is equivalent to fractional 1ECSS.%\n This \\emph{fractional} 1ECSS can be $(1+\\epsilon)$-approximated in near-linear time by plugging in the dynamic minimum cut data structure of Chekuri and Quanrud ~\\cite{ChekuriQ17} to the multiplicative weight update framework (MWU).\n\nHowever, with the presence of box constraints, to use the MWU framework we would need a dynamic data structure for a much more complicated cut problem, that we call, the \\textit{minimum normalized free cut} problem (roughly, this is a certain normalization of the minimum cut problem where the costs of up to $k$ heaviest edges in the cut are ignored.)\nFor our problem, the best algorithm in the static setting we are aware of (prior to this work) is to use Zenklusen's $\\tilde O(mn^4)$-time algorithm \\cite{Zenklusen14} for the {\\em connectivity interdiction} problem.\\footnote{In the connectivity interdiction problem, we are given $G=(V,E)$ and $k \\in {\\mathbb N}$, our goal is to compute $F \\subseteq E$ to delete from $G$ in order to minimize the minimum cut in the resulting graph.}\nThis results in an $\\tilde O(kmn^4)$-time static algorithm.  Speeding up and dynamizing this algorithm seems very challenging.\nOur main technical contribution is an efficient dynamic data structure (in the MWU framework) for the $(1+\\epsilon)$-approximate minimum normalized free cut problem.\nWe explain the high-level overview of our techniques in Section~\\ref{sec:overview}.\n\n\n\n\n\n\n\\textbf{Further Related Works.} %\nThe $k$ECSS and its special cases have been studied extensively. For all $k \\geq 2$, the $k$ECSS problem is known to be APX-hard~\\cite{fernandes1998better} even on bounded-degree graphs~\\cite{csaba2002approximability} and when the edge costs are $0$ or $1$~\\cite{Pritchard10}.  Although a factor $2$ approximation for $k$ECSS has not been improved for almost $3$ decades, various special cases of $k$ECSS admit better approximation ratios (see for instance~\\cite{grandoni2018improved,fiorini2018approximating,adjiashvili2018beating}).\nFor instance, the unit-cost $k$ECSS ($c_e=1$ for all $e \\in E$) behaves very differently, admitting a $(1+ O(1\/k))$ approximation algorithm~\\cite{gabow2009approximating,laekhanukit2012rounding}.\nFor the $2$ECA problem, one can get a better than $2$ approximation when the edge costs are bounded~\\cite{adjiashvili2018beating,fiorini2018approximating}. Otherwise, for general edge costs, the factor of $2$ has remained the best known approximation ratio even for the $2$ECA problem.\n\n\nThe $k$ECSS problem in special graph classes have also received a lot of attention.\nIn Euclidean setting, a series of papers by Czumaj and Lingas led to a near-linear time approximation schemes for constant $k$~\\cite{czumaj2000fast,czumaj1999approximability}.\nThe problem is solvable in near-linear time when $k$ and treewidth are constant~\\cite{berger2007minimum,chalermsook2018survivable}. In planar graphs, 2ECSS, 2ECSSM and 3ECSSM admit a PTAS~\\cite{czumaj2004approximation,borradaile2014polynomial}.\n\n\n\\textbf{Organization.} We provide a high-level overview of our proofs in Section~\\ref{sec:overview}.\nIn Section~\\ref{sec:prelim}, we explain the background on Multiplicative Weight Updates (MWU) for completeness (although this paper is written in a way that one can treat MWU as a black box). In~\\Cref{sec: range map}, we prove our main technical component. In~\\Cref{sec:fast LP solver}, we present our LP solver. In~\\Cref{sec: rounding}, we show how to round the fractional solution obtained from the LP solver.\nDue to space limitations, many proofs are deferred to Appendix.\n\n\n\\section{Overview of Techniques}\n\\input{overview.tex}\n\\section{Preliminaries}\n\\input{prelim.tex}\n\n\\section{Range Mapping Theorem} %\n\\input{rangemapping}\n\\section{Fast Approximate LP Solver} \n\\input{algo} \n\\section{LP Rounding for $k$ECSS (Proof of \\Cref{thm: fast rounding})}\\label{sec: rounding}\n\\input{rounding.tex}\n\n\n\\section{Omitted Proofs} \n\n\n\\subsection{Polynomially Bounded Cost in Proof of \\Cref{thm: warmup}} \\label{sec:polynomially bounded cost}\nLet us assume that the costs $c_e$ are integers (but they can be exponentially large in values).\nKarger's sampling~\\cite{Karger00mincut} gives a near-linear time algorithm to create a skeleton graph $H$ so that all  cuts in $H$ are approximately preserved, and the minimum cut value is $O(\\log |E|)$.\nIt only requires an easy modification of Karger's arguments to show that we can create a skeleton $H$ such that all $k$-free minimum cuts are approximately preserved, and that the value of the minimum $k$-free cuts is $\\Theta(k \\log |E|)$.\nWe will run our static algorithm in graph $H$ instead.\nAs outlined in Karger's paper~\\cite{DBLP:journals\/mor\/Karger99}, the assumption that we do not know the value of the optimal can be resolved by enumerating them in the geometric scales, and the sampling will guarantee that the running time would not blow up by more than a constant factor.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:mwu}}  \\label{sec:mwu proof}\n\nThe proof is done via duality. The primal and dual solutions will be maintained and updated, until the point where one can argue that their values converge to each other; this implies that both the primal and dual solutions are approximately optimal.  \nRecall the primal LP is the covering LP: \n\\[\\min \\{c^T x: A x \\geq 1, x \\geq 0\\} \\]\nThe dual LP is the following packing LP: \n\\[\\max \\{ y^T \\mathbf{1}: y^T A \\leq c^T, y \\geq 0  \\} \\]  \nFor the primal LP, we maintain vectors $\\textbf{w}^{(t)} \\in {\\mathbb R}^n$, where $\\textbf{w}^{(0)}_i = 1\/c_i$ for each $i \\in [n]$. The tentative primal solution on day $t$ is $\\bar{\\textbf{w}}^{(t)} = \\textbf{w}^{(t)}\/\\textsc{MinRow}(A, \\textbf{w}^{(t)})$.\nFor the dual packing LP, we maintain vectors $\\textbf{f}^{(t)} \\in {\\mathbb R}^m$ where $\\textbf{f}^{(0)} = \\mathbf{0}$. \nThe tentative dual solution on day $t$ is defined as  $\\bar{\\textbf{f}}^{(t)} = f^{(t)}\/{\\sf cong}(\\textbf{f}^{(t)})$, where ${\\sf cong}(\\textbf{f})$ is the maximum ratio of violated constraints by $\\textbf{f}$, that is, \\begin{align} \\label{def:congestion} {\\sf cong}(\\textbf{f}) = \\max_{i \\in [n]} \\frac{(\\textbf{f}^T A)_i}{c_i}.\\end{align} Notice that \n$\\bar{\\textbf{f}}^{(t)}$ is a feasible dual solution on each day.  \n\nNow we explain the update rules on each day. Let $j(t)$ be the row that achieves $A_{j(t)} \\textbf{w}^{(t-1)} \\leq (1+\\epsilon) \\textsc{MinRow}(A,\\textbf{w}^{(t-1)})$.  \n\\begin{itemize} \n\\item Update $\\textbf{f}^{(t)}_{j(t)} \\leftarrow \\textbf{f}^{(t-1)}_{j(t)} + \\delta(t)$ where $\\delta(t) = \\min_{i \\in [n]} \\frac{c_i}{A_{j(t), i}}$ is the ``increment'' on day $t$. \n\n\\item Update $\\textbf{w}^{(t)}_i \\leftarrow \\textbf{w}^{(t-1)}_i \\exp\\left(\\epsilon \\cdot \\frac{\\delta(t) A_{j(t),i}}{c_i} \\right)$ for each $i \\in [n]$.  \n\\end{itemize}  \n\n\nDenote the primal value at time $t$ by $P(t)= c^T \\bar{\\textbf{w}}^{(t)}$ and the dual by $D(t) =  ||\\bar{\\textbf{f}}^{(t)}||_{1}$; so we have $P(t) \\geq D(t)$ for all $t$. \n\\begin{theorem}\n\\label{thm:potential}  \nLet $t^*$ be the day $t$ for which $P(t)$ is minimized and $N = \\Omega(\\frac{n}{\\epsilon^2} \\ln n)$ be the total number of days. \nThen we have that $P(t^*) \\leq (1+O(\\epsilon)) D(N)$. In particular, $\\bar{\\textbf{w}}^{(t^*)}$ and $\\bar{\\textbf{f}}^{(N)}$ are near-optimal primal and dual solutions.   \n\\end{theorem} \n\nOur proof relies on the estimates of a potential function defined as $\\Phi^{(t)} = c^T \\textbf{w}^{(t)}= \\sum_{i \\in [n]} c_i \\textbf{w}^{(t)}_i$. \n\n\\begin{lemma} \nWe have, on each day $t$, $$\\exp(\\epsilon \\cdot {\\sf cong}(\\textbf{f}^{(t)}))  \\leq \\Phi^{(t)} \\leq n \\cdot \\exp  \\left( \\epsilon (1+3\\epsilon) \\sum_{0 < t' \\leq t} \\frac{\\delta(t')}{P(t'-1)} \\right).$$\n\\end{lemma}  \n\n\n\n\\begin{proof} \nFirst we show the lower bound of $\\Phi^{(t)}$. Fix column $i \\in [n]$ such that $\\frac{ ((\\textbf{f}^{(t)})^T A)_i  }{c_i} = {\\sf cong}(\\textbf{f}^{(t)})$. Notice that the value of $c_i \\textbf{w}^{(t)}_i$ is equal to: \n\\[\\exp\\left( \\frac{\\epsilon}{c_i} \\cdot \\sum_{t' \\leq t} \\delta(t') A_{j(t'), i} \\right). \\]\n The term $\\delta(t') A_{j(t), i}$ is exactly the increase in $((\\textbf{f}^{(t)})^T A)_i$ at time $t$, so we have that \n$$c_i \\textbf{w}^{(t)}_i \\geq  \\exp\\left( \\frac{\\epsilon}{c_i} \\cdot ((\\textbf{f}^{(t)})^T A)_i  \\right) = \\exp(\\epsilon \\cdot {\\sf cong}(\\textbf{f}^{(t)})),  $$ as desired. \n\nNext, we prove the upper bound on the potential function. \nObserve that\\footnote{In particular, we use the inequality $e^\\gamma \\leq 1 + \\gamma + \\gamma^2$ for $\\gamma \\in [0,1)$ and the fact that the ratio $\\delta(t) A_{j(t),i}\/c_i$ is at most $1$.} $\\textbf{w}^{(t)}_i \\leq \\textbf{w}_i^{(t-1)} (1+ \\epsilon (1+\\epsilon)\\cdot \\frac{\\delta(t) A_{j(t),i } }{c_i})$. \nThis formula shows the increase of potential at time $t$ to be at most \n$$\\Phi^{(t)} \\leq \\Phi^{(t-1)} + \\sum_{i \\in [n]} \\epsilon (1+\\epsilon) \\cdot \\delta(t) A_{j(t), i}\\textbf{w}^{(t-1)}_i \\leq \\Phi^{(t-1)} \\exp \\left( \\frac{\\epsilon(1+\\epsilon) \\delta(t)}{\\Phi^{(t-1)}} \\cdot \\sum_{i \\in [n]}  A_{j(t), i} \\textbf{w}^{(t-1)}_i\\right)$$ \nNotice that $\\sum_{i \\in [n]} A_{j(t),i} \\textbf{w}^{(t-1)}_i = (A_{j(t)} \\textbf{w}^{(t-1)})$ is at most $(1+\\epsilon) \\textsc{MinRow}(A, \\textbf{w}^{(t-1)})$ by the choice of the update rules. The term reduces further to: \n\\[\\Phi^{(t)} \\leq \\Phi^{(t-1)} \\exp \\left(\\frac{\\epsilon(1+\\epsilon)^2 \\delta(t)}{P(t-1)} \\right)\\leq \\Phi^{(t-1)} \\exp\\left( \\frac{\\epsilon(1+3 \\epsilon) \\delta(t)}{P(t-1)}. \\right)\\]\nBy applying the fact that $\\Phi^{(0)} = n$ and the above fact iteratively, we get the desired bound. %\n\\end{proof} \n\nFinally, we argue that the lemma implies Theorem~\\ref{thm:potential}. Consider the last day $N$. Taking logarithms on both sides gives us: \n\\[{\\sf cong}(\\textbf{f}^{(N)} ) \\leq \\frac{\\ln n}{\\epsilon} + (1+3\\epsilon) \\sum_{0 < t' \\leq N} \\frac{\\delta(t')}{P(t'-1)} \\leq \\frac{\\ln n}{\\epsilon} + (1+3\\epsilon) \\frac{||\\textbf{f}^{(N)}||_1}{P(t^*)}\\]  \nThe second inequality uses the fact that $||f^{(N)}||_1 = \\sum_{t'} \\delta(t')$ and that $P(t^*) \\leq P(t)$ for all $t$. \n \n \n\\begin{claim} \n${\\sf cong}(\\textbf{f}^{(N)}) \\geq N\/n$, so this implies that ${\\sf cong}(\\textbf{f}^{(N)}) \\geq \\ln n\/ \\epsilon^2$ when $N \\geq n \\ln n\/ \\epsilon^2$.  \n\\end{claim} \n\\begin{proof}\nWe will argue that $\\sum_{i \\in [n]} \\frac{\\textbf{f}^{(t)}A}{c_i}$ increases by at least one on each day. Since this sum is at most $n {\\sf cong}(\\textbf{f}^{(t)})$, we have the desired result. \nTo see the increase, let $i$ be the column that defines $\\delta(t)$, that is $i = \\arg \\min_{i \\in [n]} c_i\/ A_{j(t), i}$. Notice that $((\\textbf{f}^{(t+1)})^T A)_i = ((\\textbf{f}^{(t)})^T A)_i+  \\delta(t) A_{j(t), i} \\geq  ((\\textbf{f}^{(t)})^T A)_i+ c_{i}$. This shows an increase of one in the above sum. \n\\end{proof} \n\nPlugging in this term, we have that: \n\\[{\\sf cong}(\\textbf{f}^{(N)}) \\leq \\epsilon {\\sf cong}(\\textbf{f}^{(N)}) + (1+3\\epsilon) \\frac{||\\textbf{f}^{(N)}||_1}{P(t^*)} \\]\nThis implies that $P(t^*) \\leq (1+6 \\epsilon) D(N)$. \n\n\\subsection{Proof of \\Cref{lem:KC for box}}\n\nLet $x$ be a feasible solution $A^{kc} x \\geq 1$. Consider $x'_i = \\min (x_i,1)$ for each $i \\in [n]$. \nWe claim that $x'$ satisfies $A x' \\geq \\kappa$. Consider the constraint $A_j x' \\geq \\kappa_j$. Let $F =\\{i \\in \\textsf{supp}(A_j): x_i > 1\\}$. If $|F| \\geq \\kappa_j$, it would imply that $A_j x' \\geq \\kappa_j$ and we are done. Otherwise, we have $|F| \\leq \\kappa_j-1$, and the KC constraints guarantee that \n\\[\\sum_{i \\in \\textsf{supp}(A_j)} x'_i = \\sum_{i \\in \\textsf{supp}(A_j) \\setminus F} x_i + |F| \\geq \\kappa_j  \\] \n\n\nConversely, let $x$ be a feasible solution $A x \\geq \\kappa, x \\in [0,1]^n$. \nConsider any KC constraint: For any $j \\in [m]$ and $F \\subseteq \\textsf{supp}(A_j), |F| \\leq \\kappa_j-1$\n\\[\\sum_{i \\in \\textsf{supp}(A_j)\\setminus F} x_i = \\sum_{i \\in \\textsf{supp}(A_j)} x_i - \\sum_{i \\in F} x_i \\geq \\kappa_j -|F|\\] \nThis implies that $x$ itself is feasible for $A^{kc} x \\geq 1$. \n\n\n\\subsection{Proof of \\Cref{thm:fast LP solver}} \\label{sec:fast lp solver full}\n\n\nBy \\Cref{lem:KC for box}, it is enough to solve kECSS LP with KC inequalities.\n\n\\subsubsection{Interpretation of MWU Framework}  \\label{sec:interpretation}\n\nWe interpret the analysis in \\Cref{sec:mwu proof} in the language of graphs. An interesting feature is that the dual variables are only used in the analysis; it is not used in the implementation at all.\n\n\nWe use $\\vecw^{\\operatorname{mwu}}$ to be the weights that the primal LP maintains. Let $\\{(C^{(t)},F^{(t)}, c_{\\min}^{(t)})\\}_{t \\leq T}$ a sequence of normalized free cuts $(C^{(t)},F^{(t)})$ and the value $c_{\\min}^{(t)} = \\min_{e \\in C^{(t)}\\setminus F^{(t)}} c(e)$ obtained by the MWU algorithm up to day $T$. For each edge $e$, we define congestion ${\\sf cong}(e) = \\frac{1}{c(e)} \\cdot \\sum_{t \\leq T \\colon e \\in C^{(t)} \\setminus F^{(t)}} c_{\\min}^{(t)}$.  The congestion of the graph is denoted as ${\\sf cong}(G) = \\max_{e \\in E} {\\sf cong}(e)$. Note that ${\\sf cong}(G)$ is precisely the same as ${\\sf cong}$ in \\Cref{def:congestion} when we restrict the LP instance to kECSS LP. Furthermore,  by definition, we have\n\n\\begin{align} \\label{eq:we and cong}\n\\forall e,  \\vecw^{\\operatorname{mwu}}(e) \\leq \\frac{1}{c(e)} \\cdot \\exp(\\epsilon {\\sf cong}(G)) \n\\end{align}\n \nSince the running time of the Range Punisher depends on the change of weights, we need to ensure that the total change (the sum-of-log (SOL) terms) is at most near-linear. We bound the SOL term using a slightly different stopping criteria: Observe that the  analysis rely crucially on the fact that congestion ${\\sf cong}(G) \\geq \\frac{1}{\\epsilon^2} \\ln m$. We could also use ${\\sf cong}(G) \\geq \\frac{1}{\\epsilon^2} \\ln m$ as a stopping condition (instead of running up to $O(\\frac{1}{\\epsilon^2} m \\log m)$ days), and the stopping condition implies the number of days is at most $O(\\frac{1}{\\epsilon^2} m \\log m)$.\n\nWe can infer ${\\sf cong}(G)$ from the weight function $\\vecw^{\\operatorname{mwu}}$ by the following.  Let $ \\phi^{\\operatorname{mwu}}(e) :=  \\frac{1}{\\epsilon} \\cdot \\ln (c(e) \\cdot \\vecw^{\\operatorname{mwu}}(e))$ for all $e \\in E$. By definition of ${\\sf cong}(e)$, we have $\\vecw^{\\operatorname{mwu}}(e) = \\frac{1}{c(e)} \\cdot \\exp (\\epsilon {\\sf cong}(e))$, and so $\\phi^{\\operatorname{mwu}}(e) = {\\sf cong}(e)$. Therefore, we have\n\\begin{align}\\label{eq:relation to congestion}\n\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = {\\sf cong}(G).\n\\end{align}\n\n\n\n\n\n\n\\subsubsection{Algorithm}\n\n\nFor the implementation, recall that we denote  $\\vecw^{\\operatorname{mwu}}$ to be the real weights on MWU framework, and $\\textbf{w}$ to be the approximate weight that the data structure maintains. \n\nWe describe extra bookkeeping from \\textsc{RangePunisher} to construct to the final solution. First, it outputs a pair of weight function $(\\vecw^{\\operatorname{mwu}},\\textbf{w}^{\\operatorname{sol}})$ where $\\vecw^{\\operatorname{mwu}}$ is the weights at the end of $\\textsc{RangePunisher}$ and $\\textbf{w}^{\\operatorname{sol}}= \\frac{\\vecw^{\\operatorname{init}}}{{\\sf val}_{\\vecw^{\\operatorname{init}}}(C,F)}$ where $\\vecw^{\\operatorname{init}}$ is the initial weight function for \\textsc{RangePunisher}, and $(C,F)$ is the first normalized mincut obtained during the range punisher.\n\n Since the range punisher maintains approximate weights, we next explain how to detect the stopping condition using approximate weights. We want to stop as soon as $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} > \\frac{1}{\\epsilon^2}  \\cdot \\ln m$. Since we can only keep the approximate weights, we can only detect the approximate value with $O(1\/\\epsilon)$-additive error as follows.  First, it keeps track of $\\phi(e) :=  \\frac{1}{\\epsilon} \\cdot \\ln (c(e) \\cdot \\vecw(e))$ for all $e \\in E$, and early stop as soon as $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon} \\cdot \\ln m$. Since $\\textbf{w}$ is $(1+\\epsilon)$-approximation to the real weight $\\vecw^{\\operatorname{mwu}}$, it implies that with respect to weight right before the stopping day, $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} \\leq \\frac{1}{\\epsilon} \\cdot \\ln m + O(\\epsilon^{-1}) = O( \\frac{1}{\\epsilon} \\ln m)$. \n\n\nThe algorithm for LP solver is described in \\Cref{alg:fast LP solver}.\n\n\n\\begin{algorithm}[H]\n\\KwIn{An undirected graph $G = (V,E)$, a cost function $c$, $\\epsilon \\in (0,1)$}\n\\KwOut{A fractional solution $\\vecw^{\\operatorname{sol}}$.}\n\\BlankLine\n\n$\\forall e \\in E, \\vecw^{\\operatorname{mwu}}(e) \\gets \\frac{1}{c(e)}$ \\;\nLet $\\tilde \\lambda$ be an $(1+\\epsilon)$-approximation to ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}}$\\;\n$\\lambda \\gets \\frac{\\tilde \\lambda }{1+\\epsilon}$\\;\n$\\vecw^{\\operatorname{best}} \\gets \\frac{\\vecw^{\\operatorname{mwu}}}{\\tilde \\lambda}$ \\; \n\\Repeat{$\\exists$ \\normalfont{a} day such that $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon^2} \\cdot \\ln m$ (and early terminate)}\n{\n  $(\\vecw^{\\operatorname{mwu}}, \\vecw^{\\operatorname{sol}}) \\gets \\textsc{RangePunish}(G,\\vecw^{\\operatorname{mwu}},\\lambda)$\\; \n  $\\lambda \\gets \\lambda (1+\\epsilon)$\\;\n \\lIf{$c^T \\vecw^{\\operatorname{best}} > c^T\\vecw^{\\operatorname{sol}}$}{ $\\vecw^{\\operatorname{best}} \\gets \\vecw^{\\operatorname{sol}}$.}\n}\n\n  \\Return{$\\textbf{w}^{\\operatorname{best}}$.}\n\\caption{\\textsc{kECSSLPSolver}($G,c,\\epsilon$)}\n\\label{alg:fast LP solver}\n\\end{algorithm}\n\n\n\\subsubsection*{Correctness}\nWe first show that \\Cref{alg:fast LP solver} punish a sequence of $(1+O(\\epsilon))$-approximate normalized free cuts with respect to $\\vecw^{\\operatorname{mwu}}$ where the weight update rule is defined in the \\textsc{PunishMin} operations. Initially, $\\vecw^{\\operatorname{mwu}}(e) = \\frac{1}{c(e)}$ for all $e \\in E$. By definition, ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\tilde \\lambda\/(1+\\epsilon), \\tilde \\lambda)$ and thus ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$. For each iteration where ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$, the range punisher (\\Cref{lem:fast range punisher}) keeps punishing $(1+O(\\epsilon))$-approximate normalized free cuts until ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\geq (1+\\epsilon)\\lambda$.\n\nBy discussion in \\Cref{sec:interpretation}, and \\Cref{thm:potential}, there must be a day $t^*$ such that in some range such that $\\frac{w ^{(t^*)}}{{\\sf val}_{w^{(t^*)}}(C^{(t^*)},F^{(t^*)})}$ is $(1+O(\\epsilon))$-approximation to the LP solution where $w^{(t^*)}$ is $\\vecw^{\\operatorname{mwu}}$ at day $t^*$.  Since each normalized cut value is within $(1+\\epsilon)$ factor from any other cut inside the same range, we can easily show that the first cut in the range is $(1+\\epsilon)$-competitive with \\textit{any} cut in the range. Therefore, \\Cref{alg:fast LP solver} outputs $(1+O(\\epsilon))$-approximate solution to kECSS LP. \n\n\\subsubsection*{Running Time} \n By \\Cref{cor:normalized mincut}, the running time for computing the value $\\tilde \\lambda$ is $\\tilde{O}(\\frac{1}{\\epsilon}\\cdot m)$. \n By \\Cref{lem:fast range punisher}, the total running time is $$ \\widetilde{O}(m \\ell + K+ \\frac{1}{\\epsilon} \\cdot \\sum_{e \\in E}\\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})), $$ where $\\ell $ is the number of iterations, and $K$ is the total number of normalized free cuts punished (including all iterations),  $\\vecw^{\\operatorname{mwu}}$ is the final weight at the end of the algorithm, and $\\vecw^{\\operatorname{init}}(e) = 1\/c(e)$ for all $e$.\n\n Since we early stop as soon as $\\norm{\\phi}_{\\infty} > \\frac{1}{\\epsilon^2} \\cdot \\ln m$,  it means that the day right before we stop we have $\\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = O(\\frac{1}{\\epsilon^2} \\cdot \\ln m)$. By the stopping condition,\n\n \\begin{align} \\label{eq:stopping cg}\n {\\sf cong}(G) \\overset{(\\ref{eq:relation to congestion})}{=} \\norm{\\phi^{\\operatorname{mwu}}}_{\\infty} = O(\\frac{1}{\\epsilon^2} \\cdot    \\ln m). \n \\end{align}\n\n\n The following three claims finish the proof.\n \n \\begin{claim}\n    $\\ell = O(\\frac{1}{\\epsilon^2} \\log m)$.\n  \\end{claim}\n  \\begin{proof}\n    Initially, we have ${\\sf OPT}_{\\vecw^{\\operatorname{mwu}}} \\in [\\lambda, (1+\\epsilon)\\lambda)$.  By \\Cref{eq:we and cong}, we have $\\vecw^{\\operatorname{mwu}}(e) \\leq \\frac{1}{c(e)} \\cdot \\exp(\\epsilon {\\sf cong}(G)) \\overset{(\\ref{eq:stopping cg})}{=} O(\\frac{1}{c(e)} \\cdot m^{O(\\frac{1}{\\epsilon})})$ for all $e \\in E$.  Let $(C^{(0)}, F^{(0)})$ be the first normalized free cut that we punish. Let $\\lambda_0$ be the value of that cut. We have that each edge is increase by at most a factor of $m^{O(\\frac{1}{\\epsilon})}$, and thus the cut at day right before the stopping happens must be smaller than $\\lambda_0 \\cdot m^{O(\\frac{1}{\\epsilon})}$. Therefore, the number of ranges is $\\log_{1+\\epsilon}(m^{O(\\frac{1}{\\epsilon})}) = O(\\frac{1}{\\epsilon^2} \\log m)$.  \n  \\end{proof}\n\n  \\begin{claim}\n   $K = O( \\frac{1}{\\epsilon^2} m\\log m)$.\n \\end{claim}\n \\begin{proof}\n  Observe that for each normalized free cut $(C,F)$ that we punish there exists a bottleneck edge $e \\in C \\setminus F$ whose $c(e)$ is minimum. By the weight update rule, the congestion is this edge is increased by exactly $1$.  Therefore, the number of normalized free cuts is at most $O(m\\cdot {\\sf cong}(G)) \\overset{(\\ref{eq:stopping cg})}{=} O(\\frac{1}{\\epsilon^2} m\\log m)$.  \n \\end{proof}\n\n   \\begin{claim}\n   For each $e$, $\\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)})) = O(\\frac{1}{\\epsilon} \\log m)$. \n \\end{claim}\n \\begin{proof}\n   Recall that the initial weight $\\textbf{w}^{\\text{init}}(e) = 1\/c(e)$ for all $e$. Therefore,\n   $$ \\forall e \\in E,   \\log (\\frac{\\vecw^{\\operatorname{mwu}}(e)}{\\vecw^{\\operatorname{init}}(e)}))  \\overset{(\\ref{eq:we and cong})}{\\leq}  \\epsilon {\\sf cong}(G) \\overset{(\\ref{eq:stopping cg})}{\\leq} O(\\frac{1}{\\epsilon} \\cdot \\log m).$$\n \\end{proof}\n\n\\subsection{Proof of~\\Cref{lem:spase graph}} \\label{sec:spase graph}\n\nIt suffices to prove the following lemma. \n\n\\begin{lemma} \\label{lem:sparsex}\nGiven a feasible solution $x$ to kECSS, and a non-negative cost function $  : E \\rightarrow \\mathbb{R}_{\\geq 0}$, and $\\epsilon > 0$, there is an algorithm that runs in $\\tilde{O}(m)$ time, and w.h.p., outputs another feasible solution $y$ to kECSS such that\n\\begin{itemize}%\n\\item $\\sum_{e \\in E} c_ey_e \\leq (1+\\epsilon) \\sum_{e \\in E} c_ex_e$.\n\\item $\\operatorname{support}(y) \\subseteq \\operatorname{support}(x)$.\n\\item $|\\operatorname{support} (y)| = O\\left(\\frac{k n \\log n}{\\epsilon^2}\\right)$.\n\\end{itemize}\n\\end{lemma}\n\n\nWe devote the rest of this subsection to proving~\\Cref{lem:sparsex}.\n\n\nLet $x$ be a near-optimal $k$ECSS fractional solution obtained by Theorem~\\ref{thm:intro:fractional $k$ECSS}. \nCompute the solution $y$ using Lemma~\\ref{lem:sparsex}. \nCreate a graph $G'$ by keeping only edges in the support of $y$. \n\n\nBefore proving the lemma, we first develop an extension to the sparsification theorem from the paper of Benczur and Karger.\n\n\nWe follow the definitions by \\cite{BenczurK15, ChekuriQuanrud18}.\n\\begin{definition} [Edge stength]  Let $G = (V,E,w)$ be a weighted undirected graph.\n\n\\begin{itemize}%\n\\item $G$ is $k$-\\textit{connected} if every cut in $G$ has weight at least $k$.\n\\item A $k$-\\textit{strong component} is a maximal non-empty $k$-connected vertex-induced subgraph of $G$.\n\\item The \\textit{strength} of an edge $e$, denoted as $\\kappa_e$ is the maximum $k$ such that both endpoints of $e$  belong to some  $k$-strong component.\n\\end{itemize}\n\\end{definition}\n\n\\begin{lemma}[\\cite{BenczurK15}]\n$$ \\sum_{e \\in E} \\frac{w_e}{\\kappa_e} \\leq n - 1 $$\n\\end{lemma}\n\n\\begin{lemma}[\\cite{BenczurK15}] \\label{lem:approx-strength}\nIn $\\tilde{O}(m)$ time, we can compute approximate stength $\\tilde \\kappa_e$ for each edge $e \\in E$ such that $\\tilde \\kappa_e \\leq \\kappa_e$ and   $\\sum_{e \\in E} \\frac{w_e}{\\tilde \\kappa_e} = O(n) $\n\\end{lemma}\n\n\nGiven a cut $C$ and a subset $S\\subseteq C$ of \nits edges, where $|S|\\leq k-1$, we say $C\\S$ \nis a \\emph{constrained cut}. The next theorem states that all constrained cuts would have \ntheir weights closed to their original weights \nafter the sampling.\n\n\n\\begin{theorem} [Extension to Compression Theorem \\cite{BenczurK15}] \\label{thm:compression-thm}\nGiven $G = (V,E,w)$, let $p : E \\rightarrow [0,1]$ be a probability function over edges of $G$. We construct a random weighted graph  $H = (V,E_H,w')$ as follows.  For each edge $e \\in E$, we independently add edge $e$ into $E_H$ with weight $w'_e = w_e\/p_e $, with probability $p_e$.  For $\\delta \\geq \\Omega( k d \\log n)$, if $p_e \\geq \\min\\{1, \\delta \\frac{w_e}{\\kappa_e} \\}$ for all $e \\in E$,\nthen with high probability $\\left(\\text{over } 1 - \\frac{1}{n^d}\\right)$,  every constrained cut in $H$ has weight between $(1-\\epsilon)$ and $(1+\\epsilon)$ times its value in $G$.\n\\end{theorem}\n\n\\begin{proof} \n\nThis theorem follows almost closely the proof of Benczur-Karger.\nWe sketch here the part where we need a minor modification.\n\nThe proof of Benczur-Karger roughly has two components. The first reduces the analysis for general case to the ``weighted sum'' of the ``uniform'' cases where the minimum cut is large, i.e. edge weights are at most $1$ and minimum cut at least $D=  \\Omega\\left(kd\\log n\\right)$.\nThis first component works exactly the same in our case.\n\nNow in each uniform instance which is the second component of Benczur-Karger,  the probabilistic arguments can be made in the following way: For each cut $C$, since edges are sampled independently, we can use Chernoff bound to upper bound the probability that each cut $C$ deviates more than $(1+\\epsilon)$ factor (after sampling).\nLet $\\mu_C$ denote this probability.\nTherefore, the bad event that there is a cut deviating too much is upper bounded by $\\sum_{C} \\mu_C$.\n\nBenczur-Karger analyzes this probability by constructing an auxiliary experiment: Imagine each edge is deleted with probability $p$, then the sum is exactly the expected number of ``empty cuts'' in the resulting graph.\nThey upper bound this by using the term ${\\mathbb E}[2^R]$ where $R$ is the (random) number of connected components in the resulting graph.\nThey show (using a coupling argument) that ${\\mathbb E}[2^R] = O\\left(n^2 p^D\\right)$, which vanishes whenever $D = \\Omega\\left(d\\log n\\right)$.\nHere is where we need to slightly change the proof. The bad even that we need to bound is not just all the cuts $\\left(\\displaystyle \\sum_{C} \\mu_C\\right)$, but also all the constraint cuts. Let $\\mu_{C \\setminus S}$ be the probability of the bad event that the constraint cut $C \\setminus S$ is deviating too much. We want to bound\n$$\n    \\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}.\n$$\n\n\nWe will create, by enumerating, $m \\choose k$ different graphs $H$ so that each $H$ has at most $k$ edges removed from $G$. Note that all constrained cuts are now defined in these graphs $H$. In the original sampling, if an edge $G$ \nis removed, then we remove it similarly in all graphs $H$ (ignoring it is present in $H$ or not).  \n\n\nGiven that there are $R$ connected components in $H$, there are $O(2^R)$ empty cuts.\nWe consider $m \\choose k$ different graphs derived from $H$ by exhaustively remove\na subset $S \\subseteq E$ of $k$ edges. Some edges in $S$ might already be removed in $H$, so some configurations will be identical.\nWe now count the empty cuts in these $m \\choose k$ graphs. \nTo upper bound $\\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}$, we just need to compute the total number of ``empty cuts'' in all these graphs $H$. \n\nIn each $H$, there are at most $R + k$ connected components. Hence, each graph\nhas at most $2^{R + k}$ empty cuts. Sum up this number among all the graphs,\nwe get that $$\n    \\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k-1} \\mu_{C \\setminus S}\n    \\leq \\mathbb E \\left[{m \\choose k} 2^{R+k}\\right].\n$$\n\n\nSince ${\\mathbb E}[2^R] = O(n^2 p^D)$, we get that\n$$\n\\sum_{C} \\sum_{S \\subseteq C, |S| \\leq k} \\mu_{C \\setminus S} = O\\left(\n{m \\choose k} 2^k n^2 p^D\\right) = O\\left( {\\left ( \\frac{2em}{k} \\right )}^k n^2 p^D\\right),\n$$\n\n which again vanishes if $D$ is large enough (at least $\\Omega(k d \\log n)$).\n\n\n\\end{proof}\n\nWe are now ready to prove \\Cref{lem:sparsex}. In fact the same proof in \\cite{ChekuriQuanrud18} can be applied once we have \\Cref{thm:compression-thm}. \n\n\\begin{proof}[Proof of \\Cref{lem:sparsex}]\nWe first use \\Cref{lem:approx-strength} to compute approximate edge strength $\\tilde \\kappa_e$ for each edge $e \\in E$ so that $\\tilde \\kappa_e \\leq \\kappa$ and  $\\sum_{e \\in E} \\frac{w_e}{\\tilde \\kappa_e} = O(n) $ in $\\tilde{O} (m)$ time.\nLet $\\delta = \\Theta(k d\\log n)$ for some large constant $d$.  Let $\\text{cost}(x) = \\sum_{e \\in E} c_e x_e\\ $.  For each edge $e \\in E$ let $p_e = \\min \\{1, \\frac{ \\delta x_e}{\\epsilon^2 \\tilde \\kappa_e} \\}$, and $q_e = \\min \\{1, \\frac{\\delta c_e x_e}{\\epsilon^2 \\text{cost}(x) } \\} $, and define $r_e = \\max (p_e,q_e)$.\n\nWe will focus on $x$ from the perspective of kECSS LP with knapsack constraints.\n\nWe construct a random graph $H = (V, E', x')$ using $r$ as a probability function over edges of $G$ and we $x$ as weight function of the graph as follows. For each edge $e \\in E$, we independently sample edge $e$ into $E'$ with weight $x'_e = x_e\/r_e $ with probability $r_e$.    Since\n$$\n    r_e =  \\max (p_e,q_e) \\geq p_e =  \\min \\{1, \\frac{ \\delta x_e}{\\epsilon^2 \\tilde \\kappa_e} \\} \\geq  \\min\\{1, \\delta \\frac{x_e}{\\kappa_e} \\}\n$$\nfor sufficiently large constant $d$, by \\Cref{thm:compression-thm},\nwe get w.h.p.,\n$$\n    \\forall C \\in \\mathcal{C} \\forall S \\in C, |S| \\leq k-1, \\sum_{e\\in C \\setminus S} x'_e \\geq (1-\\epsilon) \\sum_{e\\in C \\setminus S} x_e \\geq (1-\\epsilon)(k - |S|).\n$$\n\nObserve that $$ \\sum_{e \\in E} r_e \\leq \\sum_{e \\in E} p_e + \\sum_{e \\in E} q_e = O( \\frac{n\\delta}{\\epsilon^2}   + \\frac{\\delta}{\\epsilon^2}) = O(\\frac{n\\delta}{\\epsilon^2} )$$\n\nBy Chernoff bound, we have  $$ P(\\sum_{e\\in E} c_e x'_e \\geq (1+\\epsilon) \\sum_{e\\in E} c_e x_e ) \\leq \\exp(-\\Omega(\\delta)) $$\nand,\n$$ P( |E'| \\geq (1+\\epsilon) O( \\frac{n\\delta}{\\epsilon^2})) \\leq \\exp(-\\delta\/\\epsilon^2) $$\nBy the union bound, we have the followings  w.h.p.\n\\begin{align*}\n\\sum_{e\\in C \\setminus S} x'_e \\geq   (1-\\epsilon) (k - |S|), \\quad  \\forall C \\in \\mathcal{C}, \\forall S \\in C, |S| \\leq k-1, \\\\\n  %\n  |\\operatorname{support}(x')| \\leq O( \\frac{n\\delta}{\\epsilon^2})\\quad \\text{ and } \n\\sum_{e\\in E} c_e x'_e \\leq (1+\\epsilon)  \\sum_{e\\in E} c_e x_e\n\\end{align*}\n\nTherefore, $y' = (1+\\epsilon)x'$ is a feasible solution to kECSS. Also,  $ |\\operatorname{support}(y')| \\leq O( \\frac{n\\delta}{\\epsilon^2})$, and $\\sum_{e\\in E} c_e y'_e \\leq (1+\\epsilon)^2 \\sum_{e\\in E} c_e x_e$. \nFinally, we can get $(1+\\epsilon') \\sum_{e\\in E} c_e x_e$ by a proper  scaling factor for $\\epsilon$.\n\n\n\n\\end{proof}\n\n\n\\subsection{Proof of \\Cref{thm: KV}} \\label{sec: proof KV}\n\nFor the first part of the theorem, \nlet $x$ denote the optimal solution in the relaxed LP of $k$ECSS of graph $H$. \nWe create a fractional solution $z$ in $D[H]$\nas follows: for every edge $e \\in E$ in $H$, \nif $e_1$ and $e_2$ are the two opposite directed edges in $D[H]$ derived from $e$, we set \n$z_{e_1}=z_{e_2}=x_e$. It is clear that \n$c(z) = 2c(x)$. We just need to argue that \n$z$ is feasible in the relaxed $k$-arboresences \nproblem. Consider a cut $C \\in \\mathcal{C}$ \n(where $r \\in C$ and $C \\neq V$). As $x(C) \\geq k$, $\\sum_{e \\in \\delta^{+}(C)} z_e \\geq k$. \nFurthermore, by Lemma~\\ref{lem:KC for box}, $x$ satisfies \nthe boxing constraint, that is, \n$0 \\leq x_e \\leq 1$ for all edges $e \\in H$, \nimplying that $0 \\leq z_e \\leq 1$ for all directed edges $e \\in D[H]$. This shows that $z$ is feasible and the first part of the theorem is proved. \n\nFor the second part, consider a feasible solution for $k$-arborescences in $D[H]$. If any of the two opposite directed edges is part of the $k$-arborescences, we include its corresponding \nundirected edge in $H$ as part of our induced solution. \nClearly, the cost of the induced solution cannot be higher and it is a feasible $k$ECSS solution, since it guarantees that the cut value is at least $k$ for all cuts. \n\n\n\\subsection{Polynomially bounded costs}\n\\label{subsec:bound_cmax}\n\nSince Gabow's algorithm for arborescences has the running time depending on $c_{\\max}$, the maximum cost of the edges, we discuss here how to ensure that $c_{\\max}$ is polynomially bounded.\n\nLet $x$ be the LP solution obtained from our LP solver. Denote by $C^* = \\sum_{e \\in E} c_e x_e$, so we have that $C^*$ is between ${\\sf OPT}\/2$ and ${\\sf OPT}$, where ${\\sf OPT}$ is the optimal integral value.\n\nFirst, whenever we see an edge $e \\in E$ with $c_e > 2 C^*$, we remove such an edge from the graph $G$.\nFor each remaining edge $e \\in E$, we round the capacity $c_e$ up to the next multiple of $M = \\lceil \\epsilon C^*\/|E| \\rceil$.\nSo, after this rounding up, we have the capacities in $\\{M, 2M, \\ldots, C^*\\}$, and we can then scale them down by a factor of $M$ so that the resulting capacities $c'_e$ are between $1$ and $O(|E|\/\\epsilon)$.\nIt is an easy exercise to verify that any $\\alpha$-approximation algorithm for $(G,c')$ can be turned into an $\\alpha(1+\\epsilon)$-approximation algorithm for $(G,c)$.\n\n\n\\subsection{Step 1: Static Algorithm}\nWe show that the minimum normalized free cut problem can be solved efficiently in the static setting. For  convenience, we often use the term cut to refer to a set of edges instead of a set of vertices.\n\nDefine the objective function of our problem as, for any cut $C$,\n$${\\sf val}_{\\textbf{w}}(C) = \\min_{F\\subseteq C: |F| \\leq k-1} \\frac{\\textbf{w}(C \\setminus F)}{k-|F|}.$$\nFor any weight function $\\textbf{w}$, denote by ${\\sf OPT}_{\\textbf{w}} = \\min_C {\\sf val}_{\\textbf{w}}(C)$. In this paper, the graph $G$ is always fixed, while $\\textbf{w}$ is updated dynamically by the algorithm (so we omit the dependence on $G$ from the notation ${\\sf val}$ and ${\\sf OPT}$).\nWhen $\\textbf{w}$ is clear from context, we sometimes omit the subscript $\\textbf{w}$.\n\nWe show that the truncation technique can be used to establish a connection between our problem and minimum cut.\n\n\n\n\n\\begin{lemma}\n\\label{lem: baby mapping}\nWe are given a graph $G=(V,E)$, weight function $\\textbf{w}$,  integer $k$, and $\\epsilon >0$.\nFor any threshold $\\rho \\in ({\\sf OPT}_{\\textbf{w}}, (1+\\epsilon){\\sf OPT}_{\\textbf{w}}]$,\n\\begin{itemize}\n    \\item any optimal normalized free cut in $(G,\\textbf{w})$ is a $(1+\\epsilon)$-approximate minimum cut in $(G,\\textbf{w}_{\\rho})$, and\n\n    \\item any minimum cut $C^*$ in $(G,\\textbf{w}_{\\rho})$ is a $(1+\\epsilon)$-approximation for the minimum normalized free cut.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFirst, consider any cut $C$ with ${\\sf val}(C) = {\\sf OPT}$.\nLet $F \\subseteq C$ be an optimal set of  free edges for $C$, so we have $\\textbf{w}_{\\rho}(C \\setminus F) \\leq \\textbf{w}(C \\setminus F) = (k-|F|) {\\sf OPT}$. Moreover, $\\textbf{w}_{\\rho}(F) \\leq |F| \\rho$. This implies that\n\\begin{equation}\n\\textbf{w}_{\\rho}(C) = \\textbf{w}_{\\rho}(C \\setminus F) + \\textbf{w}_{\\rho}(F) < k \\rho\n\\label{eq:exact map upper}\n\\end{equation}\nNext, we prove that any cut in $(G,\\textbf{w}_{\\rho})$ is of value at least $k{\\sf OPT}$ (so the cut $C$ is a $(1+\\epsilon)$ approximate minimum cut).\nAssume for contradiction that there is a cut $C'$ such that $\\textbf{w}_{\\rho}(C') < k{\\sf OPT}$. Let $F' \\subseteq C'$ be the set of $\\rho$-heavy edges.\nObserve that $|F'| \\leq k-1$ since otherwise the total weight $\\textbf{w}_{\\rho}(C')$ would have already exceeded $k{\\sf OPT}$.\nThis implies that $\\textbf{w}(C' \\setminus F') = \\textbf{w}_{\\rho}(C' \\setminus F') < (k-|F'|) {\\sf OPT}$ and that\n\t$${\\sf val}(C') \\leq \\frac{\\textbf{w}(C'\\setminus F') }{(k-|F'|)} < {\\sf OPT}$$\n\twhich is a contradiction. Altogether, we have proved the first part of the lemma.\n\n\nTo prove the second part of the lemma, consider a minimum cut $C^*$ in $(G,\\textbf{w}_{\\rho})$, we have that $\\textbf{w}_{\\rho}(C^*) < \\textbf{w}_{\\rho}(C) < k \\rho$ (from~\\Cref{eq:exact map upper}). Again, the set of heavy edges $F^* \\subseteq C^*$ can contain at most $k-1$ edges, so we must have $\\textbf{w}(C^* \\setminus F^*) < (k-|F^*|) \\rho \\leq (k-|F^*|)(1+\\epsilon) {\\sf OPT}$, implying that ${\\sf val}(C^*) < (1+\\epsilon) {\\sf OPT}$.\n\\end{proof}\n\nWe remark that this reduction from the minimum normalized free cut problem to the minimum cut problem does not give an exact correspondence, in the sense that a minimum cut in $(G,\\textbf{w}_{\\rho})$ cannot be turned into a minimum normalized free cut in $(G,\\textbf{w})$. In other words, the approximation factor of $(1+\\epsilon)$\nis unavoidable.\n\n\n\\begin{theorem}\n\\label{thm: warmup}\nGiven a graph $G=(V,E)$ with weight function $\\textbf{w}$ and integer $k$, the minimum normalized free cut problem can be $(1+\\epsilon)$ approximated by using $O( \\frac{1}{\\epsilon} \\cdot \\log n)$ calls to the exact minimum cut algorithm.\n\\end{theorem}\n\n\\begin{proof}\nWe assume that the minimum normalized free cut of $G$ is upper bounded by some value $M$ which is polynomial in $n = |V(G)|$ (we show how to remove this assumption in \\Cref{sec:polynomially bounded cost}). %\nFor each $i$ such that $(1+\\epsilon)^i \\leq M$, we compute the minimum cut $C_i$ in $(G,\\textbf{w}_{\\rho_i})$\nwhere $\\rho_i = (1+\\epsilon)^i$ and return one with minimum value ${\\sf val}(C_i)$. Notice that there must be some $i^*$ such that $\\rho_{i^*} \\in ({\\sf OPT}_{\\textbf{w}}, (1+\\epsilon){\\sf OPT}_{\\textbf{w}}]$ and by the lemma, we must have that $C_{i^*}$ is a $(1+\\epsilon)$-approximate solution for the normalized free cut problem.\n\\end{proof}\nBy using any  near-linear time minimum cut algorithm e.g.,~\\cite{Karger00mincut}, the collorary follows.\n\n\\begin{corollary} \\label{cor:normalized mincut}\nThere exists a $(1+\\epsilon)$ approximation algorithm for the minimum normalized free cut problem that runs in time $\\tilde{O}(|E|\/\\epsilon)$.\n\\end{corollary}\n\n\n\n\\subsection{Step 2: Dynamic Algorithm}\n\n\nThe next idea we use is from Chekuri and Quanrud~\\cite{ChekuriQ17}. One of the key concepts there is that it is sufficient to solve a ``range punishing'' problem in near-linear time; for completeness we  prove this sufficiency in Appendix.\nIn particular, the following proposition is a consequence of their work:\n\n\\begin{definition}\nA \\textbf{range punisher}\\footnote{Our range punisher corresponds to an algorithm of\nChekuri and Quanrud~\\cite{ChekuriQuanrud18} in one epoch.} is an algorithm that, on any input graph  $G$, initial weight function $\\textbf{w} = \\vecw^{\\operatorname{init}}$, real numbers $\\epsilon$, and $\\lambda \\leq {\\sf OPT}_{\\vecw^{\\operatorname{init}}}$, iteratively applies {\\sc PunishMin} on $(G,\\textbf{w})$  until the optimal becomes at least ${\\sf OPT}_{\\textbf{w}} \\geq (1+\\epsilon)\\lambda$.\n\\end{definition}\n\nThe following proposition connects a fast range punisher to a fast LP solver.\n\n\n\\begin{proposition}\n\\label{prop: CQ epoch}\nIf there exists a range punisher running in time $$\\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}(e)}{\\vecw^{\\operatorname{init}}(e)})\\right)$$ where $K$ is the number of cuts punished, then, there exists a fast dynamic punisher, and consequently the $k$ECSS LP can be solved in near-linear time.\n\\end{proposition}\n\n\n\nThis proposition applies generally in the MWU framework independent of problems. That is, for our purpose of solving $k$ECSS LP, we need a fast range punisher for the minimum normalized free cut problem. For Chekuri and Quanrud~\\cite{ChekuriQ17}, they need such algorithm for the minimum cut problem (therefore a fast LP solver for the Held-Karp bound).\n\n\n\\begin{theorem}[\\cite{ChekuriQ17}, informal]\n\\label{thm: CQ range punisher}\nThere exists a fast range punisher for the minimum cut problem.\n\\end{theorem}\n\n\nOur key technical tool in this paper is a more robust reduction from the range punishing of normalized free cuts to the one for minimum cuts. This reduction works for all edge weights and is suitable for the dynamic setting.\nThat is, it is a strengthened version of Lemma~\\ref{lem: baby mapping} and is summarized below (see its proof in \\Cref{sec: range map}).\n\n\\begin{theorem}[Range Mapping Theorem]\n\\label{lem: full mapping}\nLet $(G=(V,E),\\textbf{w})$ be a weighted graph.\nLet $\\lambda > 0$ and $\\rho = (1+\\gamma)\\lambda$.\n\n\n\\begin{enumerate}\n\n\n\n  %\n\n    \\item If the value of optimal normalized free cut is in $[\\lambda, (1+\\gamma)\\lambda)$, then the value of minimum cut in $(G,\\textbf{w}_{\\rho})$ lies in $[k\\rho\/(1+\\gamma), k \\rho)$.\n\n    %\n    \\item For any cut $C$ where $\\textbf{w}_{\\rho}(C) <k \\rho$, then\n    $\\frac{\\textbf{w}(C \\setminus F)}{k-|F|} < (1+\\gamma) \\lambda$ where $F$ contains all $\\rho$-heavy edges in $C$. In particular, ${\\sf val}(C) < (1+\\gamma) \\lambda$.\n    %\n\\end{enumerate}\n\\end{theorem}\n\nGiven the above reduction, we can implement range punisher fast. We present its full proof in \\Cref{sec:fast LP solver} and sketch the argument below.\n\n\\begin{theorem}\n\\label{thm: free cut range punisher}\nThere exists a fast range punisher for the minimum normalized free cut problem.\n\\end{theorem}\n\\begin{proof}(sketch)\nWe are given $\\lambda$ and weighted graph $(G,\\textbf{w}): \\textbf{w}=\\vecw^{\\operatorname{init}}$ such that ${\\sf OPT}_{\\vecw^{\\operatorname{init}}} \\geq \\lambda$.  Our goal is to punish the normalized free cuts until the optimal value in $(G,\\textbf{w})$ becomes at least $(1+\\epsilon)\\lambda$. We first invoke \\Cref{thm: warmup} to get a $(1+\\epsilon)$-approximate cut, and if the solution is already greater than $(1+\\epsilon)^2\\lambda$, we are immediately done (this means ${\\sf OPT} > (1+\\epsilon)\\lambda$).\n\nNow, we know that ${\\sf OPT} \\leq (1+\\epsilon)^2 \\lambda \\leq (1+3\\epsilon) \\lambda$.\nWe invoke Lemma~\\ref{lem: full mapping}(1) with $\\gamma = 3\\epsilon$.\nThe minimum cut in $(G,\\textbf{w}_{\\rho})$ has size in the range $[k\\rho\/ (1+3\\epsilon), k \\rho)$. We invoke (one iteration of) Theorem~\\ref{thm: CQ range punisher} with $\\lambda' = k \\rho (1+3\/\\epsilon)$ to obtain a cut $C$ whose size is less than $k \\rho$ and therefore, by Lemma~\\ref{lem: full mapping}(1), ${\\sf val}(C) < (1+3\\epsilon) \\lambda$. This is a cut that our algorithm can punish (we ignore the detail of how we actually punish it -- we would need to do that implicitly since the cut itself may contain up to $m$ edges). We repeat this process until all cuts whose values are relevant have been punished, that is, we continue this process until the returned cut $C$ has size at least $k \\rho$.\n\nThe running time of this algorithm is\n$$\\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w}_{\\rho}(e)}{\\vecw^{\\operatorname{init}}_{\\rho}(e)})\\right) \\leq \\widetilde{O}\\left(|E| + K + \\sum_{e \\in E}\\log ( \\frac{\\textbf{w} (e)}{\\vecw^{\\operatorname{init}} (e)})\\right)$$\nNotice that we rely crucially on the property of our reduction using truncated weights.\n\\end{proof}\n\nWe remark that in the actual proof of Theorem~\\ref{thm: free cut range punisher}, there are quite a few technical complications (e.g., how to find optimal free edges for a returned cut $C$?), and we cannot invoke Theorem~\\ref{thm: CQ range punisher} in a blackbox manner. We refer to \\Cref{sec:fast LP solver} for the details. %\n\n\n\n\\subsection{LP Rounding for $k$ECSS}\\label{subsec: LP rounding}\n\nMost known techniques for $k$ECSS (e.g.~\\cite{GabowGW98,laekhanukit2012rounding}) rely on iterative LP rounding, which is computationally expensive.\nWe achieve fast running time by making use of the 2-approximation algorithm of Khuller and Vishkin~\\cite{khuller1993approximation}.\n\nRoughly speaking, this algorithm creates a directed graph $H$ from the original graph $G$ and then compute on $H$ the minimum-weight $k$ disjoint arboresences. The latter can be found by Gabow's algorithms, in either\n$\\tilde{O}(|E||V| k)$ or $\\tilde{O}(k|E| \\sqrt{|V|} \\log c_{\\max})$\ntime.\n\nTo use their algorithm, we will construct $H$ based on the support\nof the fractional solution $x$ computed by the LP solver. By the integrality of the arborescence polytope~\\cite{schrijver2003combinatorial}, an integral\nsolution is as good as the fractional solution. However, the support of $x$ can be potentially large, which causes Gabow's algorithm to take longer time. Here our idea is a sparsification of the support, by extending the celebrated sparsification theorem of Benzcur and Karger~\\cite{BenczurK15} to handle our problem, i.e., we prove the following (see~\\Cref{sec: rounding} for the proofs):\n\n\\begin{theorem}\n\\label{thm: sparsification}\nLet $G$ be a graph and $c_G$ its capacities. There exists a capacitated graph $(H, c_H)$ on the same set of vertices that can be computed in\n$\\tilde{O}(m)$\nsuch that (i) $|E(H)| = \\tilde{O}(n k )$, and (ii) for every cut $S$ and $F \\subseteq S: |F| \\leq (k-1)$, we have $c_G(S~\\setminus~F)~=~(1~\\pm~\\epsilon)~c_H(S~\\setminus~F)$.\n\\end{theorem}\n\nBenzcur and Karger's theorem corresponds to this theorem when $k = 1$.  We believe that this theorem might have further applications, e.g., for providing a fast algorithm for the connectivity interdiction problem.\nOur result implies the following (see Section~\\ref{sec: rounding} for the proof):\n\n\\begin{theorem} \\label{thm: fast rounding}\nAssume that there exists an algorithm that finds a minimum-weight $k$-arborescences in an $m$-edge $n$-node graph in time  $T_k(m,n)$. Then there exists a $(2+\\epsilon)$ approximation algorithm for $k$ECSS running in time $\\tilde{O}(m\/\\epsilon^2 + T_k(kn\/\\epsilon^2,n))$\n\\end{theorem}\n\nApplying \\Cref{thm: fast rounding} with the Gabow's algorithm (see~\\Cref{thm:fast-karbor} in \\Cref{sec: rounding}), we obtain  \\Cref{thm:intro:integral $k$ECSS}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we define a new modular functor based on Kac-Wakimoto admissible\nrepresentations\nover $\\widehat{\\mbox{\\g sl}}_{2}$.\nThe modular functor introduced by Segal \\cite{seg} assigns a finite-dimensional\nvector space\nto the data consisting of a punctured curve, a rank 2 vector bundle and a\ncollection of integral\ndominant highest weights attached to the punctures. Our modular functor does\nthe same for\nthe Segal's data (with  integral\ndominant highest weights replaced with admissible highest weights) extended by\nthe lines in the\nfibers over the punctures. As the data `` surface, vector bundle, punctures,\nlines in fibers over punctures'' evolve, so does the corresponding finite\ndimensional vector space.\nThis leads to a new $\\mbox{${\\cal D}$}-$module on the moduli space of rank 2 vector bundles\nwith  parabolic\nstructure (fixed lines in certain fibers). The main feature of this\n$\\mbox{${\\cal D}$}-$module, as opposed\nto the standard one (see Tsuchiya-Ueno-Yamada \\cite{ts_u_ya},\n or Beilinson-Feigin-Mazur \\cite{beil_feig_maz}), or Moore-Seiberg\n\\cite{moorseib}) is that\n it is singular over a certain set\nof exceptional vector bundles. The latter is closely related to the Hitchin's\nglobal nilpotent\ncone.\n\nWe also prove that our $D-$module has (in a proper sense) regular singularities\nat infinity and that dimension of the generic fiber can be calculated by the\nusual combinatorial algorithm: by pinching the surface the problem is reduced\nto the case of a  sphere with $\\geq 3$ punctures and further to a collection of\nspheres with 3 punctures. Dimension of the space attached to the datum ``3\nmodules sitting at 3 points on a sphere'' is calculated explicitly. It is a\npure\nlinear algebra calculation of dimension of the space of coinvariants of a\ncertain infnite dimensional algebra with coefficients in a certain infinite\ndimensional representation. As the  result is amusing we will record it here.\n\nFirst of all, and it is important, in the genus zero case, one can work with\nmodules at a generic level, as opposed to admissible representations which only\nexist when the  level is rational. It is in complete analogy with the usual\nWZW model, where the famous theory of Knizhnik-Zamolodchikov equations arises\nfrom a collection of the so-called Weyl modules sitting on a sphere\n(terminology is borrowed from  \\cite{kazh_luszt}). The family of Weyl modules\nis good for the purpose of studying integrable representations because each\nintegrable representation is a quotient of some Weyl module.\nThis is no longer the case as far as admissible representations are concerned.\nA family of modules suitable for our needs is that of what we call {\\em\ngeneralized Weyl modules}; the latter is defined to be  a Verma module\n quotiented out by a singular vector.\n\nGeneralized Weyl modules are naturally parametrized by the symbols\n$(V_{r}^{\\epsilon},V_{s})\\;\\; r,s\\geq 0,\\;\\epsilon\\in\\nz\/2\\nz$. Here $V_{r}$ is\nto be thought of as the $r+1-$dimensional irreducible $\\mbox{\\g sl}_{2}-$module;\nmeaning\nof $V_{r}^{\\epsilon}$ will be explained soon. It is appropriate to keep in mind\nthat the conventional Weyl module is defined to be the module induced from\n$V_{r}$. Therefore usually Weyl modules are labelled by $\\mbox{\\g sl}_{2}-$modules. In\nour\nsituation Weyl modules are those related to symbols $(V_{0}^{0},V_{s})$.\n\nAccording to Verlinde, dimensions of the spaces associated to 3 modules on a\nsphere are structure constants of Verlinde algebra. Result of calculation of\nVerlinde algebra in our situation is as follows:\n\n\\begin{eqnarray}\n\\label{intr_ourforgen}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})&=&\\\\\n(V_{r_{1}+r_{2}}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&+&\n(V_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1},V_{s_{1}}\\otimes V_{s_{2}})+\n(V_{r_{1}+r_{2}-2}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})+\\cdots\n+\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&.&\\nonumber\n\\end{eqnarray}\nRecall that the usual Verlinde algebra built on Weyl modules is as follows:\n\n\\[V_{s_{1}}\\circ V_{s_{2}}=V_{s_{1}}\\otimes V_{s_{2}},\\]\ni.e. it is the Grothendieck ring if the category of finite dimensional\nrepresentations of $\\mbox{\\g sl}_{2}$. Observe that our formula agrees with the latter\none on Weyl modules.\n\nThe first component of the right hand side of our formula is  equally easy to\ninterpret. It is known that the symbols $V_{r}^{\\epsilon}$ naturally\nparametrize\nfinite dimensional representations of the simplest rank 1 superalgebra\n$\\mbox{\\g osp}(1|2)$. The category of finite dimensional $\\mbox{\\g osp}(1|2)-$modules\nis a tensor category and (\\ref{intr_ourforgen}) reads as follows: Verlinde\nalgebra is isomorphic to the product of Grothendieck rings of the categories of\nfinite dimensional representations of $\\mbox{\\g osp}(1|2)$ and $\\mbox{\\g sl}_{2}$.\n\nIt is known in principle what to do when passing from modules  to their\nquotients, in our case\nfrom generalized Weyl modules at a generic level to admissible representations\nat a rational level: one has to replace Lie algebras with quantized universal\nenveloping algebras at roots of unity and consider Grothendieck rings\nof the corresponding semi-simple ``quotient categories''. Examples: Verlinde\nalgebra built on integrable $\\mbox{\\g sl}_{2}-$modules has to do with $\\mbox{\\g sl}_{2}$ in\nthis way, and Verlinde algebra built on minimal representations of Virasoro\nalgebra in this way has to do with 2 copies of $\\mbox{\\g sl}_{2}$. It appears that\nVerlinde algebra built on admissible representations is related to the pair\n$(\\mbox{\\g osp}(1|2),\\mbox{\\g sl}_{2})$ in exactly the same way as $Vir-$Verlinde algebra\nis related to the pair of $\\mbox{\\g sl}_{2}$'s.\n\nInterest in admissible representation  originates in the fact that the\ncharacters of admissible representations representations at a fixed level give\na\nrepresentation of the modular group. However realization of this fact\nimmediately gave rise to two puzzles:\n\n(i) Given a representation of the modular group, Verlinde formula  produces\nstructure constants of Verlinde algebra; in the case of admissible\nrepresentations some of the structure constants are negative. This does not\nmake much sense as they are supposed to count dimensions.\n\n(ii) Quantum Drinfeld-Sokolov reduction provides a functor from the category\nof $\\widehat{\\mbox{\\g sl}}_{2}-$modules to the category of $Vir-$modules, which sends\nadmissible representations to minimal representations. It should give\nan epimorphism (or some weakened version of it) of a suitably defined Verlinde\nalgebra for  $\\widehat{\\mbox{\\g sl}}_{2}$ on the well-known Verlinde algebra for\n$Vir$.\n\nWe are able to give an answer to (ii), and a partial answer to (i).\n\nAs far as (ii) is concerned, let us for simplicity step aside and consider\n$Vir-$modules at a generic (not necessarily rational) level. Then there is an\nanalogue of a generalized Weyl module -- Verma module quotiented out by a\nsingular vector -- and these are naturally parametrized by the symbols\n$(V_{r},V_{s})$.  The desired  epimorphism is given by:\n\\[(V_{r}^{\\epsilon},V_{s})\\mapsto (V_{r},V_{s})+(V_{r-1},V_{s}).\\]\n\nThis map is naturally related to the  Drinfeld-Sokolov reduction in the\nfollowing way.\nAs we have fixed the category of representations, we have triangular\ndecomposition of $\\mbox{\\g sl}_{2}$; in particular we have 2 opposite nilpotent\nsubalgebras, $\\mbox{${\\bf C}$} e$, $\\mbox{${\\bf C}$} f$. Therefore there are in fact 2 Drinfeld-Sokolov\nfunctors, $\\phi_{e}$, $\\phi_{f}$. It happens that the map above is induced\nby the direct sum $\\phi_{e}\\oplus\\phi_{f}$.\n\nAs to (i), the situation is as follows. The structure constants naturally\narrange in a tensor $\\{c_{ij}^{r}\\}$, the indices running through a set of\nrepresentations in question. Let us compare the set  $\\{c_{ij}^{r}\\}$ of the\nstructure\ncoefficients of our algebra and the set $\\{b_{ij}^{r}\\}$ of structure\ncoefficients of\nthe algebra calculated by Verlinde formula:\n\n\n\n If our $c_{ij}^{r}=0$, then $b_{ij}^{r}=0$. If $c_{ij}^{r}\\neq 0$, then\n$b_{ij}^{r}$ is ``most certainly'' zero, however in some exceptional cases it\nis\nnon-zero. The latter cases in our situation are interpreted in the following\nway. Recall that we have not only 3 modules, $i,j,r,$ but also 3 Borel\nsubalgebras, $\\mbox{\\g b}_{i},\\mbox{\\g b}_{j},\\mbox{\\g b}_{r}$, which vary. Now as $c_{ij}^{r}\\neq\n0$, the fiber of our $D-$module is $\\neq 0$ (in fact it is 1-dimensional), if\nthe\n3 Borel subalgebras are pairwise different. If however 2 of them meet, the\nfiber\nusually vanishes, but sometimes survives. It survives if and only if\n$b_{ij}^{r}\\neq 0$. If non-zero, $b_{ij}^{r}$ can be $\\pm 1$. There is no doubt\nthat $b_{ij}^{r}$ is a result of some cohomological calculation related to the\n$D-$module. Unfortunately we cannot make it more precise at the moment.\n\n\n\n\n\n\n\n\nJust as in the usual case Weyl modules on a sphere produce a trivial vector\nbundle with the flat (Knizhnik-Zamolodchikov)connection,\n in our case\nwe get a bundle with a flat connection on a space of the 2 times greater\ndimension. The extra\ncoordinates come from the flag manifold, recall that we are dealing  with\nmoduli of vector bundles\nwith parabolic structure. Horizontal sections of this connection satisfy a\nsystem of differential equations; we get twice as many equations as\nthere are KZ equations: half of them are indeed KZ equations and the other half\ncomes from singular vectors in  Verma modules over $\\widehat{\\mbox{\\g sl}}_{2}$. The\nlatter is but natural -- it is exactly one of the lessons of the pioneering\nwork \\cite{bpz}.\n This allows to put the integral formulas for solutions of\nKnizhnik-Zamolodchikov equations, which we wrote in \\cite{feig_mal}, in a\nproper context: they\ngive horizontal sections of this new connection. We conjecture that our\nmethods, in fact, provide\nall horizontal sections. The relation of our formulas to those in\n\\cite{sch_varch} is that the\nlatter are necessarily polynomials as functions on the flag manifold while ours\nare not.\n\nWe wish to acknowledge that there has been a number of works approaching\nWZW model for admissible representation from different points of view, see for\nexample \\cite{awata,feld,f_g_p_p,peters,ramg}. It would be interesting to\nrelate\n our integral formulas with those in \\cite{peters} and the new Hopf algebra of\n\\cite{ramg} to the above mentioned ``$\\mbox{\\g osp}(1|2)\\times\\mbox{\\g sl}_{2}$'' at roots of\nunity. To the best of our knowledge, Verlinde algebras proposed in these work\ndo not solve (ii) above -- those algebras are rather trivial when compared to\nthe $Vir-$analogue.  Our starting point, see  \\cite{feig_mal1}, was the work\n\\cite{awata}, where Verlinde algebra for admissible representations was\nfirst calculated (in the form equivalent but much less illuminating than the\none  described above), using\nthe language which left completely open the problem of existence of a\n$D-$module, such that dimension of the  fiber is calculated through this\nalgebra.\n\n{\\bf Acknowledgments.} Parts of this work were reported\nat the AMS meeting in Hartford, March, 1995, and at Service de Physique\nTheorique at Saclay, in November, 1994. We are grateful to J.-B. Zuber for\ninvitation and warm hospitality. Considerable part of this work was done over\nthe 2 years one of us spent at Yale. Inspiring and friendly atmosphere at the\nDepartment of Mathematics contributed a lot -- and so did the discussions\nwith Igor Frenkel, Ian Grojnowski, Gregg Zuckerman. We are grateful to Itzhak\nBars for bringing to our attention the paper \\cite{peters}, to Sanjaye\nRamgoolam\nfor sending his work, and to David Kazhdan for an interesting conversation at\nHarvard.\n\n\n\n\n\n\\section{{\\bf Notations and known results}}\n\\label{notat_known_res}\n\n\\subsection{ }\n Some notations from  commutative algebra are as follows:\n\n$\\mbox{${\\bf C}$} [t]$ is a polynomial ring, $\\mbox{${\\bf C}$} [[t]]$ is its completion by\npositive powers of $t$; $\\mbox{${\\bf C}$}[t,t^{-1}]$ is a ring of Laurent\npolynomials and $\\mbox{${\\bf C}$} ((t))$ is its completion by positive powers of $t$.\n\nBy functions on the formal (punctured) neighborhood of a non-singular point on\na curve we\nwill mean a ring isomorphic to $\\mbox{${\\bf C}$} [[t]]$ ($\\mbox{${\\bf C}$} ((t))$ resp.); to specify\nsuch an isomorphism means to pick a local coordinate $t$. The analogous meaning\nwill\nbe given to the phrase `` sections of a vector bundle\non the formal (punctured) neighborhood of a non-singular point on a curve''.\n\n\\subsection{ }\\label{defofalgebrandmodusu}\nSet $\\mbox{\\g g} = \\mbox{\\g sl}_{2}$, $\\mbox{$\\hat{\\gtg}$}=\\widehat{\\mbox{\\g sl}}_{2}=\\mbox{\\g sl}_{2}\n\\otimes\\mbox{${\\bf C}$} [z,z^{-1}]\\oplus\\mbox{${\\bf C}$} c$. Choose a basis $e,\\,h,\\,f$ of\n$\\mbox{\\g g}$\nsatisfying the standard relations $[h,e]=2e,\\;[h,f]=-2f,\\;[e,f]=h$.\nWe say that\n\n$\\mbox{\\g g}_{\\geq}=\\mbox{${\\bf C}$} e\\oplus \\mbox{${\\bf C}$} h$\nand\n$\\mbox{$\\hat{\\gtg}$}_{\\geq} =\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g b}\\oplus\\mbox{${\\bf C}$} c$ are standard Borel\nsubalgebras of $\\mbox{\\g g}$ and $\\mbox{$\\hat{\\gtg}$}$ resp;\n\n$\\mbox{\\g g}_{>}=\\mbox{${\\bf C}$} e$\nand\n$\\mbox{$\\hat{\\gtg}$}_{>} =\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g g}_{>}$ are standard ``maximal\nnilpotent subalgebras''\nof $\\mbox{\\g g}$ and $\\mbox{$\\hat{\\gtg}$}$ resp.;\n\n$\\mbox{${\\bf C}$} h$ and $\\mbox{${\\bf C}$} h\\oplus\\mbox{${\\bf C}$} c$ are standard Cartan subalgebras of $\\mbox{\\g g}$ and\n$\\mbox{$\\hat{\\gtg}$}$ resp.\n\nThe Verma module $M_{\\lambda,k}$ is a module induced from the\ncharacter\nof $\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[[z]]\\oplus \\mbox{\\g b}\\oplus\\mbox{${\\bf C}$} c$ annihilating\n$\\mbox{\\g g}\\otimes z\\mbox{${\\bf C}$}[z]\\oplus \\mbox{${\\bf C}$} e$ and sending $ h$ and $c$ to\n$\\lambda$ and $k$ resp. $k$ is often referred to as\na level. Generator of $M_{\\lambda,k}$ is usually denoted by\n$v_{\\lambda,k}$. A quotient of a Verma module is called highest weight module.\n\nThe algebra $\\mbox{$\\hat{\\gtg}$}$ is $\\nz^{2}_{+}-$graded by assigning $f\\otimes\nz^{n}\\mapsto\n(1,-n),\\;e\\otimes z^{n}\\mapsto (-1,-n)$ and so is a Verma module ( as well as\nits quotients):\n$M_{\\lambda,k}=\\oplus\n_{i,j}M_{\\lambda,k}^{i,j}$.\n\nThere is a canonical antiinvolution $\\omega :\\mbox{$\\hat{\\gtg}$}\\rightarrow\\mbox{$\\hat{\\gtg}$}$\ninterchanging $\\mbox{$\\hat{\\gtg}$}_{>}$ and\n$\\mbox{$\\hat{\\gtg}$}_{<}$ and constant on the Cartan subalgebra. For any highest weight\nmodule $V$ denote by\n$V^{c}$ and call {\\em contragredient} the module equal to the\nrestricted dual $V^{\\ast}$ as a vector space with the following action of\n$\\mbox{$\\hat{\\gtg}$}$:\n\\[<gx,y>=<x,\\omega(g)y>,\\;g\\in\\mbox{$\\hat{\\gtg}$},x\\in V^{\\ast},y\\in V.\\]\n\nIf a highest weight module $V$ is irreducible then it is isomorphic to $V^{c}$.\nA morphism of\nhighest weight modules $V_{1}\\rightarrow V_{2}$ naturally induces the morphism\nof the\ncorresponding contragredient modules: $V_{2}^{c}\\rightarrow V_{1}^{c}$.\n\nA morphism of Verma modules $M_{\\lambda,k}\\rightarrow M_{\\mu,k}$ is\ndetermined by the image of $v_{\\lambda,k}$. The image can be written as\n$Sv_{\\mu,k}$ for a uniquely determined element $S$ of the universal\nenveloping algebra of  $\\mbox{\\g g}\\otimes z^{-1}\\mbox{${\\bf C}$}[z^{-1}]\\oplus\\mbox{${\\bf C}$} f$.\nIf non-zero, the vector $Sv_{\\mu,k}$, or even $S$ for this matter, is called\n{\\em singular}.\nThe singular vector can be equivalently defined as an eigenvector of the Cartan\nsubalgebra\n of $\\mbox{$\\hat{\\gtg}$}$\n annihilated by $\\mbox{$\\hat{\\gtg}$}_{>}$. In this form definition applies to an arbitrary\n$\\mbox{$\\hat{\\gtg}$}-$module.\n\n\\subsection{Singular vector formula}\n\\label{Singular_vector_formula}\nIt follows from Kac-Kazhdan\ndeterminant formula that a singular vector generically appears in the\nhomogeneous\ncomponents\nof degree either $n(-1,m),\\;m> 0,\\,n>0$ or $n(1,m),\\;m\\geq 0,n> 0$.\nDenote\nthe corresponding singular vectors by $S_{n,m}^{1}$ and\n$S_{n,m}^{0}$ resp.\n\n\n Singular vectors $S_{nm}^{i}$ were found in ~\\cite{malff}\nin an unconventional form containing  non-integral powers of\nelements\nof $\\mbox{$\\hat{\\gtg}$}$ ( see also ~\\cite{ba_soch} for another approach):\n\n\\begin{equation}\nS_{nm}^{1}=(e\\otimes z^{-1})^{n+mt}f^{n+(m-1)t}\n(e\\otimes z^{-1})^{n+(m-2)t}\\cdots (e\\otimes z^{-1})^{n-mt}\n\\label{s_v_1},\n\\end{equation}\n\\begin{equation}\nS_{nm}^{0}=f^{n+mt}(e\\otimes z^{-1})^{n+(m-1)t}\nf^{n+(m-2)t}\\cdots f^{n-mt},\n\\label{s_v_2}\n\\end{equation}\nwhere $t=k+2$.\n\nThis form is not always convenient to calculate a singular vector. It is,\nhowever,\n a useful tool to derive properties of a singular vector. For example,\ndenoting\nby\n$\\pi:\\mbox{$\\hat{\\gtg}$}\\rightarrow\\mbox{\\g g},\\;g\\otimes z^{n}\\mapsto \\mbox{\\g g}$ the evaluation\nmap,\none uses (~\\ref{s_v_1},~\\ref{s_v_2}) to derive that (see ~\\cite{fuchs}, also\n{}~\\cite{mal} for\nthe proof in a more general quantum case):\n\n\\begin{eqnarray}\n\\pi S_{nm}^{1}=(\\prod_{i=1}^{m}\\prod_{j=1}^{N} P(-it-j))e^{N}\n\\label{p_s_v_1}\\\\\n\\pi S_{nm}^{0}=(\\prod_{i=1}^{m}\\prod_{j=0}^{N-1} P(it+j))f^{N}\n\\label{p_s_v_2},\n\\end{eqnarray}\nwhere $P(t)=ef-(t+1)h-t(t+1)$.\n\n\n\\subsection{Generalized Weyl modules and admissible representations}\n\nThe  structure of Verma modules over $\\mbox{$\\hat{\\gtg}$}$ is known in full detail\n(\\cite{mal_2}).\nOutside the critical level ($k=-2$) a Verma module\nis generically irreducible. $M_{\\lambda,k}$ happens\nto be reducible if and only if it contains a singular vector.\nIf $M_{\\lambda,k}$ is reducible then the following 2 cases arise:\n\n(i) $k$ is generic (not rational) and $M_{\\lambda,k}$  contains only one\nsingular vector;\n\n(ii)$k+2=p\/q>0$ is a ratio of 2 positive integers and $M_{\\lambda,k}$ contains\ninfinitely\nmany singular vectors.\n\nIt can of course happen that $k+2=p\/q<0$. We will not be interested\nin this case and confine to mentioning that here the situation is  in a sense\ndual to (ii).\n\n\\subsubsection{ Case (i)}\n\\label{Case_(i)}\n$M_{\\lambda,k}$ contains a unique proper submodule $M$\n generated by the singular vector. $M$ is, in fact, a Verma module.\n\n{\\bf Definition.}The irreducible quotient $V_{\\lambda,k}$\nis called {\\em generalized\nWeyl module}. $\\Box$\n\n\\bigskip\n\n\nThere arises the exact sequence\n\\begin{equation}\n\\label{exactseqforweyl}\n0\\rightarrow M\\rightarrow M_{\\lambda,k}\\rightarrow V_{\\lambda,k}\\rightarrow 0.\n\\end{equation}\n\nA simple property of Kac-Kazhdan equations \\cite{kac_kazhd} is that, given\n(\\ref{exactseqforweyl}),\n the module\n$M$ is irreducible and does not project on any generalized Weyl module. Note\nthat\nif the composition series of a $\\mbox{$\\hat{\\gtg}$}-$module only consist of generalized Weyl\nmodules\nthen this module\nbreaks into a direct sum of its components. (This can be proved by methods of\nDeodhar-Gabber-Kac\n\\cite{deodgabbkac}.)\n\nIt is an exercise on Kac-Kazhdan equations to derive that the highest weight\n$(\\lambda,k)$\n of a generalized Weyl module $V_{\\lambda.k}$ belongs to either the line\n\\begin{equation}\n\\label{param_eq_line_1}\n\\lambda=-it+j-1,\\;k=t-2,\n\\end{equation}\nfor some $i\\geq 0,j\\geq 1$, or\n  to the line\n\\begin{equation}\n\\label{param_eq_line_2}\n\\lambda=it-j-1,\\;k=t-2,\n\\end{equation}\nfor some $i,j\\geq 1$; in both cases $t$ is regarded as a parameter. Formula\n(\\ref{param_eq_line_1}) cooresponds to the case when $V_{\\lambda,k}$ is\nobtained from $M_{\\lambda,k}$ by quotienting out the singular vector\n$S^{0}_{i,j}$;\nanalogously, (\\ref{param_eq_line_2}) cooresponds to the case when\n$V_{\\lambda,k}$ is obtained from $M_{\\lambda,k}$ by quotienting out the\nsingular vector $S^{1}_{i,j}$.\n\n\n\nWe see that for a fixed level $k$  generalized Weyl modules are parametrized by\nthe\ntriples consisting of a pair of nonnegative numbers, $i$, $j$ in the formulas\nabove, and an element taking one of the 2 values needed to distinguish between\n(\\ref{param_eq_line_1}) and (\\ref{param_eq_line_2}).\nTo be more\nprecise, denote by $V_{i}$ the $i+1-$dimensional irreducible representation of\n$\\mbox{\\g g}$.\n\n\n{\\bf Notation.}\nAssign\nto $V_{\\lambda,k}$ either the symbol $(V_{i}^{0},V_{j-1}),\\; i\\geq 0,j\\geq 1$\nif $(\\lambda,k)$ satisfies (\\ref{param_eq_line_1}), or the symbol\n$(V_{i-1}^{1},V_{j-1}),\\; i,j\\geq 1$ if $(\\lambda,k)$ satisfies\n(\\ref{param_eq_line_2}). $\\Box$\n\n\\bigskip\n\n\nThis gives us a one-to-one correspondence between the set of generalized Weyl\nmodules at a fixed generic level\nand the set of symbols $(V_{i}^{\\epsilon},V_{j})$, where\n$\\epsilon$ is understood as an element of $\\nz\/2\\nz$.\n\n\n\n Observe that the conventional Weyl module of the level $k$\n is  defined to be the induced\nrepresentation\n\\[\\mbox{Ind}_{\\mbox{\\g g}[[z]]\\oplus\\mbox{${\\bf C}$} c}^{\\mbox{$\\hat{\\gtg}$}}V_{n},\\]\nwhere $\\mbox{\\g g}[[z]]$ operates on $V_{n}$ via the evaluation map\n$\\mbox{\\g g}[[z]]\\rightarrow\\mbox{\\g g}$\nand $c\\mapsto k$. From our point of view the Weyl module is a quotient of the\nVerma module\n$M_{n,k}$ by the submodule generated by the singular vector\n$f^{n+1}v_{\\lambda,k}$. In other\nwords, Weyl modules are associated to the symbols $(V_{0}^{0},V_{n})$.\nThis   partially explains appearance of $\\mbox{\\g g}-$modules in our notations.\n\n\n\n\\subsubsection{Case (ii)}\n\\label{Case_(ii)}\n A Verma module\ncontains\ninfinitely many singular vectors and is embedded in finitely many\nother\nVerma modules. Among all singular vectors in\n $M_{\\lambda,k}$ there are 2 independent ones and these\ngenerate\nthe maximal proper submodule.\nAlthough formally all such Verma modules look alike a special role is\nplayed\nby those which can only embed (non-trivially) in themselves. Highest\nweights of such modules were called by Kac and Wakimoto\n{\\em admissible} (~\\cite{kac_wak}) and are described as follows.\n\nLet $k+2=p\/q$, where $p,q$ are relatively prime positive integers.\nThe set of admissible highest weights at the level $k=p\/q-2$ is given\nby\n\\[\\Lambda_{k}=\\{\\lambda(m,n)=m\\frac{p}{q}-n-1\\, :\\;0<m\\leq q,\\,\n0\\leq n\\leq p-1\\}.\\]\n\n\nWhat is said above about the structure of Verma modules implies that\nany\nVerma module appears in the exact sequence of the form\n\\begin{equation}\n\\label{bgg_res}\n0\\leftarrow L_{\\lambda_{0},k}\\leftarrow\nM_{\\lambda_{0},k}\\stackrel{d_{0}}{\\leftarrow} M_{\\lambda_{1},k}\\oplus\nM_{\\mu_{1},k}\n\\stackrel{d_{1}}{\\leftarrow} M_{\\lambda_{2},k}\\oplus\nM_{\\mu_{2},k}\\stackrel{d_{2}}{\\leftarrow}\\cdots,\n\\end{equation}\nwhere $\\lambda_{0}$ is an admissible weight at the level $k$ and\n$L_{\\lambda_{0},k}$ is\nthe corresponding irreducible module.\n$L_{\\lambda_{0},k}$ is also called {\\em admissible}.\n The exact sequence\n(\\ref{bgg_res}) is called Bernstein - Gel`fand - Gel'fand ( BGG ) resolution.\n\nAgain cohomological arguments show (see e.g. \\cite{kac_wak}) that if the\ncomposition series of\na $\\mbox{$\\hat{\\gtg}$}-$module only consists of admissible representations then the module is\ncompletely\nreducible.\n\nThe parametrization of the set of admissible representations we are going to\nuse is as follows.\nTwo different generalized Weyl modules project onto one and the same\nadmissible representation: formula (\\ref{bgg_res}) implies that the the two\nmodules\nprojecting onto $L_{\\lambda_{0},k}$ are $M_{\\lambda_{0},k}\/M_{\\lambda_{1},k}$\nand\n$M_{\\lambda_{0},k}\/M_{\\mu_{1},k}$.\nTherefore\ntwo different triples $(V_{m}^{\\epsilon},V_{n})$\nare related to the same admissible represenation.\n Introduce the equivalence relation $\\approx$ by\n$(V_{m}^{\\epsilon},V_{n})\\approx (V_{q-1-m}^{\\epsilon+1},V_{p-2-n}),\\; 0\\leq\nm\\leq q-1,\\, 0\\leq n\\leq p-2$.\n Denote\nby $(V_{m}^{\\epsilon},V_{n})^{\\sim}$ the equivalence class of\n$(V_{m}^{\\epsilon},V_{n})$.\n\nIt easy to check that\nadmissible representations are parametrized by the equivalence  classes of the\ntriples:\n\\begin{equation}\n\\label{parametrofadmreprformula}\n\\{\\mbox{ admissible representations }\\}\\Longleftrightarrow\n\\{(V_{m}^{\\epsilon},V_{n})^{\\sim}\\}.\n\\end{equation}\n\n\n\n\\subsection{ }\n\\label{whatcanbesaidhighrank}\nConsiderable part of the above carries over to the arbitrary Kac-Moody algebra\ncase.\nHere,\nfor example, is the definition of an admissible representation.\nDrop the condition that $\\mbox{\\g g}=\\mbox{\\g sl}_{2}$, let $M_{\\lambda,k}$\nbe a Verma module over $\\mbox{$\\hat{\\gtg}$}$ and $L_{\\lambda,k}$ be its irreducible quotient.\nCall $(\\lambda,k)$\nadmissible if $M_{\\lambda,k}$ satisfies the following projectivity condition:\nif composition\nseries of a $\\mbox{$\\hat{\\gtg}$}-$module $W$ contains $L_{\\lambda,k}$ then $M_{\\lambda,k}$\nnon-trivially maps\nin $W$.\n\n Unfortunately we do not have a reasonable definition of a generalized Weyl\nmodule\nin the higher rank case. This is one of the reasons\nfor which we have to  confine mostly to the $\\mbox{\\g sl}_{2}-$case.\n\n\n \\subsection{Loop modules }\n\\label{Loop_modules}\n\n We will also be using $\\mbox{$\\hat{\\gtg}$}-$modules different from Verma modules\nor corresponding irreducible ones.\n\nDenote by $\\mbox{${\\cal F}$}_{\\alpha \\beta} $ a $\\mbox{\\g g}-$ module with the basis\n$F_{i},\\;i\\in\\nz$\nand the action given by\n\\[eF_{i}=-(\\alpha+i-\\beta)F_{i+1},\\;hF_{i}=(2\\alpha+2i-\\beta)F_{i},\\;\nfF_{i}=(-\\alpha-i)F_{i-1}.\\]\nThe space $\\mbox{${\\cal F}$}_{\\alpha \\beta} ^{\\mbox{${\\bf C}$}^{\\ast}}=\\mbox{${\\cal F}$}_{\\alpha \\beta}\\otimes\\mbox{${\\bf C}$}\n[z,z^{-1}]$ is endowed with\nthe natural $\\mbox{$\\hat{\\gtg}$}-$module structure. The elements $F_{ij}=\nF_{i}\\otimes z^{j},\\;i,j\\in\\nz$\nserve as a natural basis in it.\n\nRecall (see \\ref{Singular_vector_formula}) that   $S^{1}_{nm},\\;S^{0}_{nm}$\nstand for a singular\nvector of degree $n(-1,m)$ or $n(1,m)$ resp. in a Verma module. The following\nformulas are proved\nby using (\\ref{p_s_v_1},\\ref{p_s_v_2}):\n\n\\begin{eqnarray}\n\\label{actonloops1}\nS^{1}_{nm}F_{n,nm}=\\{&\\prod_{i=1}^{m}\\prod_{j=1}^{n}&(-it-j-\\alpha+\\beta)(-it-j-\\alpha)\\}\n\\{\\prod_{s=1}^{n}\n(\\alpha+i)\\}F_{00}\\\\\nS^{2}_{nm}F_{-n,nm}=\\{&\\prod_{i=1}^{m}\\prod_{j=1}^{n}&(it+j-\\alpha+\\beta)(-it-j-\\alpha)\\}\n\\{\\prod_{s=1}^{n}\n(\\alpha-\\beta-i)\\}F_{00},\\label{actonloops2}\n\\end{eqnarray}\nwhere $t=k+2$.\n\n\n\n\\section{Construction of the modular functor.}\n\nAlthough most of our results have to do with $\\mbox{\\g sl}_{2}$,\n up to some point it is no extra effort to work in greater\ngenerality. So until sect.\\ref{The_spaces_of_coinvariants}, $\\mbox{\\g g}$ will stand\nfor $\\mbox{\\g sl}_{n}$ unless\notherwise stated.\n\\subsection{ Algebra $\\mbox{$\\hat{\\gtg}$}^{A}$ and categories of $\\mbox{$\\hat{\\gtg}$}^{A}$-modules }\n\\subsubsection{ }\n\\label{algebra_hgta}\nLet $\\mbox{${\\cal C}$}$ be a smooth compact algebraic curve and $\\rho:\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}$ be\na rank\n$n$ vector bundle  with a flat connection. The connection relates to a section\n$s$ of any\nbundle $\\mbox{${\\cal A}$}$ associated with $\\mbox{${\\cal E}$}$ the section $dc$ of $\\Omega\\otimes\\mbox{${\\cal A}$}$ where\n$\\Omega$\nis the sheaf of differential forms over $\\mbox{${\\cal C}$}$. A typical example of $\\mbox{${\\cal A}$}$ is\nthe bundle\n$\\mbox{End}\\mbox{${\\cal E}$}$ of fiberwise endomorphisms of $\\mbox{${\\cal E}$}$. The sheaf of sections of\n$\\mbox{End}\\mbox{${\\cal E}$}$\nis naturally a sheaf of Lie algebras over $\\mbox{${\\cal C}$}$.\n\n\nFor a point $P\\in \\mbox{${\\cal C}$}$ let $\\mbox{\\g g}^{P}$ be the algebra of sections of\n$\\mbox{End}\\mbox{${\\cal E}$}$ over the formal\nneighborhood of $P$. For a finite subset $\\bar{A}=\\{ P_{1},P_{2},\\ldots ,\nP_{m}\\}\n\\subset \\mbox{${\\cal C}$}$ set $\\mbox{\\g g}^{\\bar{A}}= \\oplus_{i=1}^{m}\\mbox{\\g g}^{P_{i}}$. Define\n$\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}$ to\nbe the central extension of $\\mbox{\\g g}^{\\bar{A}}$ by the cocycle\n \\[ <x,y>=\\sum_{i=1}^{m}\\mbox{Res}_{P_{i}}\\mbox{Tr}dx\\cdot y.\\]\nIn particular, we obtain the splitting\n\\begin{equation}\n\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}=\\mbox{\\g g}^{\\bar{A}}\\oplus\\mbox{${\\bf C}$}\\cdot c.\n\\label{splitting}\n\\end{equation}\n\n\n\nConsider a finite set $A=\\{(P_{1},\\mbox{\\g b}_{1}),\\ldots ,(P_{m},\\mbox{\\g b}_{m})\\}$\n  where $P_{i}\\in\\mbox{${\\cal C}$}$ are pairwise different and $\\mbox{\\g b}_{i}$ is a Borel\nsubalgebra\nof the algebra of  traceless linear transformations of the fiber\n$\\rho^{-1}P_{i}$\n ($1\\leq i\\leq m$). Let $\\bar{A}$ be the projection of $A$ on $\\mbox{${\\cal C}$}$. Set\n $\\mbox{$\\hat{\\gtg}$}^{A}=\\mbox{$\\hat{\\gtg}$}^{\\bar{A}}$.\n\n\\subsubsection{ }\n\\label{defofocat}\nGiven $A$ as above, set $\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$.\nDenote by $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$ the subalgebra\nconsisting of sections $x(.)$ such that $x(P_{i})\\in \\mbox{\\g n}_{i},\\;1\\leq i\\leq m,$\nand by $\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}$  the subalgebra spanned by the space\n of sections $x(.)$ such that $x(P_{i})\\in \\mbox{\\g b}_{i},\\;1\\leq i\\leq m,$\n and the central\nelement $c$.\nThese are analogues of the maximal ``nilpotent'' and maximal ``solvable''\nsubalgebras for $\\mbox{$\\hat{\\gtg}$}^{A}$, c.f.\\ref{defofalgebrandmodusu}.\n\n\nDenote by $\\mbox{${\\cal O}$}^{A}_{k},\\;k\\in\\mbox{${\\bf C}$}$, the category of\nfinitely generated $\\mbox{$\\hat{\\gtg}$}^{A}$-modules satisfying\n the conditions:\n\n(i) $c$ acts as  multiplication by $k$;\n\n(ii) the action of the subalgebra $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$ is locally finite.\n\nIn much the same way as in \\ref{defofalgebrandmodusu} one defines\nVerma and generalized Weyl modules over $\\mbox{$\\hat{\\gtg}$}^{A}$:\n\n{\\bf Definition.}\n\n(i) We will say that $(\\lambda,k)$ is a highest weight of $\\mbox{$\\hat{\\gtg}$}^{A}$ if\n$\\lambda$ is a functional on $\\oplus _{i}\\mbox{\\g b}_{i}\/\\mbox{\\g n}_{i}$ and $k$ is a\nnumber.\n\n(ii) A highest weight $(\\lambda,k)$ naturally determines a character of\n$\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}$\nsending $c$ to $k$ and\n annihilating $\\mbox{$\\hat{\\gtg}$}^{A}_{>}$. Denote by $\\mbox{${\\bf C}$}_{\\lambda,k}$\nthe corresponding 1-dimensional representation.\n\n(iii)  Define the Verma module\n$M^{A}_{\\lambda ,k}$ to be the induced representation\n\\[\\mbox{Ind}_{\\mbox{$\\hat{\\gtg}$}^{A}_{\\geq}}^{\\mbox{$\\hat{\\gtg}$}^{A}}\\mbox{${\\bf C}$}_{\\lambda}. \\; \\Box\\]\n\n\n\\bigskip\n\nThere is an isomorphism\n\\[M^{A}_{\\lambda ,k}\\approx\n \\otimes _{i=1}^{m} M^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}.\\]\n\nSuppose now that each $M^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}$ has at least one\nsingular vector. If\n$k\\in \\mbox{${\\bf C}$} \\setminus \\mbox{${\\bf Q}$}$ then this\nsingular vector is unique for each $i$. Quotienting out all of\nthem one obtains the\n {\\em generalized Weyl module} $V_{\\lambda,k}^{A}$. As above there is an\nisomorphism\n\\[V_{\\lambda,k}^{A}\\approx\n\\otimes _{i=1}^{m} V^{P_{i},\\mbox{\\g b}_{i}}_{\\lambda_{i},k}.\\]\n\nIf $k$ is not a rational number then any generalized Weyl module is\nirreducible. Denote by $\\mbox{$\\tilde{\\co}$}_{k}$ the full subcategory of $\\mbox{${\\cal O}$}_{k}$ consisting\nof all $\\mbox{$\\hat{\\gtg}$}^{A}-$modules whose composition series consist of generalized\nWeyl modules. Again if $k$ is not a rational number then $\\mbox{$\\tilde{\\co}$}_{k}$ is\nsemisimple.\n\nIf $k$ is rational then there arises the admissible representation\n$L_{\\lambda,k}^{A}$ if $(\\lambda,k)$ is admissible. If the composition series\nof a module $V^{A}$ consists only of admissible representations, then $V^{A}$\nis\ncompletely reducible.\n\n\\begin{remark}\n\\label{functorwrtA}\nThere is a canonical isomorphism $\\mbox{\\g b}_{1}\/\\mbox{\\g n}_{1}\\approx\\mbox{\\g b}_{2}\/\\mbox{\\g n}_{2}$\nfor any 2 Borel subalgebras $\\mbox{\\g b}_{1},\\mbox{\\g b}_{2}$. Therefore if Borel subalgebras\nappearing in $A$ evolve, so does the projectivization of the module\n$M_{\\lambda,k}^A$\nor its quotients.\n\n  \\end{remark}\n\n\n\\subsubsection{ }\n\\label{algebra_gofA}\nLet $A$ be as in \\ref{algebra_hgta}. Let $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ be the Lie algebra\nof meromorphic sections of\n $\\mbox{End}\\mbox{${\\cal E}$}$  holomorphic outside $\\bar{A}$.\n The  maps of restriction to formal neighborhoods give rise to the Lie algebra\nmorphism\n\\begin{equation}\n\\label{globalfunct_local}\n\\mbox{\\g g}(\\mbox{${\\cal C}$},A)\\rightarrow \\mbox{\\g g}^{A}\n\\end{equation}\nThe splitting (\\ref{splitting}) provides us with the section\n$s_{A}:\\; \\mbox{\\g g}^{A}\\rightarrow \\mbox{$\\hat{\\gtg}$}^{A}$. Composition of\n(\\ref{globalfunct_local}) with $s_{A}$\ngives the linear morphism\n\\begin{equation}\n\\label{globalfunct_affine}\n\\mbox{\\g g}(\\mbox{${\\cal C}$},A)\\rightarrow\\mbox{$\\hat{\\gtg}$}^{A}.\n\\end{equation}\nThe residue theorem implies that (\\ref{globalfunct_affine}) is a Lie algebra\nmorphism\n(even though $s_{A}$ is not!).\n\nBy (\\ref{globalfunct_affine}), the standard\npullback makes each object of $M^{A}\\in\\mbox{${\\cal O}$}_{k}^{A}$ into a\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$-module.\nHence there arises the space of coinvariants\n\\[(M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}=M\/\\mbox{\\g g}(\\mbox{${\\cal C}$},A)M.\\]\n\n\\subsection{ Localization of $\\mbox{$\\hat{\\gtg}$}^{A}$-modules }\n\n\\subsubsection{ }\n\\label{whi1jetparam}\n  let us\nrecall that with an $n-$dimensional vector space $W$ one associates the flag\nmanifold $F(W)=\nGL(n,\\mbox{${\\bf C}$})\/B$ and {\\em the base affine space} $Base(W)= GL(n,\\mbox{${\\bf C}$})\/N$, where $B$\nis a Borel\nsubgroup and $N$ unipotent subgroup of $B$. The natural map $Base(W)\\rightarrow\nF(W)$ is a\nprincipal $(\\mbox{${\\bf C}$}^{\\ast})^{\\times n}$-bundle.\n\nNow return to a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $V^{A}$ and suppose for simplicity that $A$\nconsists of\n1 element $(P,\\mbox{\\g b})$. Consider a family of the data\n$\\{P, \\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}\\}$ -- let us not care about Borel subalgebras for the\nmoment. One expects that the corresponding family of vector spaces arranges\nthen in a locally trivial vector bundle. An obstacle to get this is that we\nhave defined\n$V^{P}$ up to an isomorphism but have not specified any such isomorphism. For\nexample, an attempt to\nchoose a basis in $V^{P}$ requires to choose\n(in particular) a local coordinate $z$ at $P$, such that $z(P)=0$.\nDifferent choices of $z$ are essentially different as the group\n$\\mbox{Diff}(P)$\nof diffeomorphisms of the formal neighborhood of $P$ does not\nin general act on $V^{P}$. However the subgroup $\\mbox{Diff}(P)_{1}\\subset\n\\mbox{Diff}(P)$\nof diffeomorphisms preserving the 1-jet of parameter does act on $V^{P}$. We\nsee that $V^{P}$,\nin fact, depends on the 1-jet of parameter at $P$.\n\nTo take care of Borel subalgebras,  let us\nrecall that with an $n-$dimensional vector space $W$ one associates the flag\nmanifold $F(W)=\nGL(n,\\mbox{${\\bf C}$})\/B$ and {\\em the base affine space} $Base(W)= GL(n,\\mbox{${\\bf C}$})\/N$, where $B$\nis a Borel\nsubgroup and $N$ unipotent subgroup of $B$. The natural map $Base(W)\\rightarrow\nF(W)$ is a\nprincipal $(\\mbox{${\\bf C}$}^{\\ast})^{\\times n}$-bundle.\n\n\nSimilar arguments applied to $\\mbox{\\g b}$ show that\n\nthe module $V^{A}=V^{P,\\mbox{\\g b}}$ depends on the quadruple $(P,\\mbox{\\g b},j,x)$ such that\n$j$ is a 1-jet\nof parameter at $P$ and $x\\in Base(\\mbox{${\\bf C}$}^{n})$ belongs to the preimage of $\\mbox{\\g b}$.\n\nOne concludes that we do get a locally trivial vector bundle after pull-back to\nthe space of pairs ``1-jet of parameter at $P$, element of the maximal torus of\nthe\nBorel group related to $\\mbox{\\g b}$''. Let us be more precise now.\n\n\n\n\n\\subsubsection{ }\n\\label{maingenerresults}\n\nLet $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ be a family of smooth projective curves\nand $\\rho_{S}:\\; \\mbox{${\\cal E}$}_{S}\\rightarrow \\mbox{${\\cal C}$}_{S}$ be a rank $n$ vector bundle. There\narise 2 more\nbundles:\n\n(i) the  bundle $Base(\\rho_{S}):\\; Base(\\mbox{${\\cal E}$}_{S})\\rightarrow \\mbox{${\\cal C}$}_{S}$ with the\nfiber\nover any $x\\in\\mbox{${\\cal C}$}_{S}$  equal to the base affine space\nof the vector space $\\rho_{S}^{-1}x$;\n\n(ii) the $\\mbox{${\\bf C}$}^{\\ast}-$bundle $J^{(1)}(\\mbox{${\\cal C}$}_{S})\\rightarrow \\mbox{${\\cal C}$}_{S}$ of  1-jets\nof coordinates along\nfibers of $\\bar{\\pi}$.\n\n Consider the fibered product\n$Base(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})$ and the natural map\n\\[\\pi:\\; Base(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})\\rightarrow S.\\]\n\nPick a  non empty  finite set $A_{S}$ of sections of $\\pi$   satisfying the\ncondition:\n\nfor any $s\\in S$ the natural projection of\n the set $A_{S}(s)=\\{a(s),\\;a\\in A_{S}\\}$\n on  $\\bar{\\pi}^{-1}(s)$\nis an injection.\n\nPick  an arbitrary curve, say $\\mbox{${\\cal C}$}_{s_{0}}$, from our family. Consider a\nhighest weight\nmodule  $M^{A }$ over $\\mbox{$\\hat{\\gtg}$}^{A}$, where we write $A$ instead of the lengthy\n$A_{S}(s_{0})$; what follows is obviously independent of the choice of $s_{0}$.\n\nBy \\ref{defofocat}, remark  \\ref{functorwrtA}, and \\ref{whi1jetparam}, we get a\n$\\mbox{$\\hat{\\gtg}$}^{A_{S}(s)}-$module $M^{A_{S}(s)}$ for any $s\\in S$ and the collection\n $\\{M^{A_{S}(s)},\\; s\\in S\\}$ arranges in a locally trivial vector bundle.\nWith\neach $s\\in S$ we can further associate a vector space, that is the space of\ncoinvariants\n\\[(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s))},\\]\nsee \\ref{algebra_gofA}.\n\n\n\\begin{theorem}\n\\label{existofDmod}\nSuppose the collection $\\psi=(M^{A},\\pi,A_{S})$, satisfying the conditions\nimposed above, is given.\n Then\nthere is a twisted $ \\mbox{${\\cal D}$}-$module (that is\na sheaf of modules over a certain algebra of twisted differential operators) on\n$S$ such\nthat its fiber over $s\\in S$ is $(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s)}$.\n\\end{theorem}\n\nThis theorem is an immediate consequence of \\cite{beil_feig_maz} and\n\\cite{bern_beil,bryl_kash} . Briefly the construction is as follows. Take a\nvector field  $\\xi$ on $U\\subset S$. It lifts to a meromorphic vector field\non $\\mbox{${\\cal C}$}_{S} - A_{S}(S)$ over $U$, and\nfurther to a meromorphic vector field on $\\pi^{-1}(U)\\subset\nBase(\\mbox{${\\cal E}$}_{S})\\times_{\\mbox{${\\cal C}$}_{S}}J^{(1)}(\\mbox{${\\cal C}$}_{S})$; denote this vector field by\n$\\xi^{\\ast}$.  Trivializing  the infinitesimal neighborhood of\n$A_{S}(U)\\subset\\mbox{${\\cal C}$}_{S}$ by chosing, locally with respect to $U\\subset S$,\ncoordinates in the fibers, one gets vertical components\n$\\{\\xi^{\\ast}_{vert;i}\\}$, so that $\\xi^{\\ast}_{vert;i}$ is the vertical\ncomponent in the formal neighborhood of the $i-$ section. Projecting\n$\\xi^{\\ast}_{vert;i}$ on $Base(\\mbox{${\\cal E}$}_{S})$ one gets some element of $U(\\mbox{\\g g})$,\nsay\n$u_{i}$;  projecting $\\xi^{\\ast}_{vert;i}$ on $J^{(1)}(\\mbox{${\\cal C}$}_{S})$ one gets some\nvector field, say $v_{i}$. Both $u_{i},\\; v_{i}$ act on our $\\mbox{$\\hat{\\gtg}$}^{A}$-module\n$M^{A}$:\n$u_{i}$ naturally, $v_{i}$ by means of the Sugawara construction. Going over\ndefinitions one gets that this well defines a twisted $\\mbox{${\\cal D}$}-$module with the\nfiber as in the theorem. $\\Box$\n\n\\bigskip\n\n\nDenote the constructed $\\mbox{${\\cal D}$}-$module by $\\Delta_{\\psi}(M^{A})$.\n\nIn the case when $M^{A}$ is an admissible representation the following result\nis valid.\n\n\\begin{theorem}\n\\label{smoothinadm}\nIf $n=2$ and $M^{A}$ is an admissible $\\mbox{$\\hat{\\gtg}$}^{A}-$module then\n$\\Delta_{\\psi}(M^{A})$ is {\\em holonomic}\nfor almost any vector bundle $\\mbox{${\\cal E}$}_{S}$\n(i.e.as a sheaf $\\Delta_{\\psi}(M^{A})$ is isomorphic to a sheaf of sections\n of a certain finite rank  vector bundle over some open set in $S$ ).\n\\end{theorem}\n\n{\\bf Proof.}\n\nTo prove this theorem essentially  means to show that the spaces\n $(M^{A_{S}(s)})_{\\mbox{\\g g}(\\pi^{-1}S,A_{S}(s)},\\; s\\in S,$ are all finite\ndimensional. That will be done\nin \\ref{finofcoinvhighgensubs},  Proposition \\ref{proofoflisse} in the higher\ngenus case\nand in \\ref{finofcoinvsferesubs}, Proposition \\ref{fincinvforcpadmweyl} for\n$\\cp$. . We will also give there a precise meaning\nto the phrase ``almost any vector bundle'' in Theorem\\ref{smoothinadm}. $\\Box$\n\n\\bigskip\n\nResults of \\ref{Quadratic_degeneration} will show that the standard\ncombinatorial algorithm can be used to calculate the dimension of the fiber of\nour\n$\\mbox{${\\cal D}$}-$module using the dimensions of the spaces of coinvariants on a sphere\nwith 3 punctures.\nThe latter dimensions will be calculated in \\ref{The_rational_leve_case}.\n\n\\section{The spaces of coinvariants}\n\\label{The_spaces_of_coinvariants}\n In this section we will be concerned\nwith the space of coinvariants $(M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}$\n(or  spaces closely related to it ) in the case when $M^{A}$ is either\na generalized Weyl module or an admissible representation. The standard tool to\nget finiteness results\nabout coinvariants is the notion of\n {\\em singular support}.\n\n\\subsection{Singular support and coinvariants}\n\\label{singsuppandcoinvvvvv}\n\nLet $\\mbox{\\g a}$ be a Lie algebra. Universal enveloping algebra $U\\mbox{\\g a}$ is filtered\nin\n the standard way so that the associated graded algebra is $S\\mbox{\\g a}$. One says\nthat a filtration of a finitely generated $\\mbox{\\g a}-$module $V$ is good if (i) it\nis\ncompatible with the filtration of $U\\mbox{\\g a}$, and (ii) the associated graded\nmodule $Gr\\, V$ is finitely generated as an $S\\mbox{\\g a}$-module.\n\n{\\bf Definition} Singular support, $SS V$, of $V$ is the zero set of the\nvanishing ideal of the $S\\mbox{\\g a}$-module $Gr\\, V$. $\\Box$\n\n\\bigskip\n\nObviously, $SS V$ is a conical subset of $\\mbox{\\g a}^{\\ast}$.\n\nFor a subalgebra $\\mbox{\\g n}\\subset \\mbox{\\g a}$, call  $V$ an $(\\mbox{\\g a},\\mbox{\\g n})$-module if it is\nan $\\mbox{\\g a}-$module and\n$\\mbox{\\g n}$ acts on $V$ locally nilpotently. Typical example: any module from the\n$\\mbox{${\\cal O}$}-$category is a $(\\mbox{$\\hat{\\gtg}$},\\mbox{$\\hat{\\gtg}$}_{>})-$module.\n\n\n\\begin{lemma} (\\mbox{ see \\cite{beil_feig_maz}})\n\\label{singsupp-finitenessofcoinv}\nLet $\\mbox{\\g a}$ be a Lie algebra and $\\mbox{\\g p}\\subset\\mbox{\\g a}$ be its subalgebra. Denote by\n$\\mbox{\\g p}^{\\perp}$ the\nannihilator of $\\mbox{\\g p}$ in $\\mbox{\\g a}^{\\ast}$\nLet $V$ be an $(\\mbox{\\g a},\\mbox{\\g n})$-module.\nIf $SSM\\cap\\mbox{\\g p}^{\\perp}=\\{0\\}$ and $dim\\, \\mbox{\\g a}\/\\mbox{\\g n}\\oplus\\mbox{\\g p}<\\infty$ then\n$\\mbox{dim}M_{\\mbox{\\g p}}<\\infty$.\n\\end{lemma}\n\nRecall that from now on $\\mbox{\\g g}=\\mbox{\\g sl}_{2}$ unless otherwise stated.\n\n\\subsection{ Singular support of $\\mbox{$\\hat{\\gtg}$}^{A}-$modules}\n\\label{singsuppppofhgta}\nObserve that there is an involution $\\sigma$ of $\\mbox{$\\hat{\\gtg}$}$ sending $f$ to\n$e\\otimes z^{-1}$ and\n$e\\otimes z^{-1}$ to $f$, see\n\\ref{defofalgebrandmodusu}\nfor notations. There arises the involution, also denoted by $\\sigma$, acting\non the algebras $\\mbox{$\\hat{\\gtg}$}^{A}$ and their duals. This involution is not canonical\nbut we do not have\nto care as our considerations here are purely local.\n\nDenote by $\\Omega^{A}$ the space of $\\mbox{\\g g}-$ valued differential\nforms on the formal neighborhoods of the points from $A$. There is a natural\nembedding\n$\\Omega^{A}\\hookrightarrow (\\mbox{\\g g}^{A})^{\\ast}$ (``take the traces and\n then sum up all the residues!'')\n\nWe will make use of 2 subspaces of $\\Omega^{A}$: $\\Omega^{A}_{reg}$ is all\nregular forms and\n$\\Omega^{A}_{nilp}$ is all forms with values in the nilpotent\ncone.\n\\begin{theorem}\n\\label{theoronsingsupp}\n\n(i) If $M^{A}$ is a generalized Weyl module then\n$SSM^{A}=\\Omega^{A}_{reg}\\cup\\sigma\\Omega^{A}_{reg}$.\n\n(ii) (E.Frenkel, B.F.) If $M^{A}$ is an admissible representation then\n$SSM^{A}=\\Omega^{A}_{nilp}\\cup\\sigma\\Omega^{A}_{nilp}$.\n\\end{theorem}\n\n\\begin{remark}\nIt is easy to see that although $\\sigma$ is not determined uniquely the\nspaces $\\sigma\\Omega^{A}_{reg}$, $\\sigma\\Omega^{A}_{nilp}$ are canonical.\nFor example  $\\sigma\\Omega^{A}_{reg}$ is the space of forms such that:\n\nthey have at most order 1 pole at $\\bar{A}$;\n\ntheir residue at each $P_{i}\\in \\bar{A}$ belongs to $\\mbox{\\g n}_{i}$;\n\nat each $P_{i}\\in\\bar{A}$ their constant term  belongs to $\\mbox{\\g b}_{i}$.\n\\end{remark}\n\n\\subsection{Finiteness of coinvariants --  the higher genus case}\n\\label{Finiteness_of_coinvariants-hghergen}\n\n\n\n\n\n\\subsubsection{ Hitchin's theorem.}\n\\label{hitchinstheorem}\nFirst recall a well-known result of  Hitchin, \\cite{hitch}.\nWith a vector bundle $\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}$ associate the map\n\\begin{eqnarray}\n\\label{def-fhitchmap}\nH(\\mbox{${\\cal E}$}):\\;H^{0}(\\mbox{${\\cal C}$},\\Omega\\otimes\\mbox{End}\\mbox{${\\cal E}$})\\rightarrow\n \\oplus_{i=2}^{n}H^{0}(\\mbox{${\\cal C}$},\\Omega^{\\otimes i}),\\\\\nX\\mapsto \\mbox{Tr}X^{i}\\nonumber\n\\end{eqnarray}\nCall a bundle $\\mbox{${\\cal E}$}$ {\\em exceptional} if $\\mbox{ker}H(\\mbox{${\\cal E}$})\\neq 0$. Obviously\n$\\mbox{ker}H(\\mbox{${\\cal E}$})$ is exactly the space of global differential forms\nwith values in nilpotent endomorphisms of the vector bundle $\\mbox{${\\cal E}$}$.\n\n\\begin{theorem} (\\mbox{Hitchin \\cite{hitch}})\n\\label{hitchi}\n Zero set of the map (\\ref{def-fhitchmap}) is a maximal Lagrangian\nsubmanifold in the cotangent bundle of the moduli space of vector bundles over\n$\\mbox{${\\cal C}$}$. In particular,\nexceptional\nvector bundles form a positive codimension algebraic subset of the moduli space\nof vector bundles.\n\\end{theorem}\n\nFor us, importance of Theorem \\ref{hitchi} is in  that generically a vector\nbundle does not allow\na non-trivial global differential form with coefficients in nilpotent\nendomorphisms of the bundle.\n\n\\subsubsection { Subtracting lines from rank 2 vector bundles.}\n\\label{subtrlinesfromvectbundles}\n\n An analogue of subtracting a point\nfrom a line bundle (or, better to say, from  its divisor) is an operation of\nsubtracting a line from a rank 2  vector bundle.\n\nTo a  rank 2 vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}$ one can associate a  module over\nthe sheaf of regular functions -- the\nsheaf of sections of $\\mbox{${\\cal E}$}$.; denote this sheaf by $Sect(\\mbox{${\\cal E}$})$. This establishes\na one-to-one correspondence between rank 2 vector bundles and rank 2 locally\nfree modules over the sheaf of regular functions.\n Now fix a line, $l$, in a fiber of $\\mbox{${\\cal E}$}$ over some point $P\\in\\mbox{${\\cal C}$}$. Denote by\n$S (l)$ a sheaf such that:\n\n(i) $S(l)|_{U}=Sect(\\mbox{${\\cal E}$})|_{U}$ if $P$ does not belong to $U$;\n\n(ii)  $S(l)|_{U}, P\\in U,$ is the space of meromorphic sections of $\\mbox{${\\cal E}$}$ over\n$U$ regular outside $P$, having at most order 1 pole at $P$ and such that their\nresidue at $P$ belongs to the fixed line $l$.\n\nIt is obvious that $S(l)$ is a rank 2 locally free module. Therefore it\ndefines a rank 2  vector bundle. Denote this vector bundle by $\\mbox{${\\cal E}$}(l)$.\nIf a collection of lines -- $l_{1},l_{2},...,l_{m}$ -- is subtracted, then\ndenote\nthe corresponding vector bundle by $\\mbox{${\\cal E}$}(l_{1}+\\cdots + l_{m})$.\n\nSuppose we have a moduli space of rank 2 vector bundles with parabolic\nstructure with fixed determinant. Elements of such a space are isomorphism\nclasses of the data\n(vector bundle $\\mbox{${\\cal E}$}$, fixed lines $l_{1},...,l_{m}$ in some fibers.) It is\nrather\nclear  that the map $(\\mbox{${\\cal E}$},\\; l_{1},...,l_{m})\\mapsto (\\mbox{${\\cal E}$}(l_{1}+\\cdots\n+l_{m}),\\; l_{1},...,l_{m})$ is a homeomorphism of 2 moduli spaces with\ndifferent determinants.\n\n{\\bf Definition.}  Call the data $(\\mbox{${\\cal E}$},\\; l_{1},...,l_{m})$ generic if\n $\\mbox{${\\cal E}$}(l_{i_{1}}+\\cdots +l_{i_{s}})$ is not  exceptional for\nany subset $\\{i_{1},...,i_{s}\\}\\subset\\{1,2,..., m\\}$. $\\Box$\n\n\\bigskip\n\nIt follows from Theorem \\ref{hitchi} that the set of generic vector bundles is\nopen and everywhere dense.\n\n\n\n\n\\subsubsection { Finiteness of coinvariants.}\n\\label{finofcoinvhighgensubs}\n\nSuppose we are in the situation of \\ref{defofocat}: we have an admissible\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module\n$M^{A}$ on the curve $\\mbox{${\\cal C}$}$ with a vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\mbox{${\\cal C}$}$.\nAs $A$ is a collection of borel subalgebras $\\mbox{\\g b}_{1},...,\\mbox{\\g b}_{m}$ operating\nin fixed fibers, we have\nparabolic structure -- lines $l_{1},...,l_{m}$ in the corresponding fibers\npreserved by the $\\mbox{\\g b}_{i}$'s.\nCall the data $(\\mbox{${\\cal E}$},A)$  generic if the data $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$ is\ngeneric\nin the sense of \\ref{subtrlinesfromvectbundles} above.\n\n\n\n\n\n\nRecall that we are interested in the space of coinvariants\n$M^{A}_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}$, where $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is an algebra of endomorphisms of\nthe bundle $\\mbox{${\\cal E}$}$ regular outside points from the corresponding $\\bar{A}$, see\n\\ref{algebra_gofA} and \\ref{maingenerresults}, Theorem \\ref{existofDmod}.\n\n\n\\begin{proposition}\n\\label{proofoflisse}\n Let $(\\mbox{${\\cal E}$}, A)$ be generic. Then\n\\[\\mbox{dim} M^{A}_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A)}<\\infty.\\]\n\\end{proposition}\n\n{\\bf Proof.}  One extracts from definitions that the annihilator\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}$ of the algebra $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is the space\n$\\Omega_{\\mbox{${\\cal C}$},A}(\\mbox{${\\cal E}$})$ of global meromorphic $End(\\mbox{${\\cal E}$})-$valued differential\nforms regular outside\n$\\bar{A}\\subset\\mbox{${\\cal C}$}$.\n\nBy Theorem \\ref{theoronsingsupp}(ii) we get that\n$SSM^{A}\\cap\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}=\\Omega_{nilp}(\\mbox{${\\cal E}$})\\cup\\sigma\\Omega_{nilp}(\\mbox{${\\cal E}$})$, where\n$\\Omega _{nilp}(\\mbox{${\\cal E}$})$ is the space global nilpotent transformations of $\\mbox{${\\cal E}$}$,\nand $\\sigma$ is the twist introduced in \\ref{singsuppppofhgta}.\n\nGenericity condition means that $\\Omega_{nilp}(\\mbox{${\\cal E}$})=0$, see\n\\ref{hitchinstheorem} and  \\ref{subtrlinesfromvectbundles}.\n\nOn the other hand it is easy to see that the operation of subtracting a line\ngenerates the twist $\\sigma$ on endomorphisms. (In fact one has to compose\nsubtracting of a line\nwith a reflection in the fiber, but this does not change the isomorphism class\nof the bundle.) Therefore genericity condition also implies that\n$\\sigma\\Omega_{nilp}(\\mbox{${\\cal E}$})=0$.\n\nHence we get that $SSM^{A}\\cap\\mbox{\\g g}(\\mbox{${\\cal C}$},A)^{\\perp}=0$. And as the space\n$\\mbox{$\\hat{\\gtg}$}^{A}_{>}+\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ is  of finite codimension in $\\mbox{$\\hat{\\gtg}$}^{A}$,\napplication\nof Lemma \\ref{singsupp-finitenessofcoinv}, see \\ref{singsuppandcoinvvvvv},\ncompletes the proof. $\\Box$\n\n\\bigskip\n\n\n\\bigskip\n\nIn order to study quadratic degenerations we will need the following\nstronger finiteness result. Along with the set\n$A=\\{(P_{1},\\mbox{\\g b}_{1}),...,(P_{m},\\mbox{\\g b}_{m})\\}$, consider the set $ A_{2} =\\,\n\\{(P_{m+1},\\mbox{\\g b}_{m+1}), (P_{m+2},\\mbox{\\g b}_{m+2})\\}$ such that the points\n$P_{1},...,P_{m+2}\\in\\mbox{${\\cal C}$}$ are different. Denote by $\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})$ the\nsubalgebra of $\\mbox{\\g g}(\\mbox{${\\cal C}$},A)$ consisting of functions taking values in\n$\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$ at point $P_{i}$, $i=m+1,m+2$.\n\n\\begin{proposition}\n\\label{highgen_finit_afterpionching}\n\nIf $(\\mbox{${\\cal E}$}, A\\bigsqcup A_{2})$ is generic and $M^{A}$ is admissible, then\n\\[dim\\, (M^{A})_{\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})}<\\infty.\\]\n\n\\end{proposition}\n\n{\\bf Proof.} We are again going to apply Lemma\n\\ref{singsupp-finitenessofcoinv}. Observe that\n$\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}$ consists of meromorphic forms on $\\mbox{${\\cal C}$}$ with values\nin $End(\\mbox{${\\cal E}$})$,\nregular outside $\\{P_{1},...,P_{m+2}\\}\\subset\\mbox{${\\cal C}$}$, having at most order 1 poles\nat $P_{m+1},P_{m+2}$, their residues at the latter points lying in $\\mbox{\\g b}_{1}$\n( $\\mbox{\\g b}_{2}$ resp.).\n\nBy Theorem \\ref{theoronsingsupp}(ii), $\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}\\cap\\, SSM^{A}$\nconsists of forms with values in nilpotent endomorphisms, satisfying the above\nlisted global conditions. This implies, in particular, that actually residues\nof our forms\nbelong to $\\mbox{\\g n}_{m+1},\\mbox{\\g n}_{m+2}$ at $P_{m+!}, P_{m+2}$ resp..\n\nGiven an element $\\omega\\in\\mbox{\\g g}(\\mbox{${\\cal C}$},A,A_{2})^{\\perp}\\cap\\, SSM^{A}$, subtract\nsome lines from $\\mbox{${\\cal E}$}$ so as to make $\\omega$ be everywhere regular. Genericity\ncondition implies then that $\\omega=0$, and application of   Lemma\n\\ref{singsupp-finitenessofcoinv} completes the proof. $\\Box$\n\n\\bigskip\n\n\n\n\n\n\n\n\n\n\\subsection{Finiteness of coinvariants -- the case of $\\cp$}\n\\label{Finiteness_of_coinvariants}\n\n\\subsubsection{Generic vector bundles on $\\cp$}\n\\label{genervectbumdonsph}\n\nLet $O(n)$ be the degree $n$ line bundle over $\\cp$. It is known, e.g.\n\\cite{pr_seg}, that any rank 2 vector bundle over $\\cp$ is a direct sum\n$O(r)\\oplus O(s)$ for some $r$, $s$.\n\nAs there are no moduli, it is hard to speak about generic vector bundles.\nNevertheless we will call $O(r)\\oplus O(s)$ {\\em exceptional} if $|r-s|>1$.\nHere is\na justification.\n\n\\begin{lemma}\n\\label{whyexceptional}\n\nLet $\\mbox{${\\cal E}$}= O(r)\\oplus O(s)$ and $(\\mbox{${\\cal E}$},l_{1},...,l_{m})$, $m\\geq |r-s|$, a vector\nbundle with parabolic structure. Then generically with respect to\n$l_{1},...,l_{m}$ the bundle $\\mbox{${\\cal E}$}(l_{1}+\\cdots +l_{m})$ is not exceptional:\n\n\\[\\mbox{${\\cal E}$}(l_{1}+\\cdots +l_{m})=\\left\\{\\begin{array}{lll}\nO(p+1)\\oplus O(p)&\\mbox{ if } r+s-m=2p+1\\\\\n O(p)\\oplus O(p)&\\mbox{ if } r+s-m=2p.\n\\end{array}\n\\right.\\]\n\\end{lemma}\n\nLemma \\ref{whyexceptional} seems to be common knowledge, although we failed\nto find a reference with its proof.\n\nProceed just like we did in \\ref{subtrlinesfromvectbundles}:  call  $(\\mbox{${\\cal E}$},\\;\nl_{1},...,l_{m})$ generic if\n $\\mbox{${\\cal E}$}(l_{i_{1}}+\\cdots +l_{i_{s}})$ is not  exceptional for\nany subset $\\{i_{1},...,i_{s}\\}\\subset\\{1,2,..., m\\}$.\n\n\\subsubsection{Finiteness of coinvariants}\n\\label{finofcoinvsferesubs}\n\nA specific feature of the genus zero case is that we do not necessarily have\nto consider admissible representations -- generalized Weyl modules, see\n\\ref{Case_(i)},\n will also do.\n\nLet us again consider a vector bundle $\\mbox{${\\cal E}$}$ over $\\cp$ and a $\\mbox{$\\hat{\\gtg}$}^{A}-$module\n$M^{A}$. As in \\ref{finofcoinvhighgensubs}, $A$ determines a parabolic\nstructure\non $\\mbox{${\\cal E}$}$, say $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$. Call the data $(\\mbox{${\\cal E}$},A)$ generic\nif $(\\mbox{${\\cal E}$}, l_{1},...,l_{m})$ is also.\n\n\n\\begin{proposition}\n\\label{fincinvforcpadmweyl}\nIf  $(\\mbox{${\\cal E}$},A)$ is generic and $M^{A}$ is either admissible or generalized Weyl\nmodule, then\n\n\\[\\mbox{dim } (M^{A})_{\\mbox{\\g g}(\\cp,A)}<\\infty.\\]\n\n\\end{proposition}\n\n{\\bf Proof}  is a simplified version of the proof of Proposition\n\\ref{proofoflisse} in \\ref{finofcoinvhighgensubs}. The new features are as\nfollows: to include generalized Weyl modules one uses Theorem\n\\ref{theoronsingsupp}(i) in addition to Theorem \\ref{theoronsingsupp}(ii);\ninstead of the Hitchin's theorem one uses the ``observation'' that $O(n)$ has\nno\nnon-zero global sections if $n<0$. $\\Box$\n\n\\begin{corollary}\n\\label{existofdmodforweyl}\n If $M^{A}$ is a generalized Weyl module then\nthere is a holonomic\ntwisted $D$-module living in the space $(J^{(1)}(\\cp)\\times\nJ^{(1)}(\\cp))^{\\times m}$ with the fiber $M^{A}_{\\mbox{\\g g}(\\cp,A)})$.\n\\end{corollary}\n\n{\\bf Proof.} Repeating word for word proof of  Theorem\\ref{smoothinadm} one\nderives\n from Proposition \\ref{fincinvforcpadmweyl} existence of a twisted $D-$module\non\nthe space $(Base(\\mbox{${\\bf C}$}^{2})\\times J^{(1)}(\\cp))^{\\times m}$. But for $\\mbox{\\g sl}_{2}$,\n the flag manifold is $\\cp$ and\nthe base affine space $(Base(\\mbox{${\\bf C}$}^{2})$ is  also the space of 1-jets of\nparameter $ J^{(1)}(\\cp))^{\\times m}$. $\\Box$\n\n\\bigskip\n\nAs in \\ref{finofcoinvhighgensubs}, we want to prove a generalization of\nProposition \\ref{fincinvforcpadmweyl} in order to prepare grounds for studying\nquadratic degeneration.\n\nAlong with $A=\\{(P_{1},\\mbox{\\g b}_{1}),\\ldots ,(P_{m},\\mbox{\\g b}_{m})\\}$ consider 2 sets\n$A_{1}=\\{(P_{m+1},\\mbox{\\g b}_{m+1})\\}$ and $A_{2}=\\{(P_{m+1},\\mbox{\\g b}_{m+1}),\n(P_{m+2},\\mbox{\\g b}_{m+2})\\}$\nsuch that $P_{1},...,P_{m+2}$ are different points in $\\mbox{${\\cal C}$}$.\n\nWith $A_{1}$ and $A_{2}$ associate the following 2 subalgebras of\n$\\mbox{\\g g}(\\cp,A)$:\n$\\mbox{\\g g}(\\cp,A,A_{1})$ consists of all functions taking values in\n$\\mbox{\\g n}_{m+1}=[\\mbox{\\g b}_{m+1},\\mbox{\\g b}_{m+1}]$\nat the point $P_{m+1}$;\n$\\mbox{\\g g}(\\cp,A,A_{2})$ consists of all functions taking values in\n$\\mbox{\\g n}_{i}=[\\mbox{\\g b}_{i},\\mbox{\\g b}_{i}]$\nat the point $P_{i}$, $i=m+1,\\;m+2$.\n\n\\begin{proposition}\n\\label{finofcoinv_general}\n Let   $(\\mbox{${\\cal E}$}, A_{2})$ be\ngeneric. Then\n\n(i) If $M^{A}$ is a generalized Weyl module over $\\mbox{$\\hat{\\gtg}$}^{A}$, then\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}<\\infty$;\n\n(ii)  If $M^{A}$ is an admissible representation of $\\mbox{$\\hat{\\gtg}$}^{A}$, then\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$.\n\\end{proposition}\n\n \\bigskip\n\n{\\bf Proof.} of (ii) repeats almost word for word that of Proposition\n\\ref{highgen_finit_afterpionching} in \\ref{finofcoinvhighgensubs} with\nsimplifications analogous to those\nindicated in the proof of Proposition  \\ref{fincinvforcpadmweyl}.\n\nAs to (i), its proof is again application of the same technique in a slightly\ndifferent\nform: one has to take a form $\\omega\\in\\mbox{\\g g}(\\cp,A,A_{2})^{\\perp}\\cap SSM^{A}$\nand to subtract lines from $\\mbox{${\\cal E}$}$ so as to make $\\omega$ into\na form with either one pole (at $P_{m+1}$) or 2 poles (one of them is\nagain at $P_{m+1}$) in such a way that the bundle obtained is $O(n)\\oplus\nO(n)$.\nThe 2 cases are of course distinguished by the parity of the difference\nbetween the degrees of the determinant of $\\mbox{${\\cal E}$}$ and $\\omega$. In both cases\nit is easy to prove that $\\omega=0$ using the fact that any differential form\nwith trivial coefficients has at least 2 poles. $\\Box$\n\n\\subsubsection{Holonomic $D$-module on $(\\mbox{${\\bf C}$}\\times\\mbox{${\\bf C}$})^{\\times m}$}\n\\label{holondmodoncc}\n\n We will now get rid of twisted differential operators in Corollary\n\\ref{existofdmodforweyl} under the assumption that the vector bundle\n$\\mbox{${\\cal E}$}\\rightarrow \\cp$ is trivial.\nConsider the set $A'=A\\bigsqcup (P_{\\infty},\\mbox{\\g b}_{\\infty})$. Attach to the\npoint $(P_{\\infty},\\mbox{\\g b}_{\\infty})$  the module $(V^{0}_{0},V_{0})$ known\nas the vacuum representation, see \\ref{Case_(i)} for notations.\n$(P_{\\infty},\\mbox{\\g b}_{\\infty})$ can be redefined as the module induced from the\ntrivial representation (see also \\ref{Case_(i)}) and therefore there is an\nisomorphism $ M^{A}_{\\mbox{\\g g}(\\cp,A)}\\approx M^{A'}_{\\mbox{\\g g}(\\cp,A')}$. Now consider\nthe twisted $D-$module with fiber $ M^{A'}_{\\mbox{\\g g}(\\cp,A')}$ on the space\n$(J^{(1)}(\\cp)\\times J^{(1)}(\\cp))^{\\times m+1}$. Restrict it to the space\n $(J^{(1)}(\\cp)\\times J^{(1)}(\\cp))^{\\times m}$ by having  the point\n$(P_{\\infty},\\mbox{\\g b}_{\\infty})$ fixed. The result of this operation is that the\nbundles in question trivialize: $\\cp - \\mbox{\\g b}_{\\infty}=\\cp - P_{\\infty}=\\mbox{${\\bf C}$}$ and\n$J^{(1)}(\\mbox{${\\bf C}$})=\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}$. Further pushing forward by ``integrating\nalong $\\mbox{${\\bf C}$}^{\\ast}$'' one gets a $D-$module over the space $(\\mbox{${\\bf C}$}\\times\n\\mbox{${\\bf C}$})^{m}$.\nObserving that it is appearance of the bundle $J^{(1)}(\\cp)\\rightarrow\\cp$\nwhich\nwas responsible for the twisting of the $D-$module, one argues that we get\na usual holonomic $D$-module on $(\\mbox{${\\bf C}$}\\times \\mbox{${\\bf C}$})^{m}$ with fiber $\nM^{A}_{\\mbox{\\g g}(\\cp,A)}$. In particular, we get a bundle with flat connection\nover an open subset of $(\\mbox{${\\bf C}$}\\times \\mbox{${\\bf C}$})^{m}$.\n\n\n{\\bf Notation.} Denote the constructed in this way bundle with flat connection\nby $\\Delta(M^{A})$. $\\Box$\n\n\\bigskip\n\n\n\n  We are unable to describe this open\nsubset explicitly at present. It follows from the requirement that\n$(\\mbox{${\\cal E}$}, A)$ be generic in all our finiteness results however that the diagonals\nshould\nbe thrown away meaning that $P_{i}\\neq P_{j}$ and $\\mbox{\\g b}_{i}\\neq\n\\mbox{\\g b}_{i}\\neq\\mbox{\\g b}_{j}$ for all $i\\neq j$.\n\nOne may want to write down differential equations satisfied by horizontal\nsections of this bundle. We will show in \\ref{diffeqsatbycorrfuntttt} that\nhorizontal sections satisfy a system of $2m$ differential equations of\nwhich\n$m$equations are Knizhnik-Zamolodchikov equations and the other $m$ are\nobtained from singular vectors of the Verma module projecting onto $M^{A}$.\n\nEverything said here holds true for an admissible representation. It is easy to\nsee\nthat the bundle associated with an admissible representation is a quotient of\nthe just constructed bundle for the corresponding generalized Weyl module.\n\n\n\n\n\n\n\\subsection{Calculation of the dimensions of coinvariants. Fusion algebra}\nLet $\\mbox{${\\cal E}$}\\rightarrow\\cp$ be the rank 2 trivial vector bundle and $M^{A}$ be a\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module.\nHere we will calculate the dimension of the space\n$(M^{A})_{\\mbox{\\g g}(\\cp,A)},\\;\\sharp A=3$, in the\nfollowing 2 cases:\n(i) the level $k$ is not\n rational and $M^{A}$ is a generalized Weyl module; (ii) $k+2=p\/q$, $p\\mbox{\nand }q$\n being positive integers,\nand $M^{A}$ is an admissible representation. Without loss of generality we can:\n\nfix a coordinate $z$ on $\\cp$; assume that\n$A=\\{(0,\\mbox{\\g b}_{0}),\\;(1,\\mbox{\\g b}_{1}),\\; (\\infty,\\mbox{\\g b}_{\\infty})\\}$, where\n$\\mbox{\\g b}_{0}=\\mbox{${\\bf C}$} e\\oplus\\mbox{${\\bf C}$} h,\\;\\mbox{\\g b}_{\\infty}=\\mbox{${\\bf C}$} f\\oplus\\mbox{${\\bf C}$} h$ and\n$\\mbox{\\g b}_{1}=\\mbox{${\\bf C}$}(e-h-f)\\oplus\n\\mbox{${\\bf C}$}(h+2f)$.\n\n(In fact, for any $\\mbox{\\g b}_{0}\\neq\\mbox{\\g b}_{\\infty}$  we can always choose a basis of\n$\\mbox{\\g g}$ so that\n$\\mbox{\\g b}_{0},\\mbox{\\g b}_{\\infty}$ are as above. As to $\\mbox{\\g b}_{1}$, there really is some\nfreedom but it is\neasy to see that all the calculations below are independent of the choice. We\nhave set\n$\\mbox{\\g b}_{1}=(\\exp{f})\\mbox{\\g b}_{0}(\\exp{-f})$.)\n\n\\subsubsection{ The generic level case}\n\\label{Thegenericlevelcase}\nSo by \\ref{defofocat}  we are given three irreducible generalized Weyl\n modules $V^{0}_{\\lambda_{0},k},\nV^{1}_{\\lambda_{1},k},V^{\\infty}_{\\lambda_{\\infty},k}$.\nRecall, see \\ref{Case_(i)},\nthat generalized Weyl modules are parametrized by symbols\n$(V_{m}^{\\epsilon},V_{n})$, where\n$m,n$ are nonnegative integers,\n$\\epsilon\\in\\nz\/2\\nz$ and $V_{m}$ is an $m+1$-dimensional $\\mbox{\\g g}-$module.\nTherefore we\ncan and will assume that we have\n\\[(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}),\\;i=0,1,\\infty .\\]\n\nIt is convenient to interpret the result of calculation of $\\mbox{dim }\n(\\otimes_{i}(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ in terms of\nthe\n {\\em fusion algebra}.\nThe latter is defined as follows.   Suppose\n that for any pair of generalized Weyl modules, say\n$(V_{r_{i}}^{\\alpha_{i}},V_{s_{i}}),\\;i=0,1$, there is only finite number of\n$(V_{r_{\\infty}}^{\\alpha_{\\infty}},V_{s_{\\infty}})$ such that\n\\[\\mbox{dim }\n(\\otimes_{i=0,1,\\infty}(V_{r_{i}}^{\\alpha_{i}},V_{s_{i}}))_{\\mbox{\\g g}(\\cp,A)}\\neq\n0.\\]\nNow view the symbols $(V_{m}^{\\epsilon},V_{n})$ as generators of a free abelian\ngroup. Then\nthere naturally arises an algebra (over $\\nz$) with the operation of\n  multiplication $\\circ$ defined by\n\\[(V^{\\alpha_{0}}_{r_{0}},V_{s_{0}})\\circ (V^{\\alpha_{1}}_{r_{1}},V_{s_{1}})=\n\\sum_{(r_{\\infty},s_{\\infty},\\alpha_{\\infty})}\\mbox{dim }\\{\n(\\otimes_{i=0,1,\\infty}(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}\\}\n(V^{\\alpha_{\\infty}}_{r_{\\infty}},\nV_{s_{\\infty}}).\\]\nThe algebra defined in this way is called {\\em fusion\nalgebra}. Of course structure constants of the fusion\nalgebra determine the dimensions of the spaces of coinvariants.\n\n\nOne last piece  of notation:\n in the following\ntheorem  we  formally set $(X\\oplus Y,Z)=(X,Z)+(Y,Z)$\nand $(X,Y\\oplus Z)=(X,Y)+(X,Z)$. Recall also that in the category of\n$\\mbox{\\g g}-$modules\none has\n\\[V_{r}\\otimes V_{s}\\approx V_{r+s}\\oplus V_{r+s-2}\\oplus\\cdots\\oplus\nV_{|r-s|}.\\]\n\n\\begin{theorem}\n\\label{fusalggencase}\n(i) For any triple of generalized Weyl modules\nthe space $(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ is finite\ndimensional.\n\n(ii) The fusion algebra is well-defined, multiplication being given by the\nfollowing formula\n\\begin{eqnarray}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})&=&\\nonumber\\\\\n(V_{r_{1}+r_{2}}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&+&\n(V_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1},V_{s_{1}}\\otimes V_{s_{2}})+\n(V_{r_{1}+r_{2}-2}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})+\\cdots\n+\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\otimes V_{s_{2}})&.&\\nonumber\n\\end{eqnarray}\n\\end{theorem}\n\n\\subsubsection{ }\n{\\bf Proof of Theorem\\ref{fusalggencase}.}\n\nThroughout the proof $A$ will stand for $\\{(0,\\mbox{\\g b}_{0}),(1,\\mbox{\\g b}_{1})\\}$,\n$A_{1}$ -- for\n$\\{(\\infty,\\mbox{\\g b}_{\\infty})\\}$. Along with the algebras\n$\\mbox{\\g g}(\\cp,A),\\; \\mbox{\\g g}(\\cp,A,A_{1})$ (see \\ref{Finiteness_of_coinvariants})\nintroduce the\nalgebra $\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\\subset \\mbox{\\g g}(\\cp,A)$ consisting of all\nfunctions\ntaking values in $\\mbox{\\g b}_{\\infty}$ at the point $\\infty$.\n\n Of course\n$\\mbox{\\g g}(\\cp,A,A_{1})\\subset\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})$ is an ideal and\n$\\mbox{dim }\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\/\\mbox{\\g g}(\\cp,A,A_{1})=1$. Define\n$\\bar{h}_{\\infty}$\nto be a basis element of $\\mbox{dim\n}\\bar{\\mbox{\\g g}}(\\cp,A,A_{1})\/\\mbox{\\g g}(\\cp,A,A_{1})$.\nIt is a standard (and simple) fact of Lie algebra cohomology theory that\n$\\bar{h}_{\\infty}$ acts on $(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}$.\n\n\\begin{lemma}\n\\label{cohom_for_Weyl}\n\n\n Let $M^{A}$ be a generalized Weyl module.  The element $\\bar{h}_{\\infty}$\n has a simple spectrum as an operator acting on\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})} $. Further, if\n$M^{A}=(V_{r_{1}}^{\\alpha},V_{s_{1}})\\otimes (V_{r_{2}}^{\\beta},V_{s_{2}})$\nthen the\nset of eigenvalues\nof $\\bar{h}_{\\infty}$ is the set of the highest weights of the modules\nappearing in\nthe right-hand side of Theorem\\ref{fusalggencase}(ii).\n\\end{lemma}\n\nProof of this lemma is essentially the same as that of Theorem 4.4 in\n\\cite{feig_mal1} and\nmostly consists of solving a system of 2 equations related to 2 singular\nvectors -- one\nin $(V_{r_{1}}^{\\alpha},V_{s_{1}})$, another in\n$(V_{r_{2}}^{\\beta},V_{s_{2}})$.\nWe will discuss it in \\ref{prooflemma49}. Derivation\nof Theorem \\ref{fusalggencase} from Lemma \\ref{cohom_for_Weyl} is again very\nsimilar\nto that of Theorem 3.2 from Theorem 4.4 in {\\em loc. cit}\nand uses Verma modules as follows.\n\n\\begin{lemma}\n\\label{Berma-Weyl-gener}\n\n(i) Let $(M^{A})^{\\mu}_{\\mbox{\\g g}(\\cp,A,A_{1})}\\in (M^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})} $\n be the eigenspace related to the eigenvalue\n$\\mu$ of $\\bar{h}_{\\infty}$. Then $(M^{A})^{\\mu}_{\\mbox{\\g g}(\\cp,A,A_{1})}\\approx\n(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}$.\n\n(ii) Projection of a Verma module $M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}$ onto a\ngeneralized Weyl module $ W$ induces an isomorphism\nof the coinvariants\n\\[(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup\nA_{1})}\\approx\n(M^{A}\\otimes W)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}.\\]\n\\end{lemma}\n\n{\\bf Proof of Lemma\\ref{Berma-Weyl-gener}}\n\n(i) A Verma module sitting at a point is induced from the 1-dimensional\nrepresentation of the\n algebra\nof functions on the formal disk whose value at the point belong to the\ncorresponding Borel\nsubalgebra. Therefore (i) follows from Frobenius duality.\n\n\n\n\n\n(ii)  Consider the resolution of $W$ by Verma modules (see \\ref {Case_(i)},\nformula\n(\\ref{exactseqforweyl}) ):\n\\[0\\rightarrow M\\rightarrow M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}\\rightarrow\nW\\rightarrow 0\\]\nand tensor it with $M^{A}$. There arises the long exact sequence of homology\ngroups of which\nwe consider the following part:\n\n\\[(M^{A}\\otimes M)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}\\rightarrow\n(M^{A}\\otimes M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k})_{\\mbox{\\g g}(\\cp,A\\cup\nA_{1})}\\rightarrow\n(M^{A}\\otimes W)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}\\rightarrow 0.\\]\n\nSince $M^{\\infty,\\mbox{\\g b}_{\\infty}}_{\\mu,k}$ projects onto a Weyl module, the Verma\nmodule $M$\ndoes not, see \\ref{Case_(i)}. Lemma\\ref{cohom_for_Weyl} and  now give that\n$(M^{A}\\otimes M)_{\\mbox{\\g g}(\\cp,A\\cup A_{1})}=\\{0\\}.$ $\\Box$\n\n\\bigskip\n\nTo complete the proof of Therorem\\ref{fusalggencase} observe that\nLemma \\ref{cohom_for_Weyl}  and Lemma \\ref{Berma-Weyl-gener} together is a\nreformulation of\nTherorem\\ref{fusalggencase}. $\\Box$\n\\bigskip\n\n\\begin{corollary} Let $\\sharp A=1$ and let $A_{1}$ and $A_{2}$\nbe as in \\ref{Finiteness_of_coinvariants}. The following conditions are\nequivalent\n\\label{oncompseries-gener}\n\n(i) $M^{A}$ is a direct sum of generalized Weyl module;\n\n(ii)$SSM^{A}=\\Omega^{A}_{reg}\\cup\\sigma\\Omega^{A}_{reg}$;\n\n(iii) For any Verma module $W^{A_{1}}$\n$\\mbox{dim}(M^{A}\\otimes W^{A_{1}})_{\\mbox{\\g g}(\\cp,A\\cup A_{1},A_{2})}<\\infty$.\n\\end{corollary}\n\n\\subsubsection{ }\n\\label{prooflemma49}\nHere we sketch the proof of Lemma \\ref{cohom_for_Weyl}. First of all replace\n$M^{A}$ with\nthe corresponding Verma module -- $\\bar{M}^{A}$. Then\npass from the space\n$(\\bar{M}^{A})_{\\mbox{\\g g}(\\cp,A,A_{1})}$ to its dual, that is to the space\nof $\\mbox{\\g g}(\\cp,A,A_{1})-$invariant functionals on $\\bar{M}^{A}$.\nChoose $h\\otimes (1-z^{-1})$ to be a representative of $\\bar{h}_{\\infty}$. Let\n$\\Psi$ be the\n eigenvector of $h\\otimes (1-z^{-1})$. By definition $\\Psi$ is a linear\nfunctional on\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}\\otimes M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$. It is\nan excersise\non Frobenius duality to show that such a functional exists and unique.\n\n Define $F$ to be the\nfollowing linear\nfunctional on $M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$: $F(w)=\\Psi(w\\otimes\nv_{\\lambda_{1}})$, where,\nas usual, $v_{\\lambda_{1}})$ is the vacuum vector of\n$M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$. As\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$ is $\\nz_{+}\\times\\nz_{+}-$graded (see\n\\ref{defofalgebrandmodusu}),\nwe denote by $F_{ij}$ the restriction of $F$ to the $(i,j)-$component. Direct\ncalculations show that\nwith respect to the natural action of $\\mbox{$\\hat{\\gtg}$}$ on\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$:\n\n\\begin{eqnarray}\n\\label{strofloopmodact}\n& &\\oplus_{i,j\\in\\nz}\\mbox{${\\bf C}$} F_{ij}\\approx \\mbox{${\\cal F}$}_{\\alpha\\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\\\\n& &\\mbox{where }\\alpha=\\frac{\\lambda_{\\infty}-\\lambda_{1}-\\lambda_{0}-2}{2},\\;\n\\beta=\\lambda_{1}\n\\end{eqnarray}\n\nThe functional $F$ factors through the projection\n$M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}\\rightarrow V_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$ if\nand only if\nit vanishes on the singular vector of $M_{\\lambda_{0},k}^{0,\\mbox{\\g b}_{0}}$. In\nother words, if this\nsingular vector, say $S$, has degree $(i,j)$ then the following equation holds\n\\[ S F_{ij}=0.\\]\nThe latter equation can be  written down and solved explicitly using formulas\n(\\ref{actonloops1} or\n\\ref{actonloops2}). Similar arguments go through for the module\n$M_{\\lambda_{1},k}^{1,\\mbox{\\g b}_{1}}$\ngiving another equation, say\n\\[ S' F_{i'j'}=0.\\]\n\nSimultaneous solutions to these 2 equations give the desired result.\nBy the way, as (\\ref{actonloops1},\n\\ref{actonloops2}) show, each of the expressions $ S F_{ij},\\; S' F_{i'j'}$\nsplits in a product of linear factors; therefore geometrically the solution is\na collection of intersection points of 2 families of lines in the plane. $\\Box$\n\n\n\n\n\\subsubsection {The rational level case}\n\\label{The_rational_leve_case}\nSuppose $k+2=p\/q$, $p\\mbox{ and }q$\n being positive integers.\nNow instead of 3 generalized Weyl modules sitting at 3 points in $\\cp$ we are\ngiven\n3 admissible representations sitting at 3 points on $\\cp$.\nRecall, see \\ref{Case_(ii)}, that  admissible representations\nare parametrized by symbols $(V_{m}^{\\epsilon},V_{n}),\\;0\\leq m\\leq q-1,0\\leq\nn\\leq p-2$ modulo the relation\n$(V_{m}^{\\epsilon},V_{n})=(V_{q-1-m}^{\\epsilon+1},V_{p-2-n})$. Denote\nby $(V_{m}^{\\epsilon},V_{n})^{\\sim }$  an equivalence class\nof $(V_{m}^{\\epsilon},V_{n})$. We assume that $(V_{m}^{\\epsilon},V_{n})^{\\sim}$\nsatifies\nthe same bilinear condition  $(V_{m}^{\\epsilon},V_{n})$ in\nTheorem\\ref{fusalggencase} does.\n\nThe definition of the fusion (Verlinde) algebra in this case repeats\nword for word that in \\ref{Thegenericlevelcase}.\n\n\nRecall finally that  Kazhdan-Lusztig fusion functor \\cite{kazh_luszt} gives\n\\[V_{r}\\dot{\\otimes}_{k}V_{s}= V_{|m-n|}\n\\oplus V_{|m-n|+2}\\cdots \\oplus V_{\\mbox{min}\\{2k-r-s,r+s\\}}.\\]\nThe following theorem was proved in \\cite{feig_mal1}\nin an equivalent but much less illuminating form.\n\n\\begin{theorem}\n\\label{fusalgratcase}\n(i) For any triple of admissible representations\nthe space $(V_{m_{i}}^{\\epsilon_{i}},V_{n_{i}}))_{\\mbox{\\g g}(\\cp,A)}$ is finite\ndimensional.\n\n(ii) The fusion algebra is well-defined, multiplication being given by the\nfollowing formula\n\\begin{eqnarray}\n(V_{r_{1}}^{\\alpha},V_{s_{1}})^{\\sim }\\circ(V_{r_{2}}^{\\beta},V_{s_{2}})^{\\sim\n}&=&\\nonumber\\\\\n(V_{|r_{1}-r_{2}|}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2} V_{s_{2}})^{\\sim\n}&+&\n(V_{|r_{1}-r_{2}|+1}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2}\nV_{s_{2}})^{\\sim }+\n(V_{|r_{1}-r_{2}|+2}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2}\nV_{s_{2}})^{\\sim }+\\cdots +\n\\nonumber\\\\\n(V_{N}^{\\alpha+\\beta},V_{s_{1}}\\dot{\\otimes}_{p-2} V_{s_{2}})^{\\sim }&\n&\\nonumber,\n\\end{eqnarray}\nwhere $N=\\mbox{min}\\{2q-2-r-s,r+s\\}$.\n\\end{theorem}\n It is an easy exercise to derive this theorem from Theorem\n\\ref{fusalggencase}. For future\npurposes, however, we now sketch its original proof.  Set\n$A=\\{(\\infty,\\mbox{\\g b}_{\\infty}\\}$,\n$A_{2}=\\{(0,\\mbox{\\g b}_{0}),(1,\\mbox{\\g b}_{1})\\}$. In addition to the algebras\n$\\mbox{\\g g}(\\cp,A),\\mbox{\\g g}(\\cp,A,A_{2})$\nas in \\ref{Finiteness_of_coinvariants}, we introduce an algebra\n$\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\\subset\\mbox{\\g g}(\\cp,A)$. The latter consists of all\nfunctions whose values\nat the points $0$ ($1$ resp.) belong to $\\mbox{\\g b}_{0}$ ($\\mbox{\\g b}_{1}$ resp.).\nObviously\n$\\mbox{\\g g}(\\cp,A,A_{2})\\subset\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})$ is an ideal and the quotient\nalgebra\n$\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\/\\mbox{\\g g}(\\cp,A,A_{2})$ is commutative and 2-dimensional.\nThis algebra naturally operates on the space\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}$. Let $\\bar{h}_{0},\n\\bar{h}_{1}$ be a basis of $\\bar{\\mbox{\\g g}}(\\cp,A,A_{2})\/\\mbox{\\g g}(\\cp,A,A_{2})$.\n\n\\begin{lemma}(\\mbox{\\cite{feig_mal1}})\n\\label{cohom_for_admiss}\n\n (i) $\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$ if and only if $M^{A}$ is\n  an admissible representation.\n\n(ii) Let $M^{A}$ be an admissible representation. The elements\n$\\bar{h}_{0},\n\\bar{h}_{1}$\n have simple spectra\nas  operators acting on\n$(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}$. Their eigenvalues recover the structure\nconstants of the\nfusion algebra.\n\\end{lemma}\n\n``Inserting'' Verma modules and using BGG resolution one derives Theorem\n\\ref{fusalgratcase}\nfrom Lemma\\ref{cohom_for_admiss} in  a way similar to that we used in\n\\ref{Thegenericlevelcase}.\n\nAnother important corollary of Lemma\\ref{cohom_for_admiss} is as follows.\n\n\\begin{corollary} Let $\\sharp A=1$ The following conditions are equivalent\n\\label{oncompseries-ratin}\n(i) $M^{A}$ is a sum of admissible representations;\n\n(ii)$SSM^{A}=\\Omega^{A}_{nilp}\\cup\\sigma\\Omega^{A}_{nilp}$;\n\n(iii)\n$\\mbox{dim}(M^{A})_{\\mbox{\\g g}(\\cp,A,A_{2})}<\\infty$.\n\\end{corollary}\n\n\\subsubsection {Classical and quantum $\\mbox{\\g osp}(1|2)$. Fusion algebra as a\nGrothendieck ring}\n\\label{fusalgasgrotring}\n\n{\\bf A. } $\\mbox{\\g osp}(1|2)$ is a rank 1 superalgebra -- one of the superanalogues\nof $\\mbox{\\g sl}_{2}$. It can be defined as an algebra on 2 odd generators,\n$x_{+},x_{-}$, one even generator, $h$, and relations\n\\[[x_{+},x_{-}]=h,\\; [h,x_{\\pm}]=\\pm x_{\\pm}.\\]\n\nEven part of this algebra is $\\mbox{\\g sl}_{2}$ and is generated by $x_{\\pm}^{2}$;\nodd part is $V_{1}$ as an $\\mbox{\\g sl}_{2}-$ module, its basis is $x_{+},\\; x_{-}$.\n\n From this it is easy to obtain the following classification of all simple\nfinite dimensional $\\mbox{\\g osp}(1|2)-$modules. (It is even simpler to do this in the\nway\nmodelling the $\\mbox{\\g sl}_{2}-$case -- by starting with Verma modules and then\nquotienting out a singular vector; for details see \\cite{kul_resh}). Each\n$\\mbox{\\g osp}(1|2)-$module  $W$is a sum of an even and odd part\n$W=^{even}W\\oplus^{odd}W$;\neach $^{\\cdot}W$ is an $\\mbox{\\g sl}_{2}-$module, i.e. direct sum of $V_{n}$'s. These\nare generalities. But in reality each irreducible $\\mbox{\\g osp}(1|2)-$module is of\none of the 2 following\ntypes:\n\\[ V_{n}^{0} \\mbox{ such that } ^{even}V_{n}^{0}=V_{n},\\;\n^{odd}V_{n}^{0}=V_{n-1};\\]\n\n\\[ V_{n}^{1} \\mbox{ such that } ^{even}V_{n}^{0}=V_{n-1},\\;\n^{odd}V_{n}^{0}=V_{n}.\\]\n\nThe fact that the dimensions of the even and odd parts are different by 1 is a\nconsequence  of the fact  that odd part of the algebra is $V_{1}$.\n\nWe see that each irreducible $\\mbox{\\g osp}(1|2)-$module is odd-dimesional; further\n$V_{n}^{0}$ and $V_{n}^{1}$ are isomorphic as modules and obtained from each\nother by the change of parity. This is the category of finite dimensional\nrepresentations of $\\mbox{\\g osp}(1|2)$; denote it $Rep(\\mbox{\\g osp}(1|2))$. As in the\n$\\mbox{\\g sl}_{2}-$case, one proves that $Rep(\\mbox{\\g osp}(1|2))$ is semisimple.\n\nThe universal enveloping algebra $U\\mbox{\\g osp}(1|2)$ is in fact a Hopf algebra, for\nexample the comultiplication is given by the standard formula $g\\mapsto\ng\\otimes 1+1\\otimes g,\\; g\\in\\mbox{\\g osp}(1|2)$. This makes $Rep(\\mbox{\\g osp}(1|2))$ a\ntensor category: $Rep(\\mbox{\\g osp}(1|2))\\times Rep(\\mbox{\\g osp}(1|2))\\rightarrow\nRep(\\mbox{\\g osp}(1|2))$, $A,B\\mapsto A\\otimes B$, where the $\\mbox{\\g osp}(1|2)-$module\nstructure on $A\\otimes B$ is determined through the comultiplication (and the\nrule of sign!). Decomposing the tensor product of 2 irreducible modules one\ngets the Grothendieck ring of $Rep(\\mbox{\\g osp}(1|2))$.\n\n\\begin{lemma}\n\\label{decomposptensprod}\n\\[ V_{r_{1}}^{\\alpha}\\otimes V_{r_{2}}^{\\beta}=\nV_{r_{1}+r_{2}}^{\\alpha+\\beta}+\nV_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1}+\nV_{r_{1}+r_{2}-2}^{\\alpha+\\beta}+\\cdots +\nV_{|r_{1}-r_{2}|}^{\\alpha+\\beta}.\\]\n\\end{lemma}\n{\\bf Proof.} Direct calculations show that $ V_{r_{1}}^{\\alpha}\\otimes\nV_{r_{2}}^{\\beta}$ contains one and only one singular (annihilated by $x_{+}$)\nvector of each weight from $|r_{1}-r_{2}|$ to $r_{1}+r_{2}$ and that the\nsubmodules generated by these vectors are irreducible. Proof is completed by\ncounting dimensions. $\\Box$\n\n\\bigskip\n\n Theorems \\ref{fusalggencase} and \\ref{fusalgratcase} provide us with 2\ncommutative algebras. Here\nwe interprete these algebras as Grothendieck rings of certain categories. Start\nwith the algebra\nof Theorem \\ref{fusalggencase} and denote it $\\mbox{${\\cal A}$}^{gen}$. Obviously\n$\\mbox{${\\cal A}$}^{gen}=\\mbox{${\\cal A}$}_{0}\\otimes\\mbox{${\\cal A}$}$, where $\\mbox{${\\cal A}$}$ is the Grothendieck ring of the\ncategory of\nfinite-dimensional representations of $\\mbox{\\g g}$ (its multiplication law is defined\nby the formula\npreceding Theorem \\ref{fusalggencase}) and $\\mbox{${\\cal A}$}_{0}$ is the algebra with basis\n$V_{i}^{\\alpha},\\;i\\geq 0,\\alpha\\in\\nz\/2\\nz$, multiplication being given by\n\\begin{equation}\n\\label{reprospalg}\nV_{r_{1}}^{\\alpha}\\circ V_{r_{2}}^{\\beta}=\nV_{r_{1}+r_{2}}^{\\alpha+\\beta}+\nV_{r_{1}+r_{2}-1}^{\\alpha+\\beta+1}+\nV_{r_{1}+r_{2}-2}^{\\alpha+\\beta}+\\cdots +\nV_{|r_{1}-r_{2}|}^{\\alpha+\\beta}.\n\\end{equation}\n\nComparing (\\ref{reprospalg}) with Lemma \\ref{decomposptensprod} we get the\nfollowing.\n\n\\begin{proposition}\n\\label{interptofgenlev}\n$\\mbox{${\\cal A}$}_{0}$ is the Grothendieck ring of the category of finite-dimensional\nrepresentations\nof the superalgebra $ \\mbox{\\g osp}(1|2)$.\n\\end{proposition}\n\nAppearance of $\\mbox{\\g osp}(1|2)$ here, although artificial as it may seem to be, has\ndeep reasons  behind it. To see this we will analyze the rational level case\nusing quantized enveloping algebras.\n\n\\begin{remark}\nIt follows from Lemma \\ref{decomposptensprod} that the functor\n$Rep(\\mbox{\\g osp}(1|2))\\rightarrow Rep(\\mbox{\\g sl}_{2})$, $V_{m}^{\\alpha}\\mapsto\nV_{m}\\oplus V_{m-1}$ induces  an epimorhpism of the Grothendieck rings.\n\\label{renonmorh[pring}\n\\end{remark}\n\n \\bigskip\n\n\\bigskip\n\n{\\bf B.} Both $U\\mbox{\\g sl}_{2}$ and $U\\mbox{\\g osp}(1|2)$ admit quantization,\n$U_{t}\\mbox{\\g sl}_{2}$ and $U_{t}\\mbox{\\g osp}(1|2)$ resp.. Let us remind the relevant\nformulas.\nThe Drinfeld-Jimbo (see \\cite{drinf,jimbo}) algebra $U_{t}\\mbox{\\g sl}_{2},\\; t\\in\\mbox{${\\bf C}$}$\nis defined to be an associative\nalgebra on generators $E,F,K^{\\pm 1}$ and relations\n\n\\[EF-FE=\\frac{K-K^{-1}}{t-t^{-1}},\\; KEK^{-1}=t^{2}E,\\; KFK^{-1}=t^{-2}F.\\]\n\n$U_{t}\\mbox{\\g osp}(1|2)$ is similarly defined \\cite{kul_resh} as an associative\nalgebra on generators $X_{+}, X_{-}, K^{\\pm 1}$ and relations\n\n\\[X_{+}X_{-}+X_{-}X_{+}=\\frac{K-K^{-1}}{t-t^{-1}},\\; KX_{\\pm}K^{-1}=t^{\\pm\n1}X_{\\pm}.\\]\n\nThe representation theory of $\\mbox{\\g sl}_{2}$ and $\\mbox{\\g osp}(1|2)$ ``deforms to'' the\nrepresentation theory of $U_{t}\\mbox{\\g sl}_{2}$ and  $U_{t}\\mbox{\\g osp}(1|2)$ resp. We will\n continue denoting by $V_{m}$ the $m+1-$dimensional module over\n$U_{t}\\mbox{\\g sl}_{2}$, and by $V^{0}_{m},\\; V^{1}_{m}$ the 2 $(2m+1)-$ dimensional\nmodules over $U_{t}\\mbox{\\g osp}(1|2)$. For generic $t$ these modules are irreducible,\nthe categories of finite dimensional representations, $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$, generated by these modules are semisimple.\n\nThe deformations $U_{t}\\mbox{\\g sl}_{2}$ and $U_{t}\\mbox{\\g osp}(1|2)$ are especially\nremarkable in that they afford simultaneous deformation of the Hopf algebra\nstructure. We get 2 tensor categories $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$. What has been said implies that the Grothendieck rings\nof $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$ are isomorphic to the Grothendieck rings of the\ncorresponding classical objects.\n\nIf however $t$ is a root of unity, things change dramatically. Suppose for\nsimplicity that $t$ is a primitive $l$-th root of unity, $l$ being odd. Then\n\n\\begin{equation}\n\\mbox{(i) $V_{m}$ is irreducible if and only if $m< l$;}\n\\label{irratroots1}\n\\end{equation}\n\\begin{equation}\n\\label{irratroots2}\n\\mbox{(ii) $V_{m}^{\\epsilon}$ is irreducible if and only if $m< l$.}\n\\end{equation}\n\n(Both statements are proved by direct computations.)\n\nWhat is even more important is that the categories $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$ are no longer semisimple. For example, tensor product\nof 2 irreducible representations is not semisimple. Things, however, are still\nvery\nmuch under control.\n\n\\begin{lemma}\n\\label{decompprodatroots}\nLet $t$ be a primitive $l$-th root of unity, $l$ being odd, $m,n<l$. Then\n\n\\[\\mbox{ (i) } V_{m}\\otimes V_{n}= V_{|m-n |}\n\\oplus V_{|m-n|+2}\\cdots \\oplus V_{\\mbox{min}\\{2(l-1)-m-n,m+n\\}}\\oplus W,\\]\nwhere $W$ is not semisimple.\n\\[\\mbox{ (ii) } V_{m}^{\\alpha}\\otimes V_{n}^{\\beta}= V_{|m-n |}^{\\alpha+\\beta}\n\\oplus V_{|m-n|+1}^{\\alpha+\\beta+1}\\cdots \\oplus\nV_{\\mbox{min}\\{2(l-1)-m-n,m+n\\}}^{\\alpha+\\beta}\\oplus W,\\]\nwhere $W$ is not semisimple.\n\n\n\\end{lemma}\n\n{\\bf Sketch of Proof.} (i) is well-known, see \\cite{resh_tur}. We will however\nreview both cases as at our level of brevity there will no difference between\nthem. First, direct calculations as in the proof of Lemma\n\\ref{decomposptensprod} show that  regardless of $t$ at each weight space there\ncan always be only one singular vector. Now decomposition of Lemma\n\\ref{decomposptensprod}, statements (\\ref{irratroots1}, \\ref{irratroots2})\nand this uniqueness result show that the submodules $V_{l-1+i}$ and $V_{l-1-i}$\n(or $V_{l-1+i}^{\\epsilon}$ and $V_{l-1-i}^{\\epsilon}$), $i\\leq m+n-l+1$ are\nnon-trivially tangled. Other $V_{j}$ coming from generic $t$ are still\nirreducible and appear as direct summands. $\\Box$\n\n\\bigskip\n\n{\\bf Definition.}\n\n(i)  Define $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$ to be subcategories of $Rep(U_{t}\\mbox{\\g sl}_{2})$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))$resp. consisting of direct sums of irreducible modules\n$V_{m}$ (or $V_{m}^{\\alpha}$ resp.), $m< l$.\n\n(ii) Define functors\n\\[Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}\\times Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}\\rightarrow\nRep(U_{t}\\mbox{\\g sl}_{2})^{(l)},\\; A,B\\mapsto A\\dot{\\otimes}B,\\]\n\n\\[Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}\\times Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}\\rightarrow\nRep(U_{t}\\mbox{\\g osp}(1|2))^{(l)},\\; A,B\\mapsto A\\dot{\\otimes}B,\\]\nby taking the usual tensor product and then throwing away $W$ in the right hand\nside of formulas in Lemma \\ref{decompprodatroots}. $\\Box$\n\n\\bigskip\n\nWe get tensor categories $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$.\n\n\\bigskip\n\n\\bigskip\n\n{\\bf C.} It is easy now to interpret the  fusion algebra at the rational level\nin terms of the Grothendieck rings of $Rep(U_{t}\\mbox{\\g sl}_{2})^{(l)}$ and\n$Rep(U_{t}\\mbox{\\g osp}(1|2))^{(l)}$.   In view of Lemma \\ref{decompprodatroots} and\nDefinition\nabove, Theorem \\ref{fusalgratcase} reads as follows.\n\n\\begin{proposition}\n\\label{fusalgeasgrotratlev}\nFusion algebra at the level $k+2=p\/q$ is a quotient of the tensor product\nof the Grothendieck rings of the categories  $Rep(U_{t_{1}}\\mbox{\\g sl}_{2})^{(p-1)}$\nand\n$Rep(U_{t_{2}}\\mbox{\\g osp}(1|2))^{(q)}$. Further, the fusion algebra always contains\nthe Grothendieck ring of $Rep(U_{t_{2}}\\mbox{\\g osp}(1|2))^{(q)}$ via the classes\nof symbols $V_{m}^{\\alpha}, V_{0}$.\n\\end{proposition}\n\n\n\n\n\n\\subsubsection{Kac-Moody vs. Virasoro}\n\\label{kacmoodyvsvir}\nVirasoro algebra, $Vir$, is defined to be a vector space with basis\n$\\{ L_{i},z, \\; i\\in\\nz\\}$ and bracket\n\\[[L_{i},L_{j}]=(i-j)L_{i+j}+\\delta_{i,-j}\\frac{i^{3}-i}{12}z.\\]\nRepresentation theory of Virasoro algebra is to a great extent parallel\nto that of $\\mbox{$\\hat{\\gtg}$}$. We will confine to essentials, making reference to\n\\cite{fei_fuchs}.\n\nOne defines the Verma module $M_{h,c}$, where $(h,c)$ is a highest weight, i.e.\neigenvalues of $L_{0}\\;, z$ resp. determined by the vacuum vector; $c$ is\nsometimes referred to as level. A Verma module\nis reducible if and only if it contains a singular vector.  $M_{h,c}$\ngenerically has no singular vectors.By the Kac determinant formula, there is a\nfamily of hyperbolas labelled by pairs of positive integers $m,n$ in the plane\nwith coordinates $(h,c)$ such that if $M_{h,c}$ contains a singular vector,\nthen $(h,c)$ belongs to one of  these hyperbolas; generically along hyperbolas\nthe singular vector is unique.   Denote the singular vector arising in\n$M_{h,c}$ as\n$(h,c)$ gets on hyperbola with the label $m,n$ by $S_{mn}$. There arises\nthe $Vir$-analogue of the generalized Weyl module $M_{h,c}\/<S_{mn}=0>$.\nAttach to        $M_{h,c}\/<S_{mn}=0>$ the symbol $(V_{n-1},V_{m-1})$. Further,\nfor $c$ fixed there arises a one-to-one correspondence between the\n$Vir$-analogues of generalized Weyl modules and symbols $(V_{n-1},V_{m-1})$.\nThis has all been in precise analogy with \\ref{Case_(i)}.\n\nIt has hardly been written anywhere, but is nevertheless known that the $Vir-$\nanalogue\nof the fusion algebra from \\ref{Thegenericlevelcase}, i.e. at a generic level,\nis as follows:\n\\begin{equation}\n\\label{fusalgforvir}\n(V_{n_{1}},V_{m_{1}})\\circ (V_{n_{2}},V_{m_{2}})=(V_{n_{1}}\\otimes\nV_{n_{2}},V_{m_{1}}\\otimes V_{m_{2}}).\n\\end{equation}\n(The interested reader can prove this result using methods of\n\\cite{feig_fuchs_2}; our treatment of the $\\mbox{$\\hat{\\gtg}$}$-fusion algebra in\n\\ref{Thegenericlevelcase} is also a direct analogue of these.)\n\n\nThere is a functor sending $\\mbox{$\\hat{\\gtg}$}-$modules to $Vir-$modules -- quantum\nDrinfeld-Sokolov reduction. One of the prerequisites for it is a choice of\na nilpotent subalgebra of $\\mbox{\\g sl}_{2}$. The two obvious possibilities are\n$\\mbox{${\\bf C}$} e$ and $\\mbox{${\\bf C}$} f$. Denote the corresponding functors $\\phi_{e}$ and\n$\\phi_{f}$. It can be extracted from \\cite{feig_fr} that  both functors send\ngeneralized Weyl modules to generalized Weyl modules. In our terminology one\ngets\n\n\\[\\phi_{e}:\\begin{array}{lll}\n(V_{m}^{0},V_{n})&\\mapsto& (V_{m},V_{n})\\\\\n(V_{m}^{1},V_{n})&\\mapsto& (V_{m-1},V_{n}),\n\\end{array}\\]\n\n\\[\\phi_{f}:\\begin{array}{lll}\n(V_{m}^{0},V_{n})&\\mapsto& (V_{m-1},V_{n})\\\\\n(V_{m}^{1},V_{n})&\\mapsto& (V_{m},V_{n}),\n\\end{array}\\]\nwhere the symbol $V_{-1}$, if arises, is understood as zero.\n\nThe $Vir-$analogue of admissible representations is the celebrated minimal\nrepresentations. The latter can be defined as quotients of generalized Weyl\nmodules\nby repeating word for word definition of  admissible representations from\n\\ref{Case_(ii)}. It is known that minimal representations\narise only when\n \\[c=c_{pq}=1-\\frac{6(p-q)^{2}}{pq},\\]\nwhere $p,q$ are relatively prime positive integers.\n There are again 2 generalized Weyl modules projecting on\na given minimal representation. Therefore minimal representations are labelled\nby equivalence classes of symbols $(V_{m}, V_{n})$. It can be shown that the\nequivalence relation is as follows: $(V_{m},V_{n})\\approx (V_{q-2-m},\nV_{p-2-n})$ for $c=c_{pq}$. From this and (\\ref{fusalgforvir}) one can easily\ncalculate the fusion algebra. We will not write down the relevant formulas here\nand confine to mentionaing that the algebra is related to the product of\nGrothendieck rings of 2 quantum $U_{t}(\\mbox{\\g sl}_{2})$ at appropriate roots of\nunity in much the same way as the fusion algebra for $\\mbox{$\\hat{\\gtg}$}$ is related to the\nproduct\nof Grothendieck rings of  $U_{t}(\\mbox{\\g osp}(1|2))$ and $U_{t}(\\mbox{\\g sl}_{2})$. Recall\nalso that the $Vir$-fusion algebra  was calculated in \\cite{bpz};\nmathematically acceptable exposition can be found in \\cite{feig_fuchs_2}.\n\n\n\nAnother property of the Drinfeld-Sokolov reduction is that both  $\\phi_{e}$ and\n$\\phi_{f}$ send admissible representations at the level $k=2-p\/q$ of $\\mbox{$\\hat{\\gtg}$}$ to\nminimal representations\nof $Vir$ at the level $c_{pq}$, see \\cite{fkw} .\n\n\\begin{proposition}\n\\label{homosl2vir}\n The functor $\\phi_{e}\\oplus\\phi_{f}$ determines an epimorphism of the\n$\\mbox{$\\hat{\\gtg}$}-$fusion algebra onto the $Vir-$fusion algebra at both generic and\nrational levels.\n\\end{proposition}\n\n{\\bf Proof.} The generic level case follows from Remark \\ref{renonmorh[pring}\nand formula (\\ref{fusalgforvir}) above. In the rational level case, the\nstatement follows from the fact that both, $\\mbox{$\\hat{\\gtg}$}-$ and $Vir$-, fusion algebras\nare obtained from their generic level counterparts by imposing the equivalence\nrelations and the 2 equivalence relations agree with each other.  $\\Box$\n\n\\subsection{Fusion functor.}\nThis part is an announcement, proofs will appear elsewhere\n\nSuppose we have a trivial vector bundle $\\mbox{${\\cal E}$}\\rightarrow\\cp$,\n$A=\\{(P_{1},\\mbox{\\g b}_{1}),(P_{2},\\mbox{\\g b}_{2})\\}$, $B=\\{(P_{3},\\mbox{\\g b}_{3})\\}$, so that\n$(\\mbox{${\\cal E}$},A\\bigsqcup B)$ is generic. There is a construction which to a\n$\\mbox{$\\hat{\\gtg}$}^{A}-$module associates a $\\mbox{$\\hat{\\gtg}$}^{B}-$module. This construction is a\nnatural adjustment of the Kazhdan-Lusztig tensoring \\cite{kazh_luszt} to our\nneeds.\n\nDenote by $\\mbox{\\g g}(\\cp,A,B)$ the subalgebra of $\\mbox{\\g g}(\\cp,A)$ consisting of\nfunctions taking values in $\\mbox{\\g n}_{3}=[\\mbox{\\g b}_{3},\\mbox{\\g b}_{3}]$ -- just like we did\nin \\ref{finofcoinvsferesubs}. For a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $M^{A}$, denote by\n$M^{A}_{N}$ the subspace of $(M^{A})^{\\ast}$ annihilated by\n$\\mbox{\\g g}(\\cp,A,B)^{N}$. Obviously  $M^{A}_{N}\\subset M^{A}_{N+1}$, $N\\geq 1$. Set\n\n\\[F^{A\\rightarrow B}(M^{A})=\\cup_{N\\geq 1} M^{A}_{N}.\\]\n\nOne can show that the vector space\n $F^{A\\rightarrow B}(M^{A})$ affords in a natural way a structure of an\n$\\mbox{$\\hat{\\gtg}$}^{B}-$module at the same level; this is easy to show in the spirit of\n\\cite{kazh_luszt, beil_feig_maz}. Using our methods one can show that\n\n(i) if $M^{A}$ is from the $\\mbox{${\\cal O}$}-$category, or further a generalized Weyl\nmodule, or further an admissible representation, then $F^{A\\rightarrow\nB}(M^{A})$ is also as a $\\mbox{$\\hat{\\gtg}$}^{B}-$module;\n\n(ii) the arising in this way Grothendieck rings coincide with those in Theorem\n\\ref{fusalggencase} or Theorem\n\\ref{fusalgratcase} if the level is generic or rational resp..\n\nThis generalizes the statement for the integrable representations, see\n\\cite{fink}.\n\n{\\bf Problem.} Describe the arising tensoring in the spirit of Kazhdan-Lusztig.\n\n\n \\subsection{Quadratic degeneration}\n\\label{Quadratic_degeneration}\n\n\\subsubsection { }\n\\label{setupquadrdeg}\nThe setup here will the following version of  \\ref{maingenerresults}:\n\n(i) $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ be a family of  curves over  a formal\ndisk  $S$, such that the fiber over the generic point of $S$ (``outside\norigin'')\nis a smooth projective curve, and over the origin, $O$, the fiber is a curve\n$\\mbox{${\\cal C}$}_{O}$.\nwith exactly one quadratic singularity;\n\n(ii) $\\rho_{S}:\\; \\mbox{${\\cal E}$}_{S}\\rightarrow \\mbox{${\\cal C}$}_{S}$ is a rank $2$ vector bundle.\n\nAs in \\ref{maingenerresults}, we complete these data to the localization data\nwith logarithmic singularities, say\n$\\tilde{\\psi}$.\nIn the standard way, Theorem \\ref{existofDmod} rewrites to give a $D-$module\nover $S$ with logarithmic singularities at $O$; call it\n$\\Delta_{\\tilde{\\psi}}(M^{A})$. This is because $Spec(S)$ is $\\mbox{${\\bf C}$}[[t]]$ and\nvector fields\nvanishing at $q=0$ are exactly those which can be lifted to $\\mbox{${\\cal C}$}_{S}$.\n\n\nAlong with the family $\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ consider the family\n$\\bar{\\pi}^{\\vee}:\\;\\mbox{${\\cal C}$}_{S}^{\\vee}\\rightarrow S$, obtained from\n$\\bar{\\pi}:\\;\\mbox{${\\cal C}$}_{S}\\rightarrow S$ by replacing the singular fiber $\\mbox{${\\cal C}$}_{O}$\nwith\nits normalization $\\mbox{${\\cal C}$}_{O}^{\\vee}$ (i.e. be tearing $\\mbox{${\\cal C}$}_{O}$ apart at the\nself-intersection point). There is a projection $\\mbox{${\\cal C}$}_{O}^{\\vee}\\rightarrow\n\\mbox{${\\cal C}$}_{O}$\nand the preimage of the self-intersection point $a\\in C_{O}$ consists\nof 2 points $a_{0}, a_{\\infty}\\in C_{O}^{\\vee}$.\n\nIt is obvious that the datum $\\mbox{${\\cal E}$}\\rightarrow \\mbox{${\\cal C}$}_{S}$\nis equivalent to the data  ``$\\rho^{\\vee}_{S}: \\mbox{${\\cal E}$}_{S}^{\\vee}\\rightarrow\n\\mbox{${\\cal C}$}_{S}^{\\vee}$, equivalence $(\\rho^{\\vee}_{S})^{-1}(a_{0})\\approx\n(\\rho^{\\vee}_{S})^{-1}(a_{\\infty})$''. The localization data with logarithmic\nsingularities $\\tilde{\\psi}$ rewrites to give a ``normalized'' localization\ndata\n$\\psi^{\\vee}$.\n\nIn addition fix 2 different lines $l_{0},l_{\\infty}$ in the fiber of $\\mbox{${\\cal E}$}_{S}$\nover the point $a\\in\\mbox{${\\cal C}$}_{O}$. This determines 2 Borel subalgebras,\n$\\mbox{\\g b}_{0},\\mbox{\\g b}_{\\infty}$ operating in the fiber over $a$.\n\nAfter normalization these additional data determine the line $l_{0}$ and the\nBorel subalgebra $\\mbox{\\g b}_{0}$ operating in the fiber of $\\mbox{${\\cal E}$}_{S}^{\\vee}$ over\n$a_{0}$, as well as the line $l_{\\infty}$ and the Borel subalgebra\n$\\mbox{\\g b}_{\\infty}$ operating in the fiber over $a_{\\infty}$. We also get a\ndistinguished Cartan subalgebra\n$\\mbox{\\g h}=\\mbox{\\g b}_{0}\\cap\\mbox{\\g b}_{\\infty}$. Set\n$A^{\\vee}=A\\bigsqcup\\{(a_{0},\\mbox{\\g b}_{0}),(a_{\\infty},\\mbox{\\g b}_{\\infty})\\}$.\n\nNow with a $\\mbox{$\\hat{\\gtg}$}^{A}-$module $M^{A}$ at the level $k$ and an admissible weight\n  $\\lambda\\in\\mbox{\\g h}^{\\ast}$ we associate the $\\mbox{$\\hat{\\gtg}$}^{A^{\\vee}}-$module\n$M^{A}\\otimes L^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}$.  We get a $D-$module for the\n``normalized''localization data:\n\n \\[\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\n\\begin{proposition}\n\\label{statemonquadrgen}\nGenerically with respect to $l_{0},l_{\\infty}$,\nif $\\Delta_{\\tilde{\\psi}}(M^{A})$ is smooth then\n$\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k})$ is also and there is an isomorphism\nof $D-$modules\n\\[\\Delta_{\\tilde{\\psi}}(M^{A})\\approx\n\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\\end{proposition}\n\n\n\n\\subsubsection { Proof}\n\n(i) Begin with the genus zero case.Observe that the algebra of regular\nfunctions\non the neighborhood of the point $a$ is\n$\\mbox{${\\bf C}$}[t_{0},t_{\\infty}][[t]]\/<t_{0}t_{\\infty}=t>$ where $t$ is a coordinate on\n$S$; $\\mbox{${\\cal C}$}_{O}^{\\vee}$\nin this case is just a union of 2 spheres. Therefore the set $A$ splits in two:\n$A'$ and $A''$, each of which has to do with one of the spheres.\n\nHence the algebra $\\mbox{\\g g}(\\bar{\\pi}^{-1}(s),A)$ can be degenerated into the\nfollowing one\nas $s$ ``approaches'' $O$:\n\n \\[(\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))+\\mbox{\\g h})\\oplus_{\\mbox{\\g h}}\n(\\mbox{\\g h}+\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty})).\\]\n\nMeaning of the last expression is as follows: recall, see\n\\ref{Finiteness_of_coinvariants}, that $\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))$ consists\nof functions regular outside $\\bar{A}$ and sending $P_{0}$ to $\\mbox{\\g n}_{0}$;\n$\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))$ is defined similarly with\n$P_{0},\\mbox{\\g n}_{0}$ replaced with $P_{\\infty},\\mbox{\\g n}_{\\infty}$; further the algebra\n$\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))+\\mbox{\\g h}$ is the algebra of functions sending\n$P_{0}$ to $\\mbox{\\g b}_{0}$, the same is true for\n$\\mbox{\\g h}+\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))$; finally ``$\\oplus_{\\mbox{\\g h}}$''\nmeans\ndirect product over $\\mbox{\\g h}$.\n\nTherefore the coinvariants degenerate into the space\n\\[((M^{A'})_{\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))}\\otimes\n(M^{A''})_{\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))})_{\\mbox{\\g h}},\\]\nwhere $\\mbox{\\g h}$ acts by means of the diagonal embedding; this makes sense as the\nfibers are identified.\n\nBy Proposition \\ref{finofcoinv_general}, the space\n\\[(M^{A'})_{\\mbox{\\g g}(\\cp,A',(P_{0},\\mbox{\\g b}_{0}))}\\otimes\n(M^{A''})_{\\mbox{\\g g}(\\cp,A'',(P_{\\infty},\\mbox{\\g b}_{\\infty}))}\\]\nis finite dimensional. It is easy to extract from Lemma \\ref{cohom_for_Weyl}\nthat as an $\\mbox{\\g h}$-module this space is semisimple and therefore is\nisomorphic to\n\\[\\oplus_{\\lambda}((M^{A}\\otimes M_{\\lambda,k}^{P_{0},\\mbox{\\g b}_{0}}\\otimes\nM_{\\lambda,k}^{P_{\\infty},\\mbox{\\g b}_{\\infty}})_{\\mbox{\\g g}(\\mbox{${\\cal C}$}_{O}^{\\vee},A^{\\vee})}.\\]\nBy Lemma \\ref{cohom_for_admiss}, in the last formula $\\lambda$ can be chosen to\nbe admissible and the Verma modules can be replaced with the corresponding\nadmissible representations.\n\n\n\n\n\nThis proves that $\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k})$ is smooth and gives a morphism\n\n\\[\\Delta_{\\tilde{\\psi}}(M^{A})\\rightarrow\n\\oplus_{\\lambda}\\Delta_{\\psi^{\\vee}}(M^{A}\\otimes\nL^{P_{0},\\mbox{\\g b}_{0}}_{\\lambda,k}\\otimes\nL^{P_{\\infty},\\mbox{\\g b}_{\\infty}}_{\\lambda,k}).\\]\n\nThat this is an isomorphism can be shown in the standard way constructing the\ninverse map using the formal character of $L_{\\lambda,k}$, see\n\\cite{beil_feig_maz}.\n\n(ii) The higher genus case is not much different. For example, pinching makes a\ntorus into  a sphere. Therefore in this case proof is literally the same. It\nalso proves  an analogue of Lemma \\ref{cohom_for_admiss} for a torus. This\nprovides a basis for induction.\n\nIn genus $\\geq 2$\nat an appropriate place instead\nof Proposition \\ref{finofcoinv_general} one has to make reference to\nProposition\n\\ref{highgen_finit_afterpionching} and then use induction. $\\Box$\n\n\\subsubsection{Remarks}\n\\label{remonquadrdeg}\n\n(i) Meaning of Proposition \\ref{statemonquadrgen} is that the dimension of the\ngeneric\nfiber of the $D-$module $\\Delta_{\\psi}(M^{A})$ can be calculated by the usual\ncombinatorial algorithm: by pinching the surface and further inserting\ninserting all possible representations the problem is reduced to the case of a\nsphere with three punctures and in the latter case the complete results are\navailable.\n\n(ii) In the genus 0 case the  analogue of Proposition \\ref{statemonquadrgen}\nfor generalized Weyl modules\nis valid. To see this it is enough to examine  part (i) of the proof and\nconvince oneself that the only requirement on $M^{A}$ used there was that\n$M^{A}$ be generalized Weyl module; in fact at an appropriate place instead of\nLemma \\ref{cohom_for_admiss} one has to use Lemma \\ref{Berma-Weyl-gener}.\n\n(iii) Quadratic degeneration for generalized Weyl modules on the sphere allows\nto write horizontal sections of the corresponding bundle as a product of vertex\noperators. This will be explained in  sect.\\ref{screenopcorrfuncttt}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{{\\bf Screening operators and correlation functions}}\n\\label{screenopcorrfuncttt}\n\nIn this section we will study in detail the situation described in\n\\ref{holondmodoncc}: we have the trivial rank 2 bundle $\\mbox{${\\cal E}$}\\rightarrow\\cp$, a\ngeneralized Weyl module $M^{A}$, and\na holonomic $D-$module $\\Delta(M^{A})$ on the space $\\mbox{${\\bf C}$}^{m}\\times\\mbox{${\\bf C}$}^{m}$ with\nfiber\n$(M^{A})_{\\mbox{\\g g}(\\cp,A)}$. For the reasons which will become clear later\nwe replace this bundle with the dual one, its fiber being\n$((M^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. Denote the corresponding $D-$module by\n$\\Delta(M^{A})^{\\ast}$. Using our results on quadratic degeneration we rewrite\nhorizontal sections of the corresponding bundle with flat connection as  matrix\nelements of vertex operators, which serves the two-fold purpose: we find that\nthe differential equations satisfied by horizontal sections are provided by the\nsingular vectors of the corresponding Verma module and write down integral\nrepresentations for solutions to these differential equations.\n\n\n\n\n\\subsection{\\bf Vertex operators and corelation functions}\n\\label{Vertex_operators_and_corelation_functions}\nAn alternative to the language of coinvariants in the genus zero case is the\nlanguage of {\\em vertex operators}.\n\n{\\bf Definition.} A vertex operator is a $\\mbox{$\\hat{\\gtg}$}-$morphism\n\\begin{equation}\n\\label{defvertoper}\nY:\\;\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{1}\\rightarrow V_{2},\n\\end{equation}\nwhere $\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}$ is a loop module (see\n\\ref{Loop_modules}) and\n $V_{1},V_{2}\\in\\mbox{${\\cal O}$}_{k}$ are highest weight modules. $\\Box$\n\n\\bigskip\n\nIn other words, a vertex operator is an embedding\n $\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}\\hookrightarrow\n\\mbox{Hom}_{\\mbox{${\\bf C}$}}V_{1}\\rightarrow V_{2}$. The space\n$\\mbox{${\\cal F}$}_{\\alpha \\beta}^{\\mbox{${\\bf C}$}^{\\ast}}$ has the basis $\\{F_{ij}=F_{i}\\otimes\nz^{j},\\;i,j\\in\\nz\\}$,\n where $\\{F_{i},\\;i\\in \\nz\\}$ is a basis in $\\mbox{${\\cal F}$}_{\\alpha \\beta}$, see\n\\ref{Loop_modules}.\n Given a vertex operator $Y$, consider the generating function\n\\[Y(x,z)=x^{\\Delta_{1}}z^{\\Delta_{2}}\n\\sum_{i,j=-\\infty}^{\\infty}F_{ij}x^{-i}z^{-j},\\]\nthe ``monodromy\ncoefficients'' $\\Delta_{1},\\Delta_{2}$ are defined by:\n\\[\\Delta_{1}=\\frac{-\\lambda_{2}+\\lambda_{1}+\\beta}{2},\\;\n\\Delta_{2}=\\frac{-C(\\lambda_{2})+C(\\lambda_{1})+C(\\beta)}{2},\\]\nwhere $\\lambda_{i}$ is the highest weight of $V_{i}$ and\n$C(\\lambda)=\\lambda(\\lambda+2)\/2$.\n($\\Delta_{1},\\Delta_{2}$ will later appear as genuine monodromy coefficients of\na certain flat\nconnection.)\n\n The formal series\n$Y(x,z)$ is, of course, an element of $Hom_{\\mbox{${\\bf C}$}}(V_{1},V_{2}\\otimes\n x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},z^{\\pm 1}]])$. Further, for any\n$g\\in\\mbox{\\g g}$ the\ncommutator $[g\\otimes z^{n},Y(x,z)]$ is also a well-defined element of\n$Hom_{\\mbox{${\\bf C}$}}(V_{1},V_{2}\\otimes\n x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},z^{\\pm 1}]])$. For the standard\nbasis of $\\mbox{\\g g}$, see\n\n                                           \\ref{defofalgebrandmodusu}, one\nderives from the definition of a vertex operator that\n\n\\begin{equation}\n[e\\otimes z^{n},Y(x,z)]=z^{n}(-x^{2}\\frac{\\partial}{\\partial x}+\\beta x)Y(x,z),\n\\label{comm_vert_op_1}\n\\end{equation}\n\\begin{equation}\n[f\\otimes z^{n},Y(x,z)]=z^{n}\\frac{\\partial}{\\partial x}Y(x,z),\n\\label{comm_vert_op_2}\n\\end{equation}\n\\begin{equation}\n[h\\otimes z^{n},Y(x,z)]=z^{n}(2x^{2}\\frac{\\partial}{\\partial x}-\\beta x)Y(x,z).\n\\label{comm_vert_op_3}\n\\end{equation}\nWe conclude that for any $g\\in\\mbox{\\g g}$ there is a differential operator $D_{g}(x)$\nin $x$ such that\n\\begin{equation}\n[g\\otimes z^{n},Y(x,z)]=z^{n}D_{g}(x)Y(x,z),\n\\label{comm_vert_op_4}\n\\end{equation}\n\n\nSuppose now we are given a collection of vertex operators\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nThe product of the corresponding generating functions\n$Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})$ is a\nwell-defined element of\n$Hom_{\\mbox{${\\bf C}$}}(V_{1\/2},V_{m+1\/2}\\otimes\n\\prod_{i}x_{i}^{\\Delta_{i,1}}z_{i}^{\\Delta_{i,2}}\n\\mbox{${\\bf C}$}[[x_{1}^{\\pm 1},\\ldots x_{m}^{\\pm 1},z_{1}^{\\pm 1},\n\\ldots z_{m}^{\\pm 1}]])$. The matrix element\n\\[ <v^{\\ast},Y_{m}(x_{m},z_{m}\\cdots\nY_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>,\\;v\\in V_{1\/2},\nv^{\\ast}\\in V_{m+1\/2}^{\\ast}\\]\nis, therefore, a formal Laurent series in $x_{i},z_{i},\\;1\\leq i\\leq m$.\n\n{\\bf Definition} Suppose $Y_{i}(x_{i},z_{i}),\\;1\\leq i\\leq m$  are as above.\nThen the matrix\nelement\n\\begin{equation}\n\\label{defcorrfun}\n\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>\n\\end{equation}\nis called {\\em correlation function} if $V_{1\/2},\\ldots,V_{m+1\/2}$ are\nirreducible generalized\nWeyl modules, $V_{1\/2}$ is the vacuum module,\n$v$ is the highest weight vector of $V_{1\/2}$ and $v^{\\ast}$ is the dual to the\nhighest weight vector of $V_{m+1\/2}$. (The latter condition is meaningful in\nview of the weight\nspace decomposition of a highest weight module.) $\\Box$\n\n\\bigskip\n\nA correlation function has been understood as a formal power series. We will\nshow that, in fact,\nit is a holomorphic function satisfying a certain holonomic system of partial\ndifferential\nequations. In order to do that we  will interpret  vertex opeartors as\nhorizontal sections\nof a line bundle with a flat connection provided by three modules on $\\cp\\times\n\\cp$.\n\n\\subsection {{\\bf From coinvariants to vertex operator algebra}}\n\n\\subsubsection{ }\n\\label{coinv_corrfunsn}\n\nWe return to the setup of \\ref{Thegenericlevelcase}. In the cartesian product\n$\\cp\\times\\cp$\nfix coordinate system $(x,z)$. Attach to the point $x$ in the first factor the\nBorel subalgebra\n$\\mbox{\\g b}_{x}$ spanned by the vectors $e_{x}=e-xh-x^{2}f,\\; h_{x}=h+2xf$. This\nmeans, in particular,\nthat $\\mbox{\\g b}_{0}$ is the standard Borel subalgebra $\\mbox{${\\bf C}$} e\\oplus\\mbox{${\\bf C}$} h$\n(see  \\ref{defofalgebrandmodusu} ) and $\\mbox{\\g b}_{\\infty}$ is the opposite one. Set\n$A=\n\\{(0,0),(x,z),(\\infty,\\infty)\\}$. Let $V^{A}=V^{\\mbox{\\g b}_{0},0}_{0}\\otimes\nV^{\\mbox{\\g b}_{x},z}_{1}\n\\otimes V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty}$ be a generalized Weyl module over\n$\\mbox{$\\hat{\\gtg}$}^{A}$.\nConsider the space of invariants $((V^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. By Theorem\n\\ref{fusalggencase}\n this\nspace is either 0- or 1-dimensional. Suppose the latter possibility is the\ncase. Then by\nTheorem \\ref{smoothinadm} we get a line bundle with flat connection over\n$\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}^{\\ast}$ whose fiber over the point $(x,z)\\in\n\\mbox{${\\bf C}$}^{\\ast}\\times \\mbox{${\\bf C}$}^{\\ast}$ is\n$((V^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A)}$. There arises an embedding\n\\[V^{\\mbox{\\g b}_{x},z}_{1}\\hookrightarrow \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0}\\otimes\n V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty},\\mbox{${\\bf C}$}).\\]\n\nThe dual space $(V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty})^{\\ast}$ as a $\\mbox{$\\hat{\\gtg}$}-$module\nis isomorphic\nto the contragredient module $(V^{\\mbox{\\g b}_{\\infty},\\infty}_{\\infty})^{c}$,\nsee \\ref{defofalgebrandmodusu}. As the level is generic,\nthe latter module is irreducible and is, therefore, isomorphic to a certain\ngeneralized Weyl\nmodule\n$V^{\\mbox{\\g b}_{0},0}_{\\infty}$. Hence we get an embedding\n\\[V^{\\mbox{\\g b}_{x},z}_{1}\\hookrightarrow \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm 1},\nz^{\\pm 1}]]),\\]\nwhere $\\Delta_{1},\\Delta_{2}$ are monodromy coefficients of the flat\nconnection. We conclude that\nany $w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ can be looked upon as a certain generating\nfunction\n$w(x,z)=x^{\\Delta_{1}-n}z^{\\Delta_{2}-l}\\sum_{i,j\\in\\nz}w_{ij}x^{-i}z^{-j}$ of\na family of\noperators $\\{w_{ij}\\subset \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty})$, where $(n,l)$ is a bidegree of $w$ as an element of\n$V^{\\mbox{\\g b}_{x},z}_{1}$.\n\n\\begin{lemma}\n\\label{vacuum_vertoper}\nSuppose $v_{1}\\in V^{\\mbox{\\g b}_{x},z}_{1}$ is the highest weight vector. Then\n\n(i) $v_{1}(x,z)$ is a generating function of a certain vertex operator as in\n\\ref{Vertex_operators_and_corelation_functions};\n\n(ii) any vertex operator is obtained in this way.\n\\end{lemma}\n\n{\\bf Proof} is a direct and simple  calculation using definitions, see also\n \\ref{prooflemma49} formula  (\\ref{strofloopmodact}). $\\Box$\n\n\\bigskip\n\n\n\nLet us now relate correlation functions to horizontal sections of the bundle\nbuilt on the generalized Weyl module $M^{A}$, $\\Delta(M^{A})^{\\ast}$, see\nbeginning of sect.\\ref{screenopcorrfuncttt} for notations.\nSuppose that $M^{A}$ is the tensor product of ``individual'' generalized Weyl\nmodules \\[\\otimes_{i=1}^{m} V_{i}^{z_{i},\\mbox{\\g b}_{x_{i}}}.\\] Consider all possible\ncorrelation functions\n\\[<v^{\\ast}, v_{m}(x_{m},z_{m})\\cdots v_{1}(x_{1},z_{1})v>,\\]\nwhere $v_{i}(x_{i},z_{i})$ is a generating function of a vertex operator\nrelated\nto the highest weight vector $v_{i}\\in V_{i}^{z_{i},\\mbox{\\g b}_{x_{i}}}$.\n\\begin{corollary}\n\\label{betwcorrfunandhorizsecttt}\nLet $M^{A}$ be as above.\nOver a suitable  open contractible subset $U$ of $\\mbox{${\\bf C}$}^{m}\\times\\mbox{${\\bf C}$}^{m}$, there\nis\nan isomorphism between the space of horizontal sections of the  bundle\n$\\Delta(M^{A})^{\\ast}$ and the space of correlation functions\n\\[<v^{\\ast}, v_{m}(x_{m},z_{m})\\cdots v_{1}(x_{1},z_{1})v>.\\]\n\\end{corollary}\n\n{\\bf Proof.} Intertwining properties of vertex operators imply a correlation\nfunction is a horizontal section of $\\Delta(M^{A})^{\\ast}$ in a formal sense.\nThis give a map in one direction. A map in the opposite direction in provided\nby\nquadratic degeneration, see Proposition \\ref{statemonquadrgen}. $\\Box$\n\n\n\n\\subsubsection{ }\n\\label{vertop_opervertalg}\nBy Lemma \\ref{vacuum_vertoper} coinvariants recover vertex operators. In fact\nthey give us much more: the\ncollection of generating functions $w(x,z),\\;w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ affords a\nkind of\n{\\em vertex operator algebra} structure. We will not discuss the latter in\ndetail\n(see \\cite{fren_lep_meur}) and only explain how one can get exlicit formulas\nfor\n$w(x,z),\\;w\\in V^{\\mbox{\\g b}_{x},z}_{1}$ in terms of the vertex operator $v_{1}(x,z)$\nrelated to the\nhighest weight vector $v_{1}$.\n\nFor any $g\\in\n\\mbox{\\g g}$ set $g(i)=g\\otimes z^{i}\\in\\mbox{$\\hat{\\gtg}$}$. Define the {\\em current} $g(z)$ to be\n$g(z)=\\sum_{i\\in\\nz}g(i)z^{-1-i}\\in\\mbox{$\\hat{\\gtg}$}\\otimes\\mbox{${\\bf C}$}[[z^{\\pm 1}]]$. Define\n$g(z)^{(l)}$ to be\nthe $l-$th (formal) derivative of $g(z)$ with respect to $z$.\n For any  $g(z)^{(l)}$ set\n\\[(g(z)^{(l)})_{+}=(\\frac{d}{dz})^{l}\\sum_{i=0}^{\\infty}g_{-i-1}z^{i},\\;\n(g(z)^{(l)}_{-}=g(z)^{(l)}-(g(z)^{(l)})_{+}.\\]\nObserve that for any $w(x,z)\\in \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$\nand any $g\\in\\mbox{\\g g}$, the\nproducts $(g(z)^{(l)})_{-}w(x,z)$, $w(x,z)(g(z)^{(l)})_{+}$ are also\nwell-defined elements of\n$\\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$.\n\n\n\nDefine for any $g\\in\\mbox{\\g g},\\;w(x,z)\\in \\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$\n\\begin{equation}\n\\label{normalordering}\n:g(z)^{(k)}w(x,z):=(g(z)^{(k)})_{-}w(x,z)+w(x,z)(g(z)^{(k)})_{+}.\n\\end{equation}\n\n\\begin{lemma}\n\\label{descendants}\n Let $g\\in\\mbox{\\g g}$, $w\\in V^{\\mbox{\\g b}_{x},z}_{1}$, $w(x,z)$ the corresponding element\nof $\\mbox{Hom}_{\\mbox{${\\bf C}$}}(V^{\\mbox{\\g b}_{0},0}_{0},\nV^{\\mbox{\\g b}_{0},0}_{\\infty}\\otimes x^{\\Delta_{1}}z^{\\Delta_{2}}\\mbox{${\\bf C}$}[[x^{\\pm\n1},z^{\\pm 1}]])$. Then\n\n(i) $(g\\cdot w)(x,z)=[g,w(x,z)]$;\n\n(ii) $(g(-l)\\cdot w)(x,z)=(1\/(l-1)!):g(z)^{(l-1)}w(x,z):,\\;l>0$;\n\\end{lemma}\n\n{\\bf Proof} is a direct calculation of matrix elements of the operator\n$(g(-l)\\cdot w)(x,z)$\nbased on the definition of the space of coinvariants. $\\Box$\n\n\\subsection{ Differential equations satisfied by correlation functions}\n\\label{diffeqsatbycorrfuntttt}\nWe return to the setup of \\ref{Vertex_operators_and_corelation_functions} and\nconsider\na correlation function\n\\[\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},Y_{m}(x_{m},z_{m})\\cdots Y_{2}(x_{2},z_{2})Y_{1}(x_{1},z_{1})v>,\\]\ncoming from the  product of vertex operators\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nUsing Lemma \\ref{vacuum_vertoper} we assume that there are generalized Weyl\nmodules\n$V_{i},\\;1\\leq i\\leq m$ with highest weight vectors $v_{i},\\;1\\leq i\\leq m$\nsuch that\n$Y_{i}(x,z)=v_{i}(x,z)$. An advantage of this point of view is that for any\ncollection\nof elements $w_{i}\\in V_{i},\\;1\\leq i\\leq m$ we can consider the matrix element\n\\[<v^{\\ast},w_{m}(x_{m},z_{m})\\cdots w_{2}(x_{2},z_{2})w_{1}(x_{1},z_{1})v>.\\]\n\n\\begin{lemma}\n\\label{fromprimtodesc}\nFor any\n $w_{i}\\in V_{i},\\;1\\leq i\\leq m$\n\n\\[<v^{\\ast},w_{m}(x_{m},z_{m})\\cdots w_{2}(x_{2},z_{2})w_{1}(x_{1},z_{1})v>=\nD\\cdot \\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m}),\\]\nwhere $D$ is a differential operator in $x'$s with coefficients in rational\nfunctions in $z'$s.\n\\end{lemma}\n\n{\\bf Proof.} Start  with the function\n\\[<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(g(-l)v_{i})(x_{i},z_{i})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;l>0.\\]\nBy Lemma \\ref{descendants} (ii) it rewrites as\n\\[\n<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(g(z)^{(l-1)}_{-}v_{i}(x_{i},z_{i})-\nv_{i}(x_{i},z_{i})g(z)^{(l-1)}_{+})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;l>0.\\]\nThen commute all $g_{i},\\; i<0$ through to the right and all $g_{i},\\; i\\geq 0$\nto the left in\na standard way, c.f.\\cite{fren_resh}  and use commutation relations\n(\\ref{comm_vert_op_1},\\ref{comm_vert_op_2},\\ref{comm_vert_op_3}). The case\n$l=0$ is treated in\n a similar\nand\nsimpler way using Lemma \\ref{descendants} (i). Further argue by induction using\nagain\nLemma \\ref{descendants}. $\\Box$\n\n\\bigskip\n\nBy definition each $V_{i}$ is a quotient of a Verma module and therefore there\nare elements, singular vectors in the corresponding Verma module (see\n\\ref{defofalgebrandmodusu})\n$S_{i}\\in U(\\mbox{$\\hat{\\gtg}$})$ such that $S_{i}v_{i}=0,\\;1\\leq i\\leq m$. On the other\nhand,\nby Lemma \\ref{fromprimtodesc} there are differential operators $D_{i},\\;1\\leq\ni\\leq m$ such that\n\\begin{eqnarray}\n& &D_{i}<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{2}(x_{2},z_{2})v_{1}(x_{1},z_{1})v>=\\nonumber\\\\\n& &<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots\nv_{i+1}(x_{i+1},z_{i+1})(S_{i}v_{i})(x_{i},z_{i})\nv_{i-1}(x_{i-1},\nz_{i-1})\\cdots v_{1}(x_{1},z_{1})v>,\\;1\\leq i\\leq m.\\nonumber\n\\end{eqnarray}\nWe arrive to the following result.\n\n\\begin{lemma}\n\\label{firsthalfequat}\nThe correlation function\n\\[\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=\n<v^{\\ast},v_{m}(x_{m},z_{m})\\cdots v_{2}(x_{2},z_{2})v_{1}(x_{1},z_{1})v>\\]\nsatisfies the system of equations\n\\begin{equation}\n\\label{firsthalfequat_eq}\nD_{i}\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})=0,\\;1\\leq i\\leq m.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\bigskip\n\nObserve that, although there are in general no explicit formulas for $D_{i}$,\nthe fact that\n$[D_{i},D_{j}]=0$ is an obvious consequence of the definition.\n\nWe have obtained $m$ equations our function of $2m$ variables satisfies. The\nrest is, of course,\n the Knizhnik-Zamolodchikov equations. Let us write them down explicitly.\n Recall that we can look upon\n$\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ as a function of $z_{1},\\ldots\nz_{m}$ with\ncoefficients in a completed tensor product of $m$ $\\mbox{\\g g}-$modules.\n(The variables $x_{1}\\ldots x_{m}$\nare responsible for that, see\n(\\ref{comm_vert_op_1},\\ref{comm_vert_op_2},\\ref{comm_vert_op_3}).)\n For any $A=\\sum_{s}a_{i}\\otimes b_{s}\n\\in\\mbox{\\g g}\\otimes\\mbox{\\g g}$ denote by $A_{ij},1\\leq i,j\\leq m$\nan operator  acting on the $m-$fold tensor product of $\\mbox{\\g g}$-modules by the\nformula\n\\[A_{ij}\\cdot w_{1}\\otimes\\cdots w_{m}=\\sum_{s}w_{1}\\otimes\na_{s}w_{i}\\otimes\\cdots b_{s}w_{j}\n\\otimes\\cdots w_{m}.\\]\nThe formula (\\ref{comm_vert_op_4}) implies that $A_{ij}$ is a differential\noperator\n in $x_{i},x_{j}$.\nSet $\\Omega=ef+fe+h^{2}\/2$.\n\n\\begin{lemma} (\\cite{knizh_zam})\n\\label{knzameq}\n\nThe correlation function\n$\\Psi=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ satisfies the  system of\nKnizhnik-Zamolodchikov\n equations\n\\begin{equation}\n\\label{knzameq_eq}\n(k+2)\\frac{\\partial}{\\partial z_{i}}\\Psi=\\sum_{j\\neq\ni}\\frac{\\Omega_{ij}}{z_{i}-z_{j}}\\Psi,\\;\n1\\leq i\\leq m.\n\\end{equation}\n\\end{lemma}\n\nThere is no need to prove this lemma here as one can repeat\nword for word the known proofs.\n However we point out that if one considers a highest weight module $V$ as a\nmodule\nover the semi-direct product of $\\mbox{$\\hat{\\gtg}$}$ and the Virasoro algebra $Vir$ then $V$\nis annihilated\nby the element $d\/dz-L_{-1}$, where $L_{-1}$ is one of the Sugawara elements.\nOne then shows that\n  the singular vectors\n$(d\/dz-L_{-1})v_{i}$, where $v$ is a highest weight vector of $V_{i}$, give\nrise to\nthe equations (\\ref{knzameq_eq}) in exactly the same way the singular vectors\n$S_{i}$ gave\nrise to the equations (\\ref{firsthalfequat_eq}). An immediate consequence of\nthis proof is\nthat the system of equations (\\ref{knzameq_eq},\\ref{firsthalfequat_eq}) is\nconsistent.\n\n\n\n\n\\subsection{Screening operators and integral representations of correlation\nfunctions}\n\\label{scr_and_int_repr}\n\\subsubsection{ }\n\\label{motivation}\nSuppose a  function $\\Psi_{old}=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$\n is the matrix element of the product\nof vertex operators\n\\[\\Psi_{old}=<v^{\\ast},\\pi\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{o}>,\\]\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m,\\]\nsatisfying the same conditions as the expression  in (\\ref{defcorrfun}),\nsee\\ref{Vertex_operators_and_corelation_functions}, except\nthat instead of assuming that $V_{m+1\/2}$ is a generalized Weyl module we\nassume that\n$V_{m+1\/2}$ is a contragredient Verma module, see \\ref{defofalgebrandmodusu}.\n(Why ``old'' will become clear in a moment.) It is easy to see that\n $\\Psi_{old}=\\Psi(x_{1},\\ldots ,x_{m},z_{1},\\ldots z_{m})$ satisfies the same\nsystem of equations\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). Suppose in addition that there is a\nprojection\n $\\pi:\\; V_{m+1\/2}\\rightarrow W$ onto another contragredient Verma module $W$.\nDenoting\nby $w^{\\ast}$ an element dual to the highest weight vector $w\\in W$ one can\nconsider the\nmatrix element\n\\[\\Psi_{new}=<w^{\\ast},\\pi\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>.\\]\nWe again observe that $\\Psi_{old}$ is a solution to the same system\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). This new solution can be calculated\n as follows.\n\nThere arises the dual map $\\pi^{\\ast}:\\; W^{\\ast}\\rightarrow V_{m+1\/2}^{\\ast}$\nand by definition\n   there is an element $S$ of $U(\\mbox{$\\hat{\\gtg}$}_{>})$\nsuch that $\\pi^{\\ast}(w^{\\ast})= Sv^{\\ast}$. We now take the definition of\n$\\Psi_{new}$, replace\nin it $\\pi^{\\ast}(w^{\\ast})$ with $Sv^{\\ast}$ and get\n\n\\begin{equation}\n\\label{replachwvect}\n\\Psi_{new}=<S\\cdot v^{\\ast},\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>.\n\\end{equation}\n\n\n Then we  commute $S$ through to the right.\n The intertwining properties of vertex operators tell us\nthat\n\\begin{equation}\n\\label{fromoldtonew}\n\\Psi_{new}=S^{t}\\cdot\\Psi_{old},\n\\end{equation}\nwhere $^{t}$ signifies the canonical\nantiinvolution an a Lie algebra\n($g_{1}g_{2}\\cdots g_{n}\\rightarrow g_{n}g_{n-1}\\cdots g_{1}$)\nand the action  is determined by the following condition: if $g\\in\\mbox{\\g g}$ then\n\\[(g\\otimes\nz^{n})\\cdot\\Psi_{old}=\\sum_{i=1}^{m}D_{g}(x_{i})z_{i}^{n}\\Psi_{old},\\]\nsee (\\ref{comm_vert_op_4}).\n\nWe intend to use (\\ref{fromoldtonew}) in the case when $\\pi$ and therefore $S$\ndo not exist!\n\n\\subsubsection{Screening operators }\nLet $V_{\\lambda_{\\infty},k}$ be a highest weight module and\n$v\\in V_{\\lambda_{\\infty},k}$ a highest weight vector.\n If the obvious integrality conditions are satisfied\nthen the vectors $f^{\\lambda_{\\infty}+1}v,\\;( e\\otimes\nz^{-1})^{k-\\lambda_{\\infty}+1}v$ are\nsingular and give rise to embeddings of the type $W\\hookrightarrow\nV_{\\lambda_{\\infty},k}$.\nNow take 3 highest weight modules $V_{\\lambda_{i},k},\\; i=0,1,\\infty$ attach\nthem\nto 3 point in $\\cp$ and consider the space of coinvariants\n\\[(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}.\\]\nOf course an embedding $W\\hookrightarrow V_{\\lambda_{\\infty},k}$ gives rise to\na map\n\\[(W^{\\mbox{\\g b}_{\\infty},\\infty}\\otimes_{i=0,1}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}\n\\hookrightarrow\n(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}.\\]\nIt is remarkable that even if the embedding $W\\hookrightarrow\nV_{\\lambda_{\\infty},k}$ does not\nexist the last map still does. In the language of vertex operators this\nphenomenon was explained\nin great detail in \\cite{feig_mal}.\n\nTherefore with each of the formal singular vectors --\n$f^{\\lambda_{\\infty}+1}v\\mbox{ or }( e\\otimes z^{-1})^{k-\\lambda_{\\infty}+1}v$\n-- we have\nassociated an operator acting on coinvariants. Call these operators {\\em\nscreenings} and denote\nthem $R_{1}$ and $R_{0}$ respectively.\n\nLet us calculate the action of the  screenings explicitly. By definition\n\\newline\n$R_{j}(\\otimes_{i=0,1,\\infty}V_{\\lambda_{i},k}^{\\mbox{\\g b}_{i},i})_{\\mbox{\\g g}(\\cp,\\{0,1,\\infty\\})}$ only\ndepends on $V_{\\lambda_{\\infty},k}$ so we will be simply writing\n$R_{j}(V_{\\lambda_{\\infty},k})$.\nNow formulas for the related singular vectors\n ($f^{\\lambda_{\\infty}+1}v,\\;( e\\otimes z^{-1})^{k-\\lambda_{\\infty}+1}v$) and a\nvery simple\ncalculation using the formulas (\\ref{param_eq_line_1},\\ref{param_eq_line_2})\ngive the following\nresult:\n\\begin{eqnarray}\n\\label{act_scr_vert_op_1}\nR_{1}((V_{m}^{0},V_{n}))=(V_{m-1}^{1},V_{n})\\\\\n\\label{act_scr_vert_op_2}\nR_{1}((V_{m}^{1},V_{n}))=(V_{m+1}^{0},V_{n})\\\\\n\\label{act_scr_vert_op_3}\nR_{0}((V_{m}^{0},V_{n}))=(V_{m+1}^{1},V_{n})\\\\\n\\label{act_scr_vert_op_4}\nR_{0}((V_{m}^{1},V_{n}))=(V_{m-1}^{0},V_{n})\n\\end{eqnarray}\n\nSuppose we are given 2 generalized Weyl modules and a vertex operator acting\nbetween them.\nSuppose in addition that this vertex operator is related to a highest weight in\nthe third\ngeneralized Weyl module, say $(V_{m}^{\\epsilon},V_{n})$.\nTheorem \\ref{fusalggencase} tells us that given such a vertex operator\nour screenings give us all the others of the type $(V_{i}^{\\alpha},V_{n})$ --\nwe cannot only\nchange the value of $n$.\nBut then there is the standard screening operator -- $S$ -- which takes care of\n$n$, see\ne.g. \\cite{feig_fr_1}. So these three -- $R_{1},R_{2},S$ -- provide us with all\nvertex operators.\n This has\nan important application to the calculation of correlation functions.\n\nStart with a simple correlation function given by the product of vertex\noperators, each of which\nis characterized by the condition $m=0$. Then applying $S$ an appropriate\nnumber\n of times one gets all\nvertex operators and, hence,\nall correlation functions in spirit of Varchenko-Schechtman, see\\cite{awtsyam}.\n\nNow take a Varchenko-Schechtman\ncorrelation function $\\Psi_{old}$. It comes from a product of vertex operators:\n\\[\\Psi_{old}=<v^{\\ast}, Y(x_{m},z_{m})\\cdots Y_{1}(x_{1},z_{1})v_{o}>,\\]\n\n\\[Y_{i}:\\;\\mbox{${\\cal F}$}_{\\lambda_{i} \\mu_{i}}^{\\mbox{${\\bf C}$}^{\\ast}}\\otimes V_{i-1\/2}\\rightarrow\nV_{i+1\/2},\\;\n1\\leq i\\leq m.\\]\n\nLet $W_{i},\\;0\\leq i\\leq m$, be words on 2 letters $R_{1}$ and $R_{2}$.\nReplacing $V_{i+1\/2}$\nwith $W_{i}(V_{i+1\/2})$ we get a new correlation function $\\Psi_{new}$. Doing\nthis with all\n$\\Psi_{old}$ and sufficiently many $W_{i},\\;0\\leq i\\leq m$ we get all solutions\nto\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}). In principle all these solutions\ncan be written\ndown explicitly. It is especially simple to do so in the case when we keep\n$V_{i+1\/2},\\;0\\leq i\\leq m-1,$ and only change $V_{m+1\/2}$.\n\nSo assume that $\\Psi_{old}$ is as above and replace $V_{m+1\/2}$ with\n$R_{j}(V_{m+1\/2}),\\; j=0,1$.\nThen by (\\ref{replachwvect}) one is to expect that\n\\[\\Psi_{new}=<X^{\\alpha}\\cdot v^{\\ast},\\circ Y(x_{m},z_{m})\\cdots\nY_{1}(x_{1},z_{1})v_{0}>,\\]\nwhere $X$ is either $e$ or $f\\otimes z$ if $j=1$ or 0 resp., and $\\alpha$ is\neither $\\lambda+2$\nor $k-\\lambda+2$ resp., where $\\lambda$ is\n the highest weight of $V_{m+1\/2}$.\n\nOf course if $\\alpha$ is not a nonnegative integer  then the last formula does\nnot make much sense.\nNevertheless using it and (\\ref{fromoldtonew}) as a motivation we arrive to\n\\[\\Psi_{new}=X^{\\alpha}\\Psi_{old}.\\]\nNow the left-hand side of the last equality does make sense:\n$X$ is a first order differential operator,\nsee \\ref{motivation}, therefore  we can set in a rather straightforward manner\n\\[X^{\\alpha}\\Psi_{old}=\\int t^{-\\alpha-1}\\{\\exp(-Xt)\\Psi_{old}\\}dt\\]\nand get a nice integral operator, for details see \\cite{feig_mal}.\n\nThis procedure can be easily iterated to provide the functions\n\\begin{equation}\n\\label{our_solut}\n\\int \\prod_{i=1}^{n}t_{i}^{-\\alpha_{i}-1}\\{\\exp(-X_{1}t_{1})\\exp(-X_{2}t_{2})\n\\cdots \\exp(-X_{n}t_{n})\\Psi_{old}\\}\\prod_{i=1}^{n}dt{i},\n\\end{equation}\nwhere $X_{1},X_{2},...$ is either $e,f\\otimes z,e,...$ or $f\\otimes\nz,e,f\\otimes z,...$.\n\n\\begin{lemma}\n\\label{our_sol_stae}\nFunctions (\\ref{our_solut}) are solutions to\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}).\n\\end{lemma}\n\n{\\bf Proof} is  same as the proof of the analogous statement in\n\\cite{feig_mal1}. In fact it is an easy exrcise to make the heuristic arguments\nwhich have lead us to the formula (\\ref{our_solut}) into a precise proof.\n$\\Box$\n\n\\bigskip\n\nIntegrating functions (\\ref{our_solut}) with respect to $x'$s\n(or doing something similar but more esoteric) one is supposed to get the\nDotsenko-Fateev correlation functions for the Virasoro algebra. It would be\ninteresting to do\nthis explicitly and compare the result with the calculations in \\cite{f_g_p_p}.\n\n\\begin{conjecture}\n\n(i) Formulas (\\ref{our_solut}) provide all solutions to the system\n(\\ref{firsthalfequat_eq},\\ref{knzameq_eq}).\n\n(ii) If the level $k$ is rational, then there arises a subbundle of  the bundle\nin question, the one with fiber $((L^{A})^{\\ast})^{\\mbox{\\g g}(\\cp,A}$, where\n$L^{A}$ is the corresponding admissible representation. We conjecture that in\nthis case formulas (\\ref{our_solut}) actually give horizontal sections of the\nlatter bundle.\n\n\n \\end{conjecture}\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}