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+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\nLet $X$ be a smooth projective variety over the field of complex numbers.  By the diagonal subscheme of $X$, denoted by $\\Delta_X$, one means the image of the embedding $\\delta: X\\rightarrow X\\times X$ given by $\\delta(x)=(x,x)$, where $x\\in X$.  This subscheme plays a central role in intersection theory.  In fact, to get hold of the fundamental classes of any subschemes of a variety $X$, it's enough to know the fundamental class of the diagonal $\\Delta_X$ of $X$, (cf. \\cite{P}).  \n \nIn this paper, we talk about the diagonal property and the weak point property of some varieties.  Broadly speaking, the diagonal property of a variety $X$ is a property which demands a special structure of the diagonal $\\Delta_X$ and therefore very significant to study from the viewpoint of intersection theory.  Moreover, being directly related to the diagonal subscheme $\\Delta_X$, this property imposes strong conditions on the variety $X$ itself.  For example, this property is responsible for the existence or non-existence of cohomologically trivial line bundles on $X$.  The weak point property is also very much similar to diagonal property but a much weaker one.  Both of these notions were introduced in \\cite{PSP}.  Many mathematicians have studied about the diagonal property and the weak point property of varieties, (cf. \\cite{F}, \\cite{FP}, \\cite{LS}).  In this paper, we introduce these two notions for an ind-variety, that is an inductive system of varieties and showed that the ind-varieties of higher rank divisors of integral slope on a smooth projective curve $C$  satisfy these properties.  Also, we show that some Hilbert schemes associated to good partitions of a polynomial satisfy the diagonal property.\n\nBefore mentioning the results obtained in this paper more specifically, let us fix some notations which we are going to use repeatedly.  We denote by $\\mathbb{C}$ the field of complex numbers.  In this paper, by $C$ we always mean a smooth projective curve over $\\mathbb{C}$. The notation $\\mathcal{O}_C$ is reserved for the structure sheaf over $C$.  For a given divisor $D$ on $C$, by $\\mathcal{O}_C(D)$ we mean the corresponding line bundle over $C$ and denote its degree by $deg(D)$.  By $Sym^d(C)$ we denote the $d-$th symmetric power of the curve $C$. For a given positive integer $n$ and a locally free sheaf (equivalently, a vector bundle) $\\mathcal{F}$ over $C$, by $\\mathcal{F}^n$ we mean the direct sum of $n$ many copies of $\\mathcal{F}$.  By $Quot^{d}_{\\mathcal{G}}$ we denote the Quot scheme parametrizing all torsion quotients of $\\mathcal{G}$ having degree $d$, $\\mathcal{G}$ being any coherent sheaf on $C$.  For a given polynomial $P(t)\\in \\mathbb{Q}[t]$, we denote the Quot scheme parametrizing all quotients of $\\mathcal{G}$ having Hilbert polynomial $P(t)$ by $Quot^{P}_{\\mathcal{G}}$.\n \nLet us now go through the chronology of this paper in a bit more detail.  The manuscript is arranged as follows.  In Section \\ref{sec: dp and wpp}, we recall the definitions of the  diagonal property and the weak point property for a smooth projective variety and talk about a relation between these two properties.  Moreover, for a smooth projective curve $C$ over $\\mathbb{C}$, we recall a couple of relevant results about the variety $Sym^d(C)$ and the Quot scheme $Quot^{d}_{\\mathcal{O}_C^n}$.  In Section \\ref{sec: Higher rank divisors}, we recall the definition of $(r,n)-$divisors on $C$ \\& the ind-variety made out of such divisors.  We then precisely define, what we mean by the diagonal property and the weak point property of an ind-variety and prove the following theorems followed by that.\n\\begin{theorem}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$.  Also let $r\\geq 1$ and $n$ be two integers.  Then the ind-variety of $(r,n)-$ divisors having integral slope on $C$ has the weak point property. \n\\end{theorem} \n\\begin{theorem}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$ and $n$ any given integer. Then the ind-variety of $(1,n)-$ divisors on $C$ has the diagonal property.\n\\end{theorem}           \nIn Section \\ref{sec: Hilbert scheme}, we deal with the Hilbert scheme associated to a polynomial $P$ and its good partition.  E.~Bifet has dealt with these schemes in \\cite{B}.  Moreover, he showed that the Quot scheme $Quot_{\\mathcal{O}_C^r}^P$ can be written as disjoint union of some smooth, the torus $\\mathbb{G}_m^r-$invariant, locally closed vector bundles over the mentioned Hilbert schemes.  Here, we talk about the diagonal property of such Hilbert schemes.  Towards that, we prove the following lemma.\n\\begin{lemma}\nLet $n$ be a given positive integer.  Then any partition of $n$ is also a good partition of $n$.\n\\end{lemma}  \nIn the lemma stated above, we interpret the integer $n$ as a constant polynomial and therefore it makes sense to talk about good partition of $n$.  Using this, we finally prove the following theorem, which not only provides some Hilbert scheme associated to good partitions of a polynomial but also gives a lower bound on the number of such Hilbert schemes. Precisely, we obtain:\n\\begin{theorem}\nLet $n$ be a positive integer.  Then there are atleast as many Hilbert schemes associated to the constant polynomial $n$ and its good partitions satisfying the diagonal property as there are conjugacy classes of the symmetric group $S_n$ of $n$ symbols.\n\\end{theorem} \n           \n\n\\section{On the diagonal property and the weak point property of a variety}\\label{sec: dp and wpp} \nIn this section, we recall the notions of the diagonal property and the weak point property of a variety and talk about relations between these two properties.  Moreover, for a smooth projective curve $C$ over $\\mathbb{C}$, we recall a couple of relevant results about the variety $Sym^d(C)$ and the Quot scheme $Quot^{d}_{\\mathcal{O}_C^n}$.   \n\nLet us begin with the precise definitions of the diagonal property and the weak point property of a variety.\n\\begin{definition}\\label{dp}\nLet $X$ be a variety over the field of complex numbers.  Then $X$ is said to have the diagonal property if there exists a vector bundle $E\\rightarrow X\\times X$ of rank equal to the dimension of $X$, and a global section $s$ of $E$ such that the zero scheme $Z(s)$ of $s$ coincides with the diagonal $\\Delta_X$ in $X\\times X$.  \n\\end{definition} \n\\begin{definition}\\label{wpp}\nLet $X$ be a variety over the field of complex numbers.  Then $X$ is said to have the weak point property if there exists a vector bundle $F\\rightarrow X$ of rank equal to the dimension of $X$, and a global section $t$ of $F$ such that the zero scheme $Z(s)$ of $s$ is a reduced point of $X$.\n\\end{definition}\n\\begin{remark}\\label{dp implies wpp}\nIt can be noted immediately that for a variety, having the weak point property is in fact a weaker condition than having the diagonal property.  To prove this precisely, let's stick to the notations of Definition \\ref{dp} and \\ref{wpp}.  Let us choose a point $x_0\\in X$.  Then $Z(s|_{X\\times \\{x_0\\}})=\\{x_0\\}$.  Therefore, the diagonal property implies the weak point property. \n\\end{remark}\nWe now quickly go through some results related to the diagonal property and the weak point property of two varieties which arise very naturally from a given curve $C$.  To be specific, we look upon the varieties $Sym^d(C)$ and $Quot^{d}_{\\mathcal{O}_C^n}$.  We mention a couple of results in this context.  These are due to \\cite{BS}.\n\\begin{theorem}\\label{symm prod_dp}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$.  Then, the $d-$th symmetric product $Sym^d(C)$ of the curve $C$ has the diagonal property for any positive integer $d$.\n\\end{theorem}\n\\begin{proof}\nSee \\cite[Theorem 3.1, p. 447]{BS}.\n\\end{proof}\n\\begin{theorem}\\label{Quot scheme_wpp}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$. Let $d$ and $n$ be two given positive integer such that $n|d$.  Then the Quot scheme $Quot^{d}_{\\mathcal{O}_C^n}$ parametrizing the torsion quotients of $\\mathcal{O}_C^n$ of degree $d$ has the weak point property.    \n\\end{theorem}\n\\begin{proof}\nSee \\cite[Theorem 2.2, p. 446-447]{BS}.\n\\end{proof}  \n\\begin{remark}\\label{positivity required in the hypothesis}\nLet us discuss about the hypothesis of Theorem \\ref{Quot scheme_wpp}.  Firstly, positivity of the integer $n$ is necessary as we are talking about the sheaf $\\mathcal{O}_C^n$.  Moreover, if we assume that $d$ is a positive integer and $n|d$, then there exists a positive integer $r$ such that $d=nr$.  The positivity of this integer $r$ is heavily used in the proof.  Indeed, the authors first showed that to prove Theorem \\ref{Quot scheme_wpp}, it is enough to show that the Quot scheme $Quot^{d}_{L^n}$ has the weak point property for some degree $r$ line bundle $L$ over $C$.  Now the line bundle $L$ is taken to be the line bundle $\\mathcal{O}_C(rx_0)$, where $x_0\\in X$.  Now positivity of $r$ gives the natural inclusion $i:\\mathcal{O}_C \\hookrightarrow \\mathcal{O}_C(rx_0)$.  This in turn gives the following short exact sequence:\n\\begin{equation}\\label{ses_wpp of Quot scheme}\n0\\rightarrow \\mathcal{O}_C^n \\rightarrow \\mathcal{O}_C(rx_0)^n \\rightarrow T \\rightarrow 0.\n\\end{equation}   \nNow the torsion sheaf $T$ as in \\eqref{ses_wpp of Quot scheme} lies in the sheaf $Quot^{d}_{{\\mathcal{O}_C(rx_0)}^n}$, the sheaf they wanted to work on to prove the required result.  So, positivity of $d$ has a huge role to play in the proof. \n\\end{remark}\n\\begin{remark}\nIt is worthwhile to note a connection between Theorem \\ref{symm prod_dp} \\& \\ref{Quot scheme_wpp}.  If we take, $r=1$, then Theorem \\ref{Quot scheme_wpp} says that for any positive integer $d$, the Quot scheme $Quot^{d}_{\\mathcal{O}_C^n}$ has the weak point property.  As, $Sym^d(C)\\cong Quot^{d}_{\\mathcal{O}_C}$, by Remark \\ref{dp implies wpp}, Theorem \\ref{Quot scheme_wpp} follows from Theorem \\ref{symm prod_dp} for $r=1$ case.\n\\end{remark}\n\\section{Higher rank divisors on a curve, corresponding ind-varieties and the diagonal \\& the weak point property}\n\\label{sec: Higher rank divisors}\nIn this section, we recall the definition of higher rank divisors on a curve, corresponding ind-varieties and quasi-isomorphism between them.  Then we introduce the notion of the diagonal property and the weak point property for an ind-variety in general and prove some results about the ind-varieties of higher rank divisor in particular.   \n \nLet us denote by $K$ the field of rational functions on $C$, thought as a constant $\\mathcal{O}_C-$module.\n\\begin{definition}\n\tA divisor of rank $r$ and degree $n$ over $C$ is a coherent sub $\\mathcal{O}_C-$module of $K^{\\oplus r}$ having rank $r$ and degree $n$.  This is denoted by $(r,n)-$divisor.\n\\end{definition}\n\\begin{remark}\nSince we take $C$ to be smooth, these $(r,n)-$ divisors coincide with the matrix divisors defined by A. Weil, (cf. \\cite{W}).\n\\end{remark}\nLet us denote the set of all $(r,n)-$divisors on $C$ by $Div^{r,n}$.  Let $D$ be an effective divisor of degree $d$ over $C$.  Then corresponding to $D$, let us define the following subset of $Div^{r,n}$, denoted by $Div^{r,n}(D)$ as follows:\n\\begin{definition}\n\t$Div^{r,n}(D):=\\{E \\in Div^{r,n} \\mid E \\subseteq \\mathcal{O}_{C}(D)^{\\oplus r}\\}$.\n\\end{definition}\nThen clearly we have:\n\\begin{equation*}\nDiv^{r,n}=\\bigcup_{D\\geq 0}Div^{r,n}(D) .\n\\end{equation*}\nAlso, the elements of $Div^{r,n}(D)$ can be identified with the rational points of the Quot scheme $Quot_{\\mathcal{O}_C(D)^r}^m$ where $m=r\\cdot degD-n$.  Therefore taking $D=\\mathcal{O}_C$, we can say that the elements of $Div^{r,n}(\\mathcal{O}_C)$ can be identified with the rational points of the Quot scheme $Quot_{\\mathcal{O}_C^r}^{-n}$.\n\nLet us now recall what one means by a inductive system of varieties.\n\\begin{definition}\n\tAn ind-variety $\\mathbf{X}=\\{X_\\lambda,f_{\\lambda \\mu}\\}_{\\lambda,\\mu \\in \\Lambda}$ is an inductive system of complex algebraic varieties $X_\\lambda$ indexed by some filtered ordered set $\\Lambda$.  That is to say, an ind-variety is a collection $\\{X_\\lambda\\}_{\\lambda \\in \\Lambda}$ of complex algebraic varieties, where $\\Lambda$ is some filtered ordered set, along with the morphisms $f_{\\lambda \\mu}: X_{\\lambda}\\rightarrow X_{\\mu}$ of varieties for every $\\lambda \\leq \\mu$ such that the following diagrams commute for every $\\lambda \\leq \\mu \\leq \\nu$.\n\\begin{equation*}\n\\xymatrix{\nX_{\\lambda} \\ar[rd]_{f_{\\lambda \\nu}}\\ar[r]^{f_{\\lambda \\mu}} & X_{\\mu}\n\t\t\\ar[d]^{f_{\\mu \\nu}}\\\\ \n\t\t & X_{\\nu}}\n\\end{equation*}\t\n\\end{definition}\nTaking the indexing set $\\Lambda$ to be the set of effective divisors on $C$, we have the inclusion \n\\begin{equation}\\label{E6}\nDiv^{r,n}(D_\\alpha)\\rightarrow Div^{r,n}(D_\\beta),\n\\end{equation}\ninduced by the closed immersion $\\mathcal{O}_{C}(D_{\\alpha})^{\\oplus r}\\hookrightarrow \\mathcal{O}_{C}(D_{\\beta})^{\\oplus r}$ for any pair of effective divisors $D_\\alpha,D_\\beta$ satisfying $D_\\alpha\\leq D_\\beta$.\n\\begin{definition}\\label{Div_ind-variety}\n\tThe ind-variety determined by the inductive system consisting of the varieties $Div^{r,n}(D)$ and the closed immersions as in (\\ref{E6}) is denoted by ${\\mathbf{Div}}^{r,n}$.\n\\end{definition}\n\nNow we are going to consider another ind-variety.  Given any effective divisor $D$ on $C$, we consider a complex algebraic variety $Q^{r,n}(D)$ defined as follows.\n\\begin{definition}\\label{Quot schemes as constituent of ind-variety}\n\t$Q^{r,n}(D):=Quot^{n+r\\cdot degD}_{\\mathcal{O}_C^r}$.\n\\end{definition}\nLet $D_1$ and $D_2$ be any two effective divisors  with $D_2\\geq D_1$.   Denoting $D_2-D_1$ as $D$, we have the following structure map denoted by $\\mathcal{O}_C(-D)$.\n\\begin{equation*}\n\\mathcal{O}_C(-D): Quot^{n+r\\cdot degD_1}_{\\mathcal{O}_C^r}\\rightarrow Quot^{n+r\\cdot degD_2}_{\\mathcal{O}_C^r},\n\\end{equation*}\nwhere the map $\\mathcal{O}_C(-D)$ means tensoring the submodules with $\\mathcal{O}_C(-D)$.\nElaborately, let $(\\mathcal{F},q) \\in Quot^{n+r\\cdot degD_1}_{\\mathcal{O}_C^r}$.  Therefore we have the following exact sequence:\n\\begin{equation*}\n\\xymatrix{ 0 \\ar[r]& Kerq \\ar[r] &\\mathcal{O}_C^r\\ar[r]^{q} &\\mathcal{F}\n\t\\ar[r] &0},\n\\end{equation*}\nwhere degree of $\\mathcal{F}$ is $n+r\\cdot degD_1$ and hence degree of $Kerq$ is $-n-r\\cdot degD_1$.  Tensoring this by $\\mathcal{O}_C(-D)$ we get,\n\\begin{equation*}\n\\xymatrix{ 0 \\ar[r]& Kerq\\otimes \\mathcal{O}_C(-D) \\ar[r] &\\mathcal{O}_C(-D)^r\\ar[r] &\\mathcal{F}\\otimes \\mathcal{O}_C(-D)\n\t\\ar[r] &0}.\n\\end{equation*}\nHere $deg(Kerq\\otimes \\mathcal{O}_C(-D))=r\\cdot(degD_1-degD_2)-n-r\\cdot degD_1=-n-r\\cdot degD_2$.  Now as $\\mathcal{O}_C(-D)^r$ sits inside $\\mathcal{O}_C^r$, $Kerq\\otimes \\mathcal{O}_C(-D)$ also sits inside $\\mathcal{O}_C^r$.  Therefore we now get the following exact sequence:\n\\begin{equation*}\n\\xymatrix{ 0 \\ar[r]& Kerq\\otimes \\mathcal{O}_C(-D) \\ar[r] &\\mathcal{O}_C^r\\ar[r]^{q_1} &\\mathcal{F}_1\n\t\\ar[r] &0},\n\\end{equation*} \nwhere $deg(\\mathcal{F}_1)=n+r\\cdot degD_2$. Hence $\\mathcal{F}_1 \\in Quot^{n+r\\cdot degD_2}_{\\mathcal{O}_C^r}$.  Thus $\\mathcal{O}_C(-D): Quot^{n+r\\cdot degD_1}_{\\mathcal{O}_C^r} \\rightarrow Quot^{n+r\\cdot degD_2}_{\\mathcal{O}_C^r}$ is a well defined map precisely given by $(\\mathcal{F},q) \\mapsto (\\mathcal{F}_1,q_1)$.  Therefore for $D_2\\geq D_1$ we have,\n\\begin{equation}\\label{E7}\n\\mathcal{O}_C(-D):Q^{r,n}(D_1)\\rightarrow Q^{r,n}(D_2).\n\\end{equation}\n\\begin{definition}\\label{Quot_ind-variety}\n\tThe ind-variety determined by the inductive system consisting of the varieties $Q^{r,n}(D)$ and the morphisms as in (\\ref{E7}) is denoted by ${\\mathbf{Q}}^{r,n}$.\n\\end{definition}\n\nLet us clarify what we mean by a good enough morphism in the category of ind-varieties.   \n\\begin{definition}\n\tLet $\\mathbf{X}=\\{X_D,f_{DD_1}\\}_{D,D_1\\in \\mathcal{D}}$ and $\\mathbf{Y}=\\{Y_D,g_{DD_1}\\}_{D,D_1\\in \\mathcal{D}}$ be two inductive system of complex algebraic varieties, where $\\mathcal{D}$ is the ordered set of all effective divisors on $C$.  Then by a morphism $\\mathbf{\\Phi}=\\{\\alpha,\\{\\phi_D\\}_{D\\in \\mathcal{D}}\\}$ from $\\mathbf{X}$ to $\\mathbf{Y}$ we mean an order preserving map $\\alpha:\\mathcal{D}\\rightarrow \\mathcal{D}$ together with a family of morphisms $ \\phi_D: X_D \\rightarrow Y_{\\alpha(D)}$ satisfying the following commutative diagrams for all $D,D_1\\in \\mathcal{D}$ with $D\\leq D_1$.\n\t\\begin{equation*}\n\t\\label{eq:277}\n\t\\xymatrix{X_D \\ar[d]_{f_{DD_1}}\\ar[rr]^{\\phi_D} && Y_{\\alpha(D)}\n\t\t\\ar[d]^{g_{\\alpha(D)\\alpha(D_1)}}\\\\ \n\t\tX_{D_1}\\ar[rr]^{\\phi_{D_1}}\n\t\t&& Y_{\\alpha(D_1)}\n\t}\n\t\\end{equation*}\t\n\\end{definition}\n\\begin{remark}\nNote that $\\alpha:\\mathcal{D}\\rightarrow \\mathcal{D}$ being an order preserving map, $D\\leq D_1\\Rightarrow \\alpha(D)\\leq \\alpha(D_1)$.  Therefore the map $g_{\\alpha(D)\\alpha(D_1)}:Y_{\\alpha(D)}\\rightarrow Y_{\\alpha(D_1)}$ makes sense.\n\\end{remark}\n\\begin{definition}\n\tLet $\\mathbf{X}=\\{X_D,f_{DD_1}\\}_{D,D_1\\in \\mathcal{D}}$ and $\\mathbf{Y}=\\{Y_D,g_{DD_1}\\}_{D,D_1\\in \\mathcal{D}}$ be two inductive system of complex algebraic varieties.  Then a morphism $\\mathbf{\\Phi}=\\{\\alpha,\\{\\phi_D\\}_{D\\in \\mathcal{D}}\\}$ from $\\mathbf{X}$ to $\\mathbf{Y}$ is said to be a quasi-isomorphism if\n\t\\begin{enumerate}[(a)]\n\t\t\\item $\\alpha(\\mathcal{D})$ is a cofinal subset of $\\mathcal{D}$,\n\t\t\\item given any integer $n$ there exists $D_n\\in \\mathcal{D}$ such that for all $D\\geq D_n$, $ \\phi_D: X_D \\rightarrow Y_{\\alpha(D)}$ is an open immersion and codimension of $Y_{\\alpha(D)}-\\phi_D(X_D)$ in $Y_{\\alpha(D)}$ is greater than $n$, i.e for $D\\gg 0$ the maps $\\phi_D: X_D \\rightarrow Y_{\\alpha(D)}$ are open immersion and very close to being surjective.\n\t\\end{enumerate}\n\\end{definition}\n\nNow we recall an important theorem which talks about the quasi-isomorphism between the ind-varieties defined in Definition \\ref{Div_ind-variety} and \\ref{Quot_ind-variety}.\n\n\\begin{theorem}\\label{T2}\n\tThere is a natural quasi-isomorphism between the ind-varieties ${\\mathbf{Div}}^{r,n}$ and ${\\mathbf{Q}}^{r,-n}$.\n\\end{theorem}\n\\begin{proof}\n\tSee \\cite[Remark, page-647]{BGL}.  Infact, let $D$ be an effective divisor on $C$ of degree $d$. Let $(\\mathcal{F},q)\\in Quot_{\\mathcal{O}_C(D)^r}^{rd-n}$.  Then we have the following exact sequence.\n\t\\begin{equation*}\n\t\\xymatrix{ 0 \\ar[r]& Kerq \\ar[r] &\\mathcal{O}_C(D)^r\\ar[r]^{q} &\\mathcal{F}\n\t\t\\ar[r] &0},\n\t\\end{equation*}\n\twhere $deg(\\mathcal{F})=rd-n$. Tensoring this with $\\mathcal{O}_C(-D)$ we get,\n\t\\begin{equation*}\n\t\\xymatrix{ 0 \\ar[r]& Kerq\\otimes \\mathcal{O}_C(-D) \\ar[r] &\\mathcal{O}_C^r\\ar[rr]^{q_1} &&\\mathcal{F}\\otimes \\mathcal{O}_C(-D)\n\t\t\\ar[r] &0},\n\t\\end{equation*}\n\twhere $deg(\\mathcal{F}\\otimes \\mathcal{O}_C(-D))=rd-n$.  Hence, $(\\mathcal{F}\\otimes \\mathcal{O}_C(-D),q_1)\\in Quot_{\\mathcal{O}_C^r}^{rd-n}.$  So we get a map $Quot_{\\mathcal{O}_C(D)^r}^{rd-n}\\rightarrow Quot_{\\mathcal{O}_C^r}^{rd-n}$.  Restricting this map to the rational points of $Quot_{\\mathcal{O}_C(D)^r}^{rd-n}$, we obtain a map $Div^{r,n}(D)\\rightarrow Q^{r,-n}(D)$.  This map in turn will induce the required quasi-isomorphism \n\t\\begin{equation*}\n\t{\\mathbf{Div}}^{r,n}\\rightarrow {\\mathbf{Q}}^{r,-n}.\n\t\\end{equation*}  \n\\end{proof}\n\\begin{remark}\\label{ind-variety of divisors}\nBy Theorem \\ref{T2}, we can interpret ${\\mathbf{Q}}^{r,-n}.$ as the ind-variety of $(r,n)-$ divisors on $C$.\n\\end{remark}\nNow we are in a stage to describe what we mean by the diagonal property and the weak point property of an ind-variety.  In this regard, we have couple of definitions as follows.  The notion of smoothness of an ind-variety (cf. \\cite[\\S 2, p. 643]{BGL}) motivates us to define the following two notions relevant to our context. \n\\begin{definition}\\label{indvariety_dp}\nLet $\\Lambda$ be a filtered ordered set.  Let $X=\\{X_{\\lambda}, f_{\\lambda \\mu}\\}_{\\lambda, \\mu \\in \\Lambda}$ be an ind-variety.  Then $X$ is said to have the diagonal property if there exists some $\\lambda_0\\in \\Lambda$ such that for all $\\lambda\\geq \\lambda_0$, the varieties $X_{\\lambda}$'s have the diagonal property.\n\\end{definition}\nOne can analogously define the weak point property of an ind-variety.  To be precise, we have the following definition.\n\\begin{definition}\\label{indvariety_wpp}\nLet $\\Lambda$ be a filtered ordered set.  Let $X=\\{X_{\\lambda}, f_{\\lambda \\mu}\\}_{\\lambda, \\mu \\in \\Lambda}$ be an ind-variety.  Then $X$ is said to have the weak point property if there exists some $\\lambda_0\\in \\Lambda$ such that for all $\\lambda\\geq \\lambda_0$, the varieties $X_{\\lambda}$'s have the weak point property.\n\\end{definition}\n\nLet us now associate a rational number to a given higher rank divisor.  In fact, this number helps us to find some ind-varieties having the diagonal property and weak point property. \n\\begin{definition}\\label{slope of higher rank divisor}\nFor a given $(r,n)-$divisor, the rational number $\\tfrac{n}{r}$ is said to its slope.\n\\end{definition}\n \nWe now prove a couple of theorems about the diagonal property and weak point property of ind-varieties of $(r,n)-$ divisors, when the rational number as in Definition \\ref{slope of higher rank divisor} is in fact an integer.\n\\begin{theorem}\\label{integral slope case}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$.  Also let $r\\geq 1$ and $n$ be two integers.  Then the ind-variety of $(r,n)-$ divisors having integral slope on $C$ has the weak point property. \n\\end{theorem}\n\\begin{proof}\nIt can be noted that a $(r,n)-$divisor is of integral slope if and only if $n$ is an integral multiple of $r$, by Definition \\ref{slope of higher rank divisor}.  Therefore, the ind-variety ${\\mathbf{Div}}^{r,kr}$, or equivalently ${\\mathbf{Q}}^{r,-kr}$ by Remark \\ref{ind-variety of divisors}, can be considered as the ind-variety of higher rank divisors of integral slope.\n\nLet $D$ be an effective divisor of degree $d$ on $C$.  Then we have, $Q^{r,-n}(D)=Quot_{\\mathcal{O}_C^r}^{rd-n}$ by Definition \\ref{Quot schemes as constituent of ind-variety}.  Now if $n=rk$ for some integer $k$, then $Q^{r,-rk}(D)=Quot_{\\mathcal{O}_C^r}^{rd-rk}=Quot_{\\mathcal{O}_C^r}^{r(d-k)}$.  Now let's pick an effective divisor $D_0$ of degree $d_0$ satisfying the inequality $d_0>k$.  Then for all $D\\geq D_0$ and $n=rk$, we have \n\\begin{equation}\\label{eqn_1_wpp}\ndeg(D)\\geq deg(D_0)=d_0>k, \n\\end{equation}\nand \n\\begin{equation}\\label{eqn_2_wpp}\nQ^{r,-n}(D)=Q^{r,-rk}(D)=Quot_{\\mathcal{O}_C^r}^{r(deg(D)-k)}.\n\\end{equation}\nHere $r(deg(D)-k)$ is a positive integer by \\eqref{eqn_1_wpp}.  Therefore, by Theorem \\ref{Quot scheme_wpp} and Definition \\ref{indvariety_wpp} \\& \\eqref{eqn_2_wpp}, the ind-variety ${\\mathbf{Q}}^{r,-kr}$ has the weak point property.  Hence we have the assertion.  \n\\end{proof}\n\\begin{theorem}\\label{rank one case}\nLet $C$ be a smooth projective curve over $\\mathbb{C}$ and $n$ any given integer. Then the ind-variety of $(1,n)-$ divisors on $C$ has the diagonal property.\n\\end{theorem}\n\\begin{proof}\nLet $D$ be an effective divisor of degree $d$ on $C$.  Then we have, $Q^{1,-n}(D)=Quot_{\\mathcal{O}_C}^{d-n}$.  Now let's pick an effective divisor $D_1$ of degree $d_1$ satisfying the inequality $d_1>n$.  Then for all $D\\geq D_1$, we have \n\\begin{equation}\\label{eqn_1_dp}\ndeg(D)\\geq deg(D_1)=d_1>n,\n\\end{equation}\nand  \n\\begin{equation}\\label{eqn_2_dp}\nQ^{1,-n}(D)=Quot_{\\mathcal{O}_C}^{deg(D)-n}\\cong Sym^{deg(D)-n}(C).\n\\end{equation}\nHere $deg(D)-n$ is a positive integer by \\eqref{eqn_1_dp}.  Therefore, by Theorem \\ref{symm prod_dp} and Definition \\ref{indvariety_dp} \\& \\eqref{eqn_2_dp}, the ind-variety ${\\mathbf{Q}}^{1,-n}$ of all $(1,n)-$ divisors has the diagonal property.  \n\\end{proof}\n\\begin{remark}\nIt can be noted a particular case of Theorem \\ref{integral slope case}, namely the case $r=1$, follows from Theorem \\ref{rank one case} and Remark \\ref{dp implies wpp}.\n\\end{remark}\n\n\\section{The diagonal property of the Hilbert scheme associated to a constant polynomial and its good partition}\\label{sec: Hilbert scheme}\nIn this section, we talk about the Hilbert schemes associated to a polynomial and some of its good partitions.     First we mention the importance of studying such Hilbert schemes and then show that few of these Hilbert schemes satisfy the diagonal property.  Moreover, we provide a lower bound on the number of such Hilbert schemes.\n\nLet $P(t)$ be a polynomial with rational coefficients.  We use the notation $Hilb^{P}_C$, or simply$Hilb^{P}$,  to denote the Hilbert scheme parametrizing all subschemes of $C$ having Hilbert polynomial $P(t)$.  Let $n$ be a positive integer.  Then interpreting $n$ as a constant polynomial, by $Hilb^n$ we mean the Hilbert scheme parametrizing subschemes of $C$ having Hilbert polynomial $n$.  Let us recall the notion of a good partition of a polynomial and a Hilbert scheme associated to that. \n\\begin{definition}\\label{gp_defn}\n\tLet $\\underline{P}=(P_i)_{i=1}^r$ be a family of polynomials with rational coefficients.  Then $\\underline{P}$ is said to be a good partition of $P$ if $\\sum_{i=1}^r P_i=P$ and $Hilb^{P_i}\\neq \\phi$ for all $i$.\n\\end{definition} \n\\begin{definition}\n\tThe Hilbert scheme associated to a polynomial $P$ and its good partition, denoted by $Hilb^{\\underline{P}}$ , is defined as $Hilb^{\\underline{P}}:=Hilb^{P_1}\\times_\\mathbb{C}\\cdots \\times_\\mathbb{C} Hilb^{P_r}$.\n\\end{definition}\n\\begin{remark}\\label{motivation for checking dp for Hilbert schemes}\nAt this point it is worthwhile to mention the importance of the Hilbert scheme $Hilb^{\\underline{P}}$.  Recall that by  $Quot^{P}_{\\mathcal{F}}$ we denote the Quot scheme parametrizing all quotients of $\\mathcal{F}$ having having Hilbert polynomial $P(t)$.  We have a decomposition of $Quot_{\\mathcal{O}_C^r}^P$ as follows, whenever $Quot_{\\mathcal{O}_C^r}^P$ is smooth.\n\t\\begin{equation*}\\label{E8}\n\tQuot_{\\mathcal{O}_C^r}^P=\\bigsqcup_{\\substack{\\underline{P}\\; such\\; that \\;\\underline{P}\\\\is \\;a\\; good\\; partition\\; of\\; P }}\\mathcal{S}_{\\underline{P}},\n\t\\end{equation*}\n\twhere each $\\mathcal{S}_{\\underline{P}}$ is smooth, the torus $\\mathbb{G}_m^r-$invariant, locally closed and isomorphic to a vector bundle over the scheme $Hilb^{\\underline{P}}$, (cf. \\cite[p. 610]{B}).  Therefore, the cohomology of $Quot_{\\mathcal{O}_C^r}^P$ can be given by the direct sum of the cohomologies of $Hilb^{\\underline{P}}$, where the sum varies over the good partitions of the polynomial $P$.  So to study the cohomology ring $H^{\\ast}(Quot_{\\mathcal{O}_C^r}^P)$, it is enough the cohomology rings $H^{\\ast}(Hilb^{\\underline{P}})$, $\\underline{P}$ being good partition of the polynomial $P$.  Now, to get hold of the cohomology rings $H^{\\ast}(Hilb^{\\underline{P}})$, it's nice to get hold of the structure of the Hilbert scheme $Hilb^{\\underline{P}}$.  Now, as the diagonal property and the weak point property force strong conditions on the underlying variety (cf. \\cite{PSP}), therefore to the study the cohomology of $Quot_{\\mathcal{O}_C^r}^P$ it's reasonable enough to check whether the Hilbert schemes $Hilb^{\\underline{P}}$'s posses these properties or not. \n\\end{remark}\nRemark \\ref{motivation for checking dp for Hilbert schemes} motivates us to talk about the diagonal property of the Hilbert schemes associated to a constant polynomial and some particular good partitions of the same.  Towards that, we have the following Lemma.  \n\\begin{lemma}\\label{partition_a good partition}\nLet $n$ be a given positive integer.  Then any partition of $n$ is also a good partition of $n$.\n\\end{lemma}\n\\begin{proof}\nLet $n$ be a positive integer.  An arbitrary partition of $n$ of length $r$ is given by a $r-$tuple $(n_1,n_2,\\ldots,n_r)$ such that $\\sum_{i=1}^r n_i=n$ and $n_i>0$ for all $i$.  As $n_i>0$ and $Hilb^{n_i}$ is isomorphic to the moduli space $Sym^{n_i}(C)$ of effective divisors of degree $n_i$ over $C$, we have $Hilb^{n_i}\\neq \\emptyset$ for all $i$.  Therefore, by Definition \\ref{gp_defn}, the chosen partition of $n$ is a good partition as well.  \n\\end{proof}  \nFinally, we have the following theorem which says about the diagonal property of the Hilbert schemes associated to a constant polynomial and some particular good partitions, namely the partitions, of the same.  Moreover, we provide a lower bound on the number of such Hilbert schemes.  The theorem is followed by the following lemma which follows from Definition \\ref{dp},(cf. \\cite[p. 1235]{PSP}).\n\\begin{lemma}\\label{dp for product}\nLet $X_1$ and $X_2$ be two varieties over $\\mathbb{C}$ satisfying the the diagonal property.  Then the product variety $X_1\\times X_2$ also have the diagonal property. \n\\end{lemma} \n\\begin{theorem}\nLet $n$ be a positive integer.  Then there are atleast as many Hilbert schemes associated to the constant polynomial $n$ and its good partitions satisfying the diagonal property as there are conjugacy classes of the symmetric group $S_n$ of $n$ symbols. \n\\end{theorem}\n\\begin{proof}\nFor a positive integer $n$, a partition of $n$ of length $r$ is given by a $r$-tuple $(n_1,n_2,\\ldots,n_r)$ such that $\\sum_{i=1}^r n_i=n$ and $n_i>0$ for all $i$. Then $(n_1,n_2,\\ldots,n_r)$ is also a good partition of $n$ by Lemma \\ref{partition_a good partition}. Moreover, the associated Hilbert scheme is given by $Hilb^{n_1}\\times_\\mathbb{C} Hilb^{n_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n_r}$.  As $Hilb^m\\cong Sym^m(C)$ for any positive integer $m$, by Theorem \\ref{symm prod_dp} and Lemma \\ref{dp for product}, we get that the associated Hilbert scheme $Hilb^{n_1}\\times_\\mathbb{C} Hilb^{n_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n_r}$ satisfies the diagonal property.  Therefore, given any arbitrary partition of $n$, the associated Hilbert scheme has the diagonal property.  \n\nNow let us take two distinct partition of $n$, say $(n_1,n_2,\\ldots,n_r)$ and $(n^{'}_1,n^{'}_2,\\ldots,n^{'}_s)$.  Then we have the following two mutually exclusive and exhaustive cases:\\\\\n\\textit{First Case :} $r\\neq s$\\\\\nIn this case the associated Hilbert schemes $Hilb^{n_1}\\times_\\mathbb{C} Hilb^{n_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n_r}$ and $Hilb^{n^{'}_1}\\times_\\mathbb{C} Hilb^{n^{'}_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n^{'}_s}$ are distinct as they can be written as product of different number of $Hilb^m$'s.\\\\ \n\\textit{Second Case :} $r=s$ and $n_i\\neq n^{'}_i$ for some $1\\leq i \\leq r$\\\\\nIn this case, as $n_i\\neq n^{'}_i$ for some $1\\leq i \\leq r$, therefore $Hilb^{n_i}\\neq Hilb^{n^{'}_i}$.  Hence the associated Hilbert schemes $Hilb^{n_1}\\times_\\mathbb{C} Hilb^{n_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n_r}$ and $Hilb^{n^{'}_1}\\times_\\mathbb{C} Hilb^{n^{'}_2}\\times_\\mathbb{C} \\cdots \\times_\\mathbb{C} Hilb^{n^{'}_r}$ are not same as well.\n  \nTherefore, we conclude that any two distinct partitions of $n$ gives us two distinct Hilbert schemes associated to those partitions satisfying the diagonal property.  Now as number of conjugacy classes of $S_n$ is equal to the number of partition $p(n)$ of $n$ (cf. \\cite[Lemma 2.11.3, p. 89]{H}), the assertion follows.  \n\\end{proof}\n\n\n\\section*{Acknowledgements}\nThe author would like to thank Prof. D. S. Nagaraj for the throughout encouragements.  The author also wishes to thank Indian Institute of Science Education and Research Tirupati (Award No. - IISER-T\/Offer\/PDRF\/A.M.\/M\/01\/2021) for financial support.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRevealing the rich phase structure and thereby \ndeveloping a condensed matter physics \nof Quantum Chromodynamics (QCD) in \nthe high density region\nis one of the main subjects in current nuclear physics~\\cite{Fukushima:2010bq,Fukushima:2013rx}, and \nmuch endeavor has been being made \nboth theoretically and experimentally.\nIn the high density region, for instance,\nthe first-order chiral transition line(s) with\nthe QCD critical \npoint(s)  are  expected \nto exist on the basis of theoretical \nworks~\\cite{Asakawa:1989bq,Barducci:1989wi,Kitazawa:2002jop}, and\nthe experimental search for these phase transitions~\\cite{Stephanov:1999zu}\nis one of the  main purposes of\nthe beam-energy scan program in the relativistic heavy-ion collisions (HIC)\nat RHIC, HADES and NA61\/SHINE;\nfurther \nstudies to reveal the phase structure with higher statistics\nwill be pursued in future experiments \nplanned at GSI-FAIR, NICA-MPD and J-PARC-HI~\\cite{Galatyuk:2019lcf}.\nSuch studies on the Earth will also provide us with invaluable information\non the interior structure \nof compact stars~\\cite{Kojo:2020ztt,Cierniak:2021knt,Kojo:2021wax}.\n\nAn interesting feature of the dense quark matter in yet higher density region\nis the possible realization of the color superconductivity (CSC)\ninduced by\nthe condensation of diquark Cooper pairs~\\cite{Alford:2007xm}.\nNow that the future HIC experiments are designed so as to \nenable detailed analyses of the dense matter, \nit  would be intriguing to explore the possible existence of the CSC phases \nin these experiments.\nThe search for the CSC in the HIC, however, is quite a challenge\nbecause the temperatures $T$ achieved in the HIC can become as high as\n$100$~MeV at the highest baryon density~\\cite{Ohnishi:2015fhj},\nwhich may be  much higher than the critical temperature $T_c$ of \nthe CSC, and hence an observation of \nthe  CSC phases can be unlikely in the HIC.\n\nNevertheless, the  matter created in  the HIC may be within the critical \nregion above $T_c$ where the diquark-pair fluctuations are\nsignificant, and thus {\\it precursory phenomena} of the\nCSC~\\cite{Kitazawa:2001ft,Kitazawa:2003cs,Kitazawa:2005vr}\ndo manifest themselves through appropriate observables by the HIC.\nIn this respect, it is suggestive that \nfluctuations of Cooper pairs (preformed pairs) of electrons in metals are\nknown to cause an anomalous enhancement of \nthe electric conductivity above $T_c$ \nof the superconductivity (SC)~\\cite{skocpol1975fluctuations,book_Larkin}.\nMoreover, \nsince the quark matter in the relevant density region is a strongly-coupled\nsystem~\\cite{Abuki:2001be,Kitazawa:2003cs}, \nthe CSC can have a wider critical region where\nthe precursory phenomena of the CSC are pronounced.\nIn fact, \nit has been already shown~\\cite{Kitazawa:2001ft,Kitazawa:2003cs,Voskresensky:2003wd,Kitazawa:2005vr,Kunihiro:2007bx,Kerbikov:2014ofa,Kerbikov:2020lqm}\nthat  the diquark fluctuations develop a well-defined collective mode,\nwhich is the soft mode of the CSC, \nand its collectivity and the softening nature \naffect various observables\nincluding the appearance of the ``pseudogap'' region~\\cite{Kitazawa:2003cs}\nin a rather wide range of temperature.\n\nIn the present Letter, \nwe investigate possible enhancement of the production rate of virtual\nphotons due to the precursory diquark fluctuations,\nwhich is to be  observed as the dilepton production rate (DPR) in the HIC.\nA desirable feature of \nthe electromagnetic probes, needless to say,  lies in the fact that\nthe interactions of the probes  with the medium are weak,\nand  their properties are hardly modified from what they had when created, \nin contrast to hadronic signals.\n\nHere we remark that the DPR \n{\\em in} the CSC phases below $T_c$ is known to\nshow some unique behavior~\\cite{Jaikumar:2001jq}.\nHowever, such a behavior \nbecomes weaker when $T$ goes higher and closer to $T_c$\nbecause they are caused by the finite diquark gap.\nOn the other hand, the precursory phenomena to be \ninvestigated in the present Letter\nare most enhanced at $T=T_c$, which is an attractive feature in the HIC.\n\nThe medium modification of the DPR or the \nvirtual photons\nis dictated by \n that of the  photon self-energy~\\cite{McLerran:1984ay,Weldon:1990iw,Kapusta:1991qp}.\nThe effects of the diquark fluctuations  on the photon self-energy\ncan be taken into account by\nthe Aslamasov-Larkin, Maki-Thompson and\ndensity of states terms~\\cite{Kitazawa:2005vr,Kunihiro:2007bx,Kerbikov:2020lqm}.\nIn the case of the metallic SC, \nthese terms at the vanishing energy-momentum limit \nare known to explain an anomalous enhancement of the electric\nconductivity above $T_c$~\\cite{skocpol1975fluctuations,book_Larkin}.\n\nIn the present Letter, we calculate these terms composed of\ndiquark fluctuations near but above $T_c$ of the CSC at nonzero\nenergy and momentum.\nWe show that the Ward-Takahashi (WT) identity is satisfied\nby summing up all of these terms.\nFrom the imaginary part of these terms we calculate\na virtual photon emission from \nthe diquark fluctuations that form a collective mode.\nIt is found that the virtual photons emitted from the collective mode\nhaving a spectral support in the space-like region \nin turn have the spectral support in the time-like region.\nOur numerical results show that the DPR\nis significantly enhanced at low invariant-mass region $M\\lesssim200$~MeV\nabove $T_c$ up to, say,  $T\\simeq1.5T_c$, reflecting\nthe critical enhancement of the diquark fluctuations.\nWe argue that \nan experimental measurement of dileptons and exploration of the \npossible enhancement of the DPR in that far low-mass region  \nin the HIC is quite worthwhile to do because  \nit would  give an experimental \nevidence of strong diquark correlations\nas a precurosr of \nthe phase transition to CSC in dense quark matter.\n\n\\begin{figure}[tbp]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{fig_phaseD.pdf}\n\\caption{\n  Phase diagram obtained in the massless 2-flavor NJL model Eq.~(\\ref{eq_lagrangian}).\n  The bold lines show the transition lines at $G_{\\rm C}=0.7G_S$.\n  The solid and dashed lines represent the first- and second-order\n  phase transitions.\n  The $T_c$ of the 2SC phase at $G_{\\rm C}=0.5G_S$ and $0.9G_S$ are also shown\n  by the thin-dotted lines.\n}\n\\label{fig_phase}\n\\end{figure}\n\n\n\n\n\n\\section{Model and phase diagram}\n\\label{sec:model}\n\nIn this Letter, we consider the diquark fluctuations above\n$T_c$ of the 2-flavor superconductor (2SC), which is one of the\nCSC phases that realizes at relatively low density~\\cite{Alford:2007xm}.\nWe employ the 2-flavor NJL model~\\cite{Hatsuda:1994pi,Buballa:2003qv} as\n an effective model of QCD to describe the phase transition to 2SC;\n\\begin{align}\n  \\mathcal{L} &= \\bar{\\psi} i \n  \\slashed{\\partial}\n  \\psi + \\mathcal{L}_{\\rm S} + \\mathcal{L}_{\\rm C},\n  \\label{eq_lagrangian}\n  \\\\\n  \\mathcal{L}_{\\rm S} &= \\ G_{\\rm S} [(\\bar{\\psi} \\psi)^2 + (\\bar{\\psi} i \\gamma_5 \\vec{\\tau} \\psi)^2],\n  \\label{eq_LS}\n  \\\\\n  \\mathcal{L}_{\\rm C} &= \\ G_{\\rm C} (\\bar{\\psi} i \\gamma_5 \\tau_2 \\lambda_A \\psi^C)(\\bar{\\psi}^C i \\gamma_5 \\tau_2 \\lambda_A \\psi),\n  \\label{eq_LC}\n\\end{align}\nwhere $\\mathcal{L}_{\\rm S}$ and $\\mathcal{L}_{\\rm C}$ represent the quark-antiquark \nand  quark-quark interactions, respectively, and\n$\\psi^C (x) = i \\gamma_2 \\gamma_0 \\bar{\\psi}^T (x)$.\n$\\tau_2$ and $\\gamma_A$ $(A=2,5,7)$ are \nthe antisymmetric components of the Pauli and Gell-mann matrices \nfor the flavor $SU(2)_f$ and color $SU(3)_c$, respectively.\nThe scalar coupling constant $G_{\\rm S}=5.01 \\rm{GeV^{-2}}$ and the\nthree-momentum cutoff $\\Lambda=650$~MeV are\ndetermined so as to reproduce the pion decay constant $f_{\\pi}=93 \\rm{MeV}$ and \nthe chiral condensate $\\langle \\bar{\\psi} \\psi \\rangle = (-250\\rm{MeV})^3$ in vacuum~\\cite{Hatsuda:1994pi}.\nThe current quark mass is neglected for simplicity, while \nthe diquark coupling $G_{\\rm C}$ is treated as a free parameter.\nWe employ a common quark chemical potential $\\mu$ for\nup and down quarks since the effect of isospin breaking is not large\nin the medium created in the HIC.\n\nIn Fig.~\\ref{fig_phase}, we show the phase diagram in the $T$--$\\mu$ plane\nobtained in the mean-field approximation (MFA) with the mean fields\n$\\langle \\bar{\\psi} \\psi \\rangle$ and\n$\\langle \\bar{\\psi}^C \\Gamma \\psi \\rangle~$\nwith $\\Gamma = i \\gamma_5 \\tau_2 \\lambda_A$.\nThe bold lines show the phase diagram at $G_{\\rm C}=0.7G_{\\rm S}$, where\nthe solid and dashed lines represent the first- and second-order\nphase transitions, respectively.\nThe 2SC phase is realized in the dense region at relatively low temperatures.\nIn the figure, the phase boundary of the 2SC for $G_{\\rm C}=0.5G_{\\rm S}$ and $0.9G_{\\rm S}$\nis also shown by the thin-dotted lines.\n\nIn MFA, the phase transition to 2SC is of second order\nas shown in Fig.~\\ref{fig_phase}.\nIt is known that the transition becomes\nfirst order due to the effect of gauge fields (gluons)\nin asymptotically high density region~\\cite{Matsuura:2003md, Giannakis:2004xt, Noronha:2006cz, Fejos:2019oxz}.\nOn the other hand, the fate of the transition at lower densities\nhas not been settled down to the best of the authors' knowledge.\nIn the present study we thus assume that the transition is second or\nweak first order having the formation of the soft mode discussed below.\n\n\n\n\n\n\\section{The soft mode of 2SC}\n\n\\subsection{Propagator of diquark field}\n\\label{sec:propagator}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figure_softmode-crop.pdf}\n\\caption{\n  Diagrammatic representation of the T-matrix Eq.~(\\ref{eq:Xi})\n  in the RPA.}\n\\label{fig_softmode}\n\\end{figure}\n\nA characteristic feature of the second-order phase transition is that\nthe fluctuation amplitude of the order parameter diverges at $T=T_c$.\nTo see such a divergence at the $T_c$ of the 2SC,\nlet us consider the imaginary-time propagator of the diquark field\n$\\Delta(x)=\\bar{\\psi}^C(x) \\Gamma \\psi(x)$,\n\\begin {align}\n  \\mathcal{D}(k) \n  = -\\int_0^{1\/T} d\\tau \\int d^3\\bm{x}\n  \\langle T_\\tau \\Delta^\\dagger (x) \\Delta(0)   \\rangle e^{i \\nu_l \\tau} e^{-i \\bm{k}\\cdot\\bm{x}}, \n  \\label{eq:D}\n\\end {align}\nwhere $k=(\\bm{k}, i\\nu_l)$ is the four momentum of the diquark field\nwith $\\nu_l$ the Matsubara frequency for bosons, \n$\\tau$ is the imaginary time, and $T_\\tau$ denotes the imaginary-time\nordering.\nIn the random-phase approximation (RPA), Eq.~(\\ref{eq:D}) is given by\n$\\mathcal{D}(k) = \\mathcal{Q}(k)\/(1+G_{\\rm C}\\mathcal{Q}(k))$ with\nthe one-loop quark-quark correlation function\n\\begin {align}\n  \\mathcal{Q} (k) = - 8 \n  \\int_p\n  {\\rm Tr} [\\mathcal{G}_0 (k-p) \\mathcal{G}_0(p)], \n  \\label{eq:Q}\n\\end {align}\nwhere $\\mathcal{G}_0(p)\n= 1\/[(i\\omega_m + \\mu)\\gamma_0 - \\bm{p} \\cdot \\bm{\\gamma}]$ is the free quark propagator with $p=(\\bm{p},i\\omega_m)$ and \nthe Matsubara frequency for fermions $\\omega_m$, \n${\\rm Tr}$ denotes the trace over the Dirac indices,\nand $\\int_p = T\\sum_m \\int d^3 \\bm{p}\/(2\\pi)^3$.\nWe also introduce the T-matrix to describe the diquark fluctuation \n\\begin{align}\n  \\tilde\\Xi(k)\n  = \\frac1{G_{\\rm C}^{-1}+\\mathcal{Q}(k)}\n  = G_{\\rm C} - G_{\\rm C}\\mathcal{D}(k)G_{\\rm C},\n  \\label{eq:Xi}\n\\end{align}\nwhich is diagrammatically represented in Fig.~\\ref{fig_softmode}.\n\nThe retarded Green functions \n$D^R(\\bm{k},\\omega)$, $Q^R(\\bm{k},\\omega)$ and $\\Xi^R(\\bm{k},\\omega)$\ncorresponding to Eqs.~(\\ref{eq:D})--(\\ref{eq:Xi}), respectively, are\nobtained by the analytic continuation $i\\nu_l\\to\\omega+i\\eta$.\nThe imaginary part of $Q^R (\\bm{k}, \\omega)$ is calculated to be~\\cite{Kitazawa:2005vr}\n\\begin{align}\n  &{\\rm Im} Q^R (\\bm{k}, \\omega) =\n  -\\frac{2T}{\\pi k} [(\\omega + 2\\mu)^2 - k^2]\n  \\nonumber  \\\\\n  &\\qquad\\qquad\n  \\times \\bigg\\{\n  \\log \\frac{{\\rm cosh}(\\omega+k)\/4T}{{\\rm cosh}(\\omega-k)\/4T} \n  -\\frac{\\omega}{2T}\n  \\theta (k-|\\omega+2\\mu|)\n  ~\\bigg\\} \\ .\n  \\label{eq_ImQ}\n\\end{align}\nIts real part is then constructed using the Kramers-Kronig relation \n\\begin{align}\n  {\\rm Re} Q^R (\\bm{k}, \\omega) =\\frac{1}{\\pi} P \\int^{2\\Lambda-2\\mu}_{-2\\Lambda-2\\mu} d\\omega '\n  \\frac{{\\rm Im} Q^R (\\bm{k}, \\omega)}{\\omega '- \\omega} \\ ,\n  \\label{eq_ReQ}\n\\end{align}\nwhere $P$ denotes the principal value~\\cite{Kitazawa:2005vr}.\n\nThe retarded diquark propagator $D^R(\\bm{k},\\omega)$, and hence the T-matrix\n$\\Xi^R(\\bm{k},\\omega)$, has a pole at $\\omega=|\\bm{k}|=0$ at $T=T_c$;\n$[D^R(\\bm{0}, 0)]^{-1}_{T=T_c}=[\\Xi^R(\\bm{0}, 0)]^{-1}_{T=T_c}=0$.\nThis fact, known as the Thouless criterion~\\cite{thouless1960perturbation},\nis confirmed by comparing the denominator of $D^R(\\bm{k},\\omega)$\nwith the gap equation for the diquark field.\nThe criterion shows that the diquark field has a massless collective mode\nat $T=T_c$.\nFurthermore, the pole of this collective mode \nmoves continuously toward the origin in the complex energy plane \nas $T$ is lowered to $T_c$, and hence the collective mode \nhas a vanishing excitation energy toward $T_c$.\nThis collective mode is called the {\\em soft mode}.\nBecause of the small excitation energy, they tend to be easily excited \nand affect various observables in the medium near $T_c$~\\cite{Kitazawa:2003cs,Kitazawa:2005vr}.\n\nAlthough we had recourse to the MFA and RPA,\nthe appearance of a soft mode is a generic feature of the second-order\nphase transition~\\cite{book_Larkin}, and \neven if the phase transition is \nof first order, \nthe development of a collective mode with the softening nature \nprior to the critical point \nis still expected for weak first-order transitions.\nTherefore, \nthe emergence of the soft mode in the diquark channel and \nthe following discussions on its effects on observables should\nhave a model-independent validity, at least qualitatively.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[bb = 0 0 477 416, scale=0.45]{DSF.pdf}\n\\caption{\n  Contour plot of the \n  dynamical structure factor $S(\\bm{k},\\omega)$\n  at $T=1.05T_c$ for $\\mu = 350$~MeV and $G_{\\rm C}=0.7G_{\\rm S}$.\n  The solid lines show the light cone.\n  The left panel is the result of RPA obtained from \n  Eqs.~(\\ref{eq_ImQ}) and (\\ref{eq_ReQ}),\n  while the right panel is the result in the TDGL approximation\n  Eq.~(\\ref{eq_softmodeapprox}).\n}\n\\label{fig_DSF}\n\\end{figure}\n\nTo detail  the properties of the soft mode,\nit is convenient to introduce the dynamical structure factor \n$S(\\bm{k},\\omega)$ given by\n\\begin{align}\n  S(\\bm{k},\\omega)\n  = - \\frac1\\pi \\frac1{1-e^{-\\beta\\omega}} {\\rm Im}D^R (\\bm{k}, \\omega).\n  \\label{eq:S}\n\\end{align}\nFigure~\\ref{fig_DSF} shows a contour map of $S(\\bm{k},\\omega)$ at $T=1.05T_c$\nfor $\\mu=350$~MeV and $G_{\\rm C}=0.7G_{\\rm S}$.\nOne sees that $S(\\bm{k},\\omega)$ has a clear spectral concentration with \na peak \naround the origin in the $\\omega$--$|\\bm{k}|$ plane,\nwhich implies a development of the collective mode having a\ndefinite dispersion relation \n$\\omega=\\omega(|\\bm{k}|)$ \nwith a small width~\\cite{Kitazawa:2001ft,Kitazawa:2005vr}.\nWe also note that the spectral concentration is confined in the space-like region,\n$\\omega(|\\bm{k}|)<|\\bm{k}|$.\nThis feature will be picked up again later when \nwe discuss the DPR that has a spectral support in the time-like region.\n\n\n\\subsection{Time-dependent Ginzburg-Landau (TDGL) approximation}\n\\label{sec:TDGL}\n\nSince the diquark fluctuations near $T_c$ have spectral concentration\nin the low energy region as we have seen above,\nwe approximate the T-matrix $\\Xi^R(\\bm{k}, \\omega)$\nin the small $\\omega$ region as\n\\begin {align}\n  \\Xi^R (\\bm{k}, \\omega) \\simeq \\frac{1}{c \\omega + G_{\\rm C}^{-1} + Q^R(\\bm{k},0)} \\ ,\n  \\label{eq_softmodeapprox}\n\\end {align}\nwith $c = \\partial Q^R(\\bm{0}, \\omega)\/ \\partial \\omega|_{\\omega=0}$.\nWe refer Eq.~(\\ref{eq_softmodeapprox}) to as the \ntime-dependent Ginzburg-Landau (TDGL) approximation,\nsince Eq.~(\\ref{eq_softmodeapprox}) corresponds to the linearlized\nTDGL approximation for the T-matrix~\\cite{Cyrot:1973}\nwithout the expansion along $|\\bm{k}|^2$.\nIn this study we do not expand $[\\Xi^R(\\bm{k}, \\omega)]^{-1}$\nwith respect to $|\\bm{k}|^2$\nfor a better description of the spectral strength\nextending along $|\\bm{k}|$ direction widely as in Fig.~\\ref{fig_DSF}.\nAn explicit calculation shows that $c$ is a complex number,\nwhile $G_{\\rm C}^{-1} + Q^R(\\bm{k},0)$ is real.\n\nIn the right panel of Fig.~\\ref{fig_DSF}, we show $S(\\bm{k},\\omega)$\nobtained by the TDGL approximation. \nBy comparing the result with the left panel, one sees \nthat the TDGL approximation Eq.~(\\ref{eq_softmodeapprox}) reproduces\nthe result obtained by the RPA quite well in a\nwide range in the $\\omega$--$|\\bm{k}|$ plane.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figure_dynamicalpotential-crop.pdf}\n\\caption{Contribution of the diquark fluctuations to the thermodynamic potential.}\n\\label{fig_potential}\n\\end{figure}\n\n\n\n\n\n\\section{Photon self-energy and dilepton production rate}\n\nThe DPR\nis given in terms of the retarded photon self-energy $\\Pi^{R \\mu\\nu}(\\bm{k}, \\omega)$ as~\\cite{McLerran:1984ay,Weldon:1990iw,Kapusta:1991qp},\n\\begin {align}\n  \\frac{d^4\\Gamma(\\bm{k}, \\omega)}{d^4k} = -\\frac{\\alpha}{12\\pi^4} \n  \\frac{1}{\\omega^2-|\\bm{k}|^2} \\frac{1}{e^{\\beta\\omega}-1} g_{\\mu\\nu}  \n       {\\rm Im} \\Pi^{R \\mu\\nu} (\\bm{k}, \\omega) ,\n       \\label{eq_RATE_kom}\n\\end {align}\nwith the fine structure constant $\\alpha$. \n\n\n\\subsection{Construction of the photon self-energy}\n\n\\begin{figure}[tbp]\n    \\begin{tabular}{cccc}\n      \\begin{minipage}[t]{0.45\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 917 424, keepaspectratio, scale=0.083]{figure_AL-crop.pdf}\n        \\subcaption{}\n        \\label{fig_AL}\n      \\end{minipage} &\n      \\begin{minipage}[t]{0.45\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 918 428, keepaspectratio, scale=0.083]{figure_MT-crop.pdf}\n       \\subcaption{}\n        \\label{fig_MT}\n      \\end{minipage} \\\\ \\\\ \n      \\begin{minipage}[t]{0.45\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 917 410, keepaspectratio, scale=0.083]{figure_DOS1-crop.pdf}\n        \\subcaption{}\n        \\label{fig_DOS1}\n      \\end{minipage} &\n      \\begin{minipage}[t]{0.45\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 917 410, keepaspectratio, scale=0.083]{figure_DOS2-crop.pdf}\n        \\subcaption{}\n        \\label{fig_DOS2}\n      \\end{minipage} \n    \\end{tabular}\n    \\caption{Diagrammatic representations of the Aslamasov-Larkin (a),\n      Maki-Thompson (b) and density of states (c,d) terms\n      in Eqs.~(\\ref{eq_AL})--(\\ref{eq_DOS}).\n      The double and wavy lines represent diquarks and\n      photons, respectively.}\n     \\label{fig_selfenergy}\n\\end{figure}\n\nWe are now in a position to discuss the way how \nthe effects of the diquark fluctuations are included in \n$\\Pi^{R \\mu\\nu}(\\bm{k}, \\omega)$.\nFor that, we start from the one-loop diagram of the diquark propagator \nshown in Fig.~\\ref{fig_potential}, which is the lowest-order\ncontribution of the diquark fluctuations to the thermodynamic potential.\nThe photon self-energy is then constructed by attaching \nelectromagnetic vertices at two points of quark lines\nin Fig.~\\ref{fig_potential}.\nThis construction guarantees the WT identity $k_\\nu \\Pi^{R\\mu\\nu}(\\bm{k},\\omega)=0$.\nThis procedure leads to four types of diagrams \nshown in Fig.~\\ref{fig_selfenergy},\nwhich are called (a) Aslamasov-Larkin (AL)~\\cite{AL:1968}, \n(b) Maki-Thompson (MT)~\\cite{Maki:1968,Thompson:1968}\nand (c,d) density of states (DOS) terms, respectively,\nin the theory of metallic SC~\\cite{book_Larkin}. \nThe respective contributions to the photon self-energy\nare denoted by $\\tilde\\Pi_{\\rm AL}^{\\mu\\nu} (k)$, $\\tilde\\Pi_{\\rm MT}^{\\mu\\nu} (k)$ and\n$\\tilde\\Pi_{\\rm DOS}^{\\mu\\nu} (k)$ in  the imaginary-time formalism, \nwhich are expressed as\n\\begin {align}\n  \\tilde\\Pi_{\\rm AL}^{\\mu\\nu} (k) &= 3\n  \\int_q\n  \\tilde\\Gamma^\\mu(q, q+k) \\tilde\\Xi(q+k) \\tilde\\Gamma^\\nu(q+k, q) \\tilde\\Xi(q),\n  \\label{eq_AL}\n  \\\\\n  \\tilde\\Pi_{\\rm MT}^{\\mu\\nu} (k) &= 3\n  \\int_q\n  \\tilde\\Xi(q) \\ \\mathcal{R}_{\\rm MT}^{\\mu\\nu}(q, k),\n  \\label{eq_MT}\n  \\\\\n  \\tilde\\Pi_{\\rm DOS}^{\\mu\\nu} (k) &= 3\n  \\int_q\n  \\tilde\\Xi(q) \\ \\mathcal{R}_{\\rm DOS}^{\\mu\\nu}(q, k),\n  \\label{eq_DOS} \n\\end {align}\nrespectively, \nwhere $q=(\\bm{q}, i\\nu_n)$ is the four momentum of a diquark field\nand the overall coefficients $3$ come from three antisymmetric channels\nof the diquark field.\nThe vertex functions $\\tilde\\Gamma^\\mu(q, k)$, $\\mathcal{R}_{\\rm MT}^{\\mu\\nu} (q, k)$ and\n$\\mathcal{R}_{\\rm DOS}^{\\mu\\nu} (q, k)$ in Eqs.~(\\ref{eq_AL})--(\\ref{eq_DOS}) are given by\n\\begin {align}\n  &\\tilde\\Gamma^\\mu (q, q+k) = 8(e_u+e_d)\n  \\int_p\n  {\\rm Tr} [\\mathcal{G}_0 (p)\\gamma^\\mu\\mathcal{G}_0 (p+k)\\mathcal{G}_0 (q-p)],\n  \\label{eq_Gamma} \\\\\n  &\\mathcal{R}_{\\rm MT}^{\\mu\\nu} (q, k) = 16e_u e_d \\ \\nonumber \\\\\n  &\\times\n  \\int_p\n  {\\rm Tr} [\\mathcal{G}_0 (p)\\gamma^\\mu\\mathcal{G}_0 (p+k)\\mathcal{G}_0 (q-p-k) \n    \\gamma^\\nu\\mathcal{G}_0 (q-p)],\n  \\label{eq_RMT} \\\\\n  &\\mathcal{R}_{\\rm DOS}^{\\mu\\nu} (q, k) = 8(e^2_u+e^2_d) \\ \\nonumber \\\\\n  &\\times\n  \\int_p\n  \\Big\\{ {\\rm Tr} [\\mathcal{G}_0 (p)\\gamma^\\mu\\mathcal{G}_0 (p+k)\\gamma^\\nu \\mathcal{G}_0 (p) \\mathcal{G}_0 (q-p)] \\nonumber \\\\\n  &\\qquad + {\\rm Tr} [\\mathcal{G}_0 (p) \\gamma^\\mu\n    \\mathcal{G}_0 (p-k) \\gamma^\\nu \\mathcal{G}_0 (p) \\mathcal{G}_0 (q-p)] \\Big\\}.\n  \\label{eq:RDOS}\n\\end {align}\nwhere \n$e_u=2|e|\/3$ and $e_d=-|e|\/3$ are the electric charges of up and down quarks,\nrespectively, with the elementary charge $e$.\n\nThe total photon self-energy in imaginary time is given by\n\\begin {align}\n\\tilde\\Pi^{\\mu\\nu} (k) &= \\tilde\\Pi_{\\rm free}^{\\mu\\nu} (k) + \\tilde\\Pi_{\\rm fluc}^{\\mu\\nu} (k), \n\\label{eq_Pi_all} \\\\\n\\tilde\\Pi_{\\rm fluc}^{\\mu\\nu} (k) &= \\tilde\\Pi_{\\rm AL}^{\\mu\\nu} (k)\n+\\tilde\\Pi_{\\rm MT}^{\\mu\\nu} (k)+\\tilde\\Pi_{\\rm DOS}^{\\mu\\nu} (k),\n\\label{eq_Pi_fluc}\n\\end {align}\nwhere $\\tilde\\Pi^{\\mu\\nu}_{\\rm fluc}(k)$ denotes the modification of \nthe self-energy due to the diquark fluctuations and\n$\\tilde\\Pi^{\\mu\\nu}_{\\rm free}(k)$ is that of the free quark system~\\cite{book_Kapusta,book_LeBellac}.\n\n\n\\subsection{Vertices}\n\\label{sec:vertex}\n\nThe vertices (\\ref{eq_Gamma})--(\\ref{eq:RDOS}) satisfy the WT identities \n\\begin {align}\n  k_\\mu \\tilde\\Gamma^\\mu (q, q+k)\n  =& e_\\Delta \\big({\\cal Q}(q+k) - {\\cal Q}(q) \\big) \n  = e_\\Delta \\bigg( \\frac{1}{\\tilde\\Xi (q+k)} - \\frac{1}{\\tilde\\Xi (q)} \\bigg) \\ ,\n  \\label{eq_ALvertex_Ward} \n  \\\\\n  k_\\mu \\mathcal{R}^{\\mu\\nu} (q, k) =& e_\\Delta \\big( \\Gamma^\\nu (q-k, q)-\\Gamma^\\nu (q, q+k) \\big),\n  \\label{eq_MTvertex_Ward}\n\\end {align}\nwith $\\mathcal{R}^{\\mu\\nu} (q, k) =  \\mathcal{R}_{\\rm MT}^{\\mu\\nu} (q, k) + \\mathcal{R}_{\\rm DOS}^{\\mu\\nu} (q, k)$\nand $e_\\Delta=e_u+e_d$ being the electric charge of diquarks.\nUsing Eqs.~(\\ref{eq_ALvertex_Ward}), (\\ref{eq_MTvertex_Ward}) and\n$\\tilde\\Gamma^\\nu (q, q+k)=\\tilde\\Gamma^\\nu(q+k,q)$,\nthe WT identity of the photon self-energy\n$k_\\nu \\tilde\\Pi^{\\mu\\nu}_{\\rm fluc}(k)=0$ is shown explicitly as \n\\begin {align}\n  k_\\mu \\tilde\\Pi_{\\rm fluc}^{\\mu\\nu} (k)\n  =& ~k_\\mu \\tilde\\Pi_{\\rm AL}^{\\mu\\nu} (k) + k_\\mu \\big\\{\\tilde\\Pi_{\\rm MT}^{\\mu\\nu} (k)+\\tilde\\Pi_{\\rm DOS}^{\\mu\\nu} (k) \\big\\} \\nonumber \\\\\n  =&-3 e_\\Delta\n  \\int_q\n  \\big[ \\tilde\\Xi(q+k)-\\tilde\\Xi(q)\\big] \\tilde\\Gamma^\\nu(q, q+k)\n  \\nonumber \\\\\n  &+3 e_\\Delta\n  \\int_q\n  \\tilde\\Xi(q) \\big[ \\tilde\\Gamma^\\nu(q-k, q) - \\tilde\\Gamma^\\nu(q, q+k)\\big] \n  \\nonumber \\\\\n  =&0.\n  \\label{eq:WT=0}\n\\end {align}\n\nSince we adopt the TDGL approximation for $\\Xi^R(\\bm{k},\\omega)$,\nthe vertices $\\tilde\\Gamma^\\mu(q,q+k)$ and $\\mathcal{R}^{\\mu\\nu}(q,k)$ have to be approximated\nto satisfy Eqs.~(\\ref{eq_ALvertex_Ward}) and (\\ref{eq_MTvertex_Ward}) \nwithin this approximation.\nFrom Eq.~(\\ref{eq_softmodeapprox}) one finds\n\\begin{align}\n  & [\\tilde\\Xi(q+k)]^{-1} - [\\tilde\\Xi(q)]^{-1} \n  \\simeq c_0 i\\nu_n + Q(\\bm{q}+\\bm{k},0) - Q(\\bm{q},0)\n  \\nonumber \\\\\n  &= c_0 i\\nu_n + \\frac{Q(\\bm{q}+\\bm{k},0)\n    - Q(\\bm{q},0)}{|\\bm{q}+\\bm{k}|^2 - |\\bm{q}|^2}\n            (|\\bm{q}+\\bm{k}|^2 - |\\bm{q}|^2)\n  \\nonumber \\\\\n  &= c_0 i\\nu_n + Q_{(1)}(\\bm{q}+\\bm{k},\\bm{q})~\n  (2\\bm{q}+\\bm{k})\\cdot \\bm{k},\n  \\label{eq:Xi-Xi}\n\\end{align}\nwhere $Q_{(1)}(\\bm{q}_1,\\bm{q}_2)\n=(Q(\\bm{q}_1,0) - Q(\\bm{q}_2,0))\/(|\\bm{q}_1|^2 - |\\bm{q}_2|^2)$\nis finite in the limit $|\\bm{q}_1-\\bm{q}_2|\\to0$\nbecause $Q(\\bm{q},\\omega)$ is a function of $|\\bm{q}|^2$.\nSubstituting Eq.~(\\ref{eq:Xi-Xi}) into Eq.~(\\ref{eq_ALvertex_Ward})\nand requiring the analyticity of $\\tilde\\Gamma^\\mu(q,q+k)$\nat $\\omega=|\\bm{k}|=0$ one finds that $\\tilde\\Gamma^0(q,q+k) = e_\\Delta c_0$ and \n\\begin {align}\n  \\tilde\\Gamma^i (q, q+k) \n  = - e_\\Delta  Q_{(1)}(\\bm{q}+\\bm{k},\\bm{q}) (2q+k)^i,\n  \\label{eq_ALvertex_approx}\n\\end{align}\nare choices that satisfy Eq.~(\\ref{eq_ALvertex_Ward}),\nwhere $i = 1, 2, 3$.\nOne can also obtain forms of $\\mathcal{R}^{\\mu\\nu}(q,k)$ satisfying\nEq.~(\\ref{eq_MTvertex_Ward}) with Eq.~(\\ref{eq_ALvertex_approx})\nin a similar manner, which, however, are not shown explicitly\nsince \nthey turn out unnecessary in this study as discussed below.\nThese vertices with Eq.~(\\ref{eq_softmodeapprox}) satisfy\nthe WT identity of $\\tilde\\Pi^{\\mu\\nu}(k)$.\nIt should be warned, however, that the uniqueness of the choice of \nEq.~(\\ref{eq_ALvertex_approx}) holds \nonly in the lowest order of $\\omega$ and $|\\bm{k}|^2$,\nand hence the non-uniqueness may affect\nthe final result in the high energy region.\n\n\n\\subsection{Dilepton production rate}\n\nIn the above construction of $\\tilde\\Gamma^\\mu (q, q+k)$ and\n$\\mathcal{R}^{\\mu\\nu}(q,k)$, \nthe spatial components of these vertices are real.\nThis fact greatly simplifies the analytic continuation from\n$\\tilde\\Pi^{ij}_{\\rm fluc}(k)$ to $\\Pi^{R ij}_{\\rm fluc}(\\bm{k},\\omega)$.\nFrom the reality of $\\mathcal{R}^{ij}(q,k)$ it is also shown that \n${\\rm Im}[\\Pi^{Rij}_{\\rm MT}(\\bm{k},\\omega)+\\Pi^{Rij}_{\\rm DOS}(\\bm{k},\\omega)]=0$~\\cite{book_Larkin},\nwhich means that the spatial components \n${\\rm Im}\\Pi^{R ij}_{\\rm fluc}(\\bm{k},\\omega)$ only come from\n$\\Pi_{\\rm AL}^{R ij}(\\bm{k},\\omega)$,\nwhile the temporal component ${\\rm Im}\\Pi^{R00}_{\\rm fluc}(\\bm{k},\\omega)$ is\ngiven by the sum of AL, MT and DOS terms.\nThe temporal component, however, is obtained from the\nspatial ones using the WT identity\n\\begin {align}\n  \\tilde\\Pi^{00} (k) = \\frac{\\bm{k}^2}{(i\\nu_l)^2} \\tilde\\Pi^{11}(k),\n  \\label{eq_WT}\n\\end {align}\nwith $k=(i\\nu_l, |\\bm{k}|, 0, 0)$.\nOne then finds that $g_{\\mu\\nu} \\tilde\\Pi_{\\rm fluc}^{\\mu\\nu}(k)$\nin Eq.~(\\ref{eq_RATE_kom}) is obtained only\nfrom $\\Pi_{\\rm AL}^{R ij}(\\bm{k},\\omega)$ as \n\\begin {align}\n&g_{\\mu\\nu} \\tilde\\Pi_{\\rm fluc}^{\\mu\\nu}(k)\n= \\frac{\\bm{k}^2}{(i\\nu_l)^2} \\tilde\\Pi_{\\rm AL}^{11}(k) - \\sum_{i=1}^{3} \\tilde\\Pi_{\\rm AL}^{ii}(k) \n\\nonumber \\\\\n&=3 \\int \\frac{d^3\\bm{q}}{(2\\pi)^3} \n\\bigg[ \\frac{\\bm{k}^2}{(i\\nu_l)^2} \\big(\\tilde\\Gamma^1 (q, q+k) \\big)^2 \n-\\sum_i \\big(\\tilde\\Gamma^i (q, q+k) \\big)^2 \\bigg] \n\\nonumber \\\\\n&\\qquad\\qquad\\times\n\\oint_C \\frac{dq_0}{2\\pi i} \\frac{\\coth \\frac{q_0}{2T}}{2}\n\\tilde\\Xi(\\bm{q}+\\bm{k}, q_0+i\\nu_l)\\tilde\\Xi(\\bm{q}, q_0)\n\\label{eq:gPi'} \\\\\n&=3 \\int \\frac{d^3\\bm{q}}{(2\\pi)^3} \n\\bigg[ \n\\frac{\\bm{k}^2}{(i\\nu_l)^2} \\big(\\tilde\\Gamma^1 (q, q+k) \\big)^2 - \\sum_i \\big(\\tilde\\Gamma^i (q, q+k) \\big)^2 \n\\bigg] \n\\nonumber \\\\\n&\\qquad\\quad \\times \\bigg\\{\nP \\int \\frac{d \\omega '}{2\\pi i} \\frac{{\\rm coth} \\frac{\\omega '}{2T}}{2}\n\\Xi^R(\\bm{q}+\\bm{k}, \\omega '+i\\nu_l) \\Xi^R(\\bm{q}, \\omega ') \n\\nonumber \\\\\n&\\qquad\\quad\\ \\ - P \\int \\frac{d \\omega '}{2\\pi i} \\frac{{\\rm coth} \\frac{\\omega '}{2T}}{2}\n\\Xi^R(\\bm{q}+\\bm{k}, \\omega '+i\\nu_l) \\Xi^A(\\bm{q}, \\omega ') \n\\nonumber \\\\\n&\\qquad\\quad\\ \\  + P \\int \\frac{d \\omega '}{2\\pi i} \\frac{{\\rm coth} \\frac{\\omega '}{2T}}{2}\n\\Xi^R(\\bm{q}+\\bm{k}, \\omega ') \\Xi^A(\\bm{q}, \\omega '-i\\nu_l) \n\\nonumber \\\\\n&\\qquad\\quad\\ \\  - P \\int \\frac{d \\omega '}{2\\pi i} \\frac{{\\rm coth} \\frac{\\omega '}{2T}}{2}\n\\Xi^A(\\bm{q}+\\bm{k}, \\omega ') \\Xi^A(\\bm{q}, \\omega '-i\\nu_l) \n\\bigg\\},\n\\label{eq:gPi}\n\\end {align}\nwhere the contour $C$ in Eq.~(\\ref{eq:gPi'}) surrounds\nthe poles of $\\coth (q_0\/2T)$\nand $\\Xi^A(\\bm{k}, \\omega) = \\tilde\\Xi(k)|_{i\\nu_l\\to\\omega-i\\eta}$ is\nthe advanced T-matrix.\nThe far right-hand side Eq.~(\\ref{eq:gPi})\nis obtained after deforming the contour $C$\navoiding the cut in $\\tilde\\Xi(q)$ on the real axis~\\cite{book_Larkin}.\nBy taking the analytic continuation\n$i\\nu_l \\rightarrow \\omega + i\\eta$ and using\n$\\Xi^A(\\bm{k}, \\omega) =[\\Xi^R(\\bm{k}, \\omega)]^*$, \nwe obtain\n\\begin {align}\n&g_{\\mu\\nu} {\\rm Im} \\Pi_{\\rm fluc}^{R\\mu\\nu}(\\bm{k}, \\omega) \n  =3 e_\\Delta^2 \\int_{-2\\Lambda-2\\mu}^{2\\Lambda-2\\mu} \\frac{d \\omega '}{2\\pi}\n  \\int \\frac{d^3\\bm{q}}{(2\\pi)^3} {\\rm coth} \\frac{\\omega '}{2T} \n\\nonumber \\\\\n&\\times \n\\big(Q_{(1)}(\\bm{q}+\\bm{k},\\bm{q})\\big)^2\n\\Bigg[ \\Bigg(\\frac{(\\bm{q}+\\bm{k})^2-\\bm{q}^2}{\\omega}\\Bigg)^2 - (2\\bm{q}+\\bm{k})^2 \\Bigg]\n\\nonumber \\\\\n&\\times \n{\\rm Im} \\Xi^R (\\bm{q}+\\bm{k}, \\omega ') \n\\Big\\{\n{\\rm Im} \\Xi^R(\\bm{q}, \\omega '+\\omega) - {\\rm Im} \\Xi^R(\\bm{q}, \\omega '-\\omega)\n\\Big\\}.\n\\label{eq:ImPi}\n\\end {align}\nTo deal with the momentum integral in Eq.~(\\ref{eq:ImPi}),\nwe introduce the ultraviolet cutoff with the same procedure\nas in Ref.~\\cite{Kitazawa:2005vr}.\nThe DPR is obtained by substituting this result\ninto Eq.~(\\ref{eq_RATE_kom}).\n\n\\begin{figure*}[tbp]\n\\begin{tabular}{ccc}\n      \\begin{minipage}[t]{0.31\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 917 424, keepaspectratio, scale=0.31]{RATE-kom_Gc0.7Gs_mu350.pdf}\n      \\end{minipage} &\n      \\begin{minipage}[t]{0.31\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 918 428, keepaspectratio, scale=0.31]{RATE-kom_Gc0.7Gs_mu400.pdf}\n      \\end{minipage} &\n      \\begin{minipage}[t]{0.31\\hsize}\n        \\centering\n        \\includegraphics[bb = 0 0 917 410, keepaspectratio, scale=0.31]{RATE-kom_Gc0.7Gs_mu500.pdf}\n      \\end{minipage} \n\\end{tabular}\n\\caption{\nDilepton production rates per unit energy and momentum \n$d^4 \\Gamma \/ d\\omega d^3k$\nat $\\bm{k}=0$ for several values of $T\/T_c$ with\n$\\mu=350$~MeV (left), $400$~MeV (middle)\nand $500$~MeV (right) and $G_{\\rm C}=0.7G_{\\rm S}$. \nThe thick-red (thin-blue) lines show the contribution of \n$\\tilde\\Pi_{\\rm fluc}^{\\mu\\nu}(k)$ ($\\tilde\\Pi_{\\rm free}^{\\mu\\nu}(k))$.\n}\n\\label{fig_RATE_0om}\n\\end{figure*}\n\n\n\n\n\n\\section{Numerical results}\n\nIn Fig.~\\ref{fig_RATE_0om}, we show the numerical results of\nthe production rate $d^4\\Gamma \/ d^4k$ \nper unit energy and momentum at $\\bm{k}=\\bm{0}$\ncalculated with use of the photon self-energy Eq.~(\\ref{eq_Pi_all})\nand Eq.~(\\ref{eq:ImPi})\nfor various values of $T$ and $\\mu$ at $G_{\\rm C} = 0.7G_{\\rm S}$.\nThe thick lines show the contribution of diquark fluctuations\nobtained from $\\Pi_{\\rm fluc}^{R\\mu\\nu}(\\bm{k},\\omega)$,\nwhile the thin lines are the results for the free quark gas.\nThe total rate is given by the sum of these two contributions.\nThe figure shows that the production rate \nis enhanced so much by the diquark fluctuations \nthat it greatly exceeds that of the free quarks\nin the low energy region $\\omega\\lesssim300$~MeV.\nThe enhancement is more pronounced as $T$ is lowered toward $T_c$,\nwhile the enhancement at $\\omega\\simeq200$~MeV\nis observed up to $T \\simeq 1.5T_c$.\nThe figure also shows that the contribution of diquark fluctuations is\nmore enhanced as $\\mu$ becomes larger.\nThis behavior is understood as the effect of the\nlarger Fermi surface for larger $\\mu$.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.43\\textwidth]{figure_pair-annihilation-crop.pdf}\n\\caption{Diagrams representing the processes of a virtual photon production.\n}\n\\label{fig_mechanism}\n\\end{figure}\n\nIt is found worth scrutinizing the underlying mechanism \nof the low-energy enhancement of the production mechanism\nof virtual photons. \nAlthough it is rather natural that $d^4\\Gamma \/ d^4k$ is enhanced\nin the  low energy region since the virtual photons are emitted from the\nsoft collective modes,\ntheir pronounced effects on the production of virtual photons\nin the {\\em time-like} region deserves an elucidation\nsince the soft mode has a dominant strength \nin the {\\em space-like} region as shown in Fig.~\\ref{fig_DSF}.\nIn our formalism, the virtual photons are dominantly emitted\nthrough the process obtained by cutting Fig.~\\ref{fig_selfenergy} (a),\ni.e. the scattering of diquarks\nshown in the left panel of Fig.~\\ref{fig_mechanism}.\nIn this process, energy-momentum of the virtual photon\n$k=(\\bm{k},\\omega)$ can be time-like, $\\omega>\\vert \\bm{k}\\vert$,\nsince \nthe absolute value of the momentum $\\bm{k}=\\bm{q}_1-\\bm{q}_2$ can be taken\narbitrarily small keeping $\\omega=\\omega_1-\\omega_2$ finite.\nThis kinematics is contrasted to the scattering of massless quarks\nshown in the right panel of Fig.~\\ref{fig_mechanism}, in which \nthe produced virtual photon is always in the space-like region\n$\\omega<\\vert \\bm{k}\\vert$.\nHowever, $\\omega=\\omega_1-\\omega_2$ of a virtual photon is restricted\nto small values due to the small energies $\\omega_1$ and $\\omega_2$\nof diquarks.\nThe sharp peak of $d^4\\Gamma \/ d^4k$ in Fig.~\\ref{fig_RATE_0om}\nis understood in this way.\n\nTo have more detailed properties of the enhancement of DPR,\nwe show in the far left panel of Fig.~\\ref{fig_RATE}\na three-dimensional plot of DPR \nin the $\\omega$--$|\\bm{k}|$ plane for several values of $T$\nat $\\mu=350$~MeV and $G_{\\rm C}=0.7G_{\\rm S}$.\nWe see that the DPR is enhanced strongly \naround the origin in the $\\omega$--$|\\bm{k}|$ plane, \nand the larger $\\omega$ and\/or  $|\\bm{k}|$, the smaller the DPR. \nThis behavior is in accordance with the mechanism explained above.\n\n\\begin{figure*}[tbp]\n\\begin{tabular}{ccc}\n\\begin{minipage}[t]{0.32\\hsize}\n\\centering\n\\includegraphics[keepaspectratio, scale=0.47]{RATE-3d.pdf}\n\\end{minipage} &\n\\begin{minipage}[t]{0.32\\hsize}\n\\centering\n\\includegraphics[keepaspectratio, scale=0.3]{RATE-M-Tratio.pdf}\n\\end{minipage} \n\\begin{minipage}[t]{0.32\\hsize}\n\\centering\n\\includegraphics[keepaspectratio, scale=0.3]{RATE-M-Tfix.pdf}\n\\end{minipage} \n\\end{tabular}\n\\caption{\n  \\textbf{Left}: Dilepton production rates $d^4 \\Gamma \/ d\\omega d^3k$\n  as a function of $\\omega$ and $|\\bm{k}|$\n  for $\\mu=350$~MeV and $T=1.01~T_c$, $1.1~T_c$ and $1.5~T_c$ \n  at $G_{\\rm C}=0.7G_{\\rm S}$.\n  The gray surface shows the light-cone. \n  \\textbf{Middle}: \n  The invariant-mass spectrum $d\\Gamma\/dM^2$ for several values of $T\/T_c$\n  at $G_{\\rm C}=0.7G_{\\rm S}$ and $\\mu =350$~MeV.\n  \\textbf{Right}: The invariant-mass spectrum $d\\Gamma\/dM^2$\n  at $(T,\\mu) =(90,350)$~MeV \n  for $G_{\\rm C}=0.9G_{\\rm S}$ (solid), $0.7G_{\\rm S}$ (dashed) and $0.5G_{\\rm S}$ (dash-dotted). \n}\n\\label{fig_RATE}\n\\end{figure*}\n\nIn the HIC experiments, the dilepton production rate is usually measured\nas a function of the invariant mass, $M$,\n\\begin {align}\n  \\frac{d\\Gamma}{dM^2}\n  = \\int d^3k \\frac{1}{2\\omega} \\frac{d^4\\Gamma}{d^4k} \\bigg|_{\\omega=\\sqrt{k^2+M^2}}\\ .\n  \\label{eq:dGdM}\n\\end {align}\nIn the middle panel of Fig.~\\ref{fig_RATE}, we show Eq.~(\\ref{eq:dGdM})\nfor several values of $T\/T_c$ at $G_{\\rm C}=0.7G_{\\rm S}$ and $\\mu = 350$~MeV.\nOne sees that the enhancement due to diquark fluctuations\nis observed in the low invariant-mass region \n$M < (150 -200)$~MeV up to $T\\simeq1.5T_c$.\nThe little $T$ dependence of DPR seen in the far low region of $M$\nmay be understood as a result of an accidental \ncancellation between the enhanced spectral function due to \nthe soft mode and the kinematical thermal effect:\nThe sharp enhancement of the former \nat low energy-momentum near $T_c$ decreases while\nthe creation probability due to the thermal effect \nincreases as $T$ goes high.\nThe contribution of the diquark fluctuations is relatively suppressed\nfor higher $T$ as the contribution of free quarks becomes larger.\n\nFinally, shown in the right panel of Fig.~\\ref{fig_RATE} is\n$d\\Gamma\/dM^2$ at fixed $(T,\\mu)=(90,350)$~MeV\n(the cross symbol in Fig.~\\ref{fig_phase})\nfor several values of $G_{\\rm C}$.\nThe panel shows that the production rate is more enhanced\nfor larger $G_{\\rm C}$ and $T_c$.\nFor $G_{\\rm C}=0.9G_{\\rm S}$ ($T_c\\simeq78$~MeV),\nthe production rate from the diquark fluctuations exceeds those of\nthe free quarks for $M\\lesssim100$~MeV.\n\n\n\\section{Discussions}\n\nIn this Letter, we have investigated the effect of diquark\nfluctuations on the DPR \nnear but above the critical temperature of the 2SC.\nThe contribution of the diquark fluctuations were \ntaken into account through the AL, MT and DOS terms\nin the photon self-energy.\nWe have found that the dilepton production rate \nis strongly enhanced in comparison with the free-quark gases \nin the low energy and low invariant-mass regions near $T_c$\nup to $T\\simeq1.5T_c$ reflecting the formation of the diquark soft mode\nassociated with the phase transition to 2SC.\n\nWe would say that it should be rewarding to try to make \nan experimental measurement of dileptons in that far low-mass region  \nand examine the possible enhancement of the DPR in the HIC; \nif the enhancement is confirmed, it may possibly give an experimental \nevidence of strong diquark correlations, which lead to\nthe phase transition to CSC in dense quark matter.\nMoreover, it is to be noted that the\nDPR with vanishing energy\/momentum is\ndirectly related to \nthe electric conductivity, as is evident \nfrom the fact that the AL, MT and DOS terms in condensed \nmatter physics are responsible for the anomalous\nenhancement of the electric conductivity (paraconductivity)\nin metals above $T_c$ but in the close \nvicinity of the superconducting phase.\n\nThere are, however, many issues to be resolved\nfor making the measurements meaningful, in the \nsense that it can help in revealing the significance of the\ndiquark fluctuations prior to the phase transition to the 2SC in the \ndense matter.\nSince observed yield of the dilepton production in the HIC is\na superposition of those with various origins in the space-time history,\nwe need to `disentangle' the observed total yield into \nthose with the respective origins.\nFor that, it is necessary to quantitatively estimate \nthe residence time around the phase boundary of the CSC, say,\nwith resort to dynamical models~\\cite{Nara:2021fuu}.\nEven when some enhancement of the DPR in the \nlow $M$ region is identified, it is to be noted that\nit may have come from a different  mechanism\ndue to medium effects~\\cite{Rapp:2009yu,Laine:2013vma,Ghiglieri:2014kma}.\nA comparison of our results with these effects \nconstitutes future projects.\n\nThe experimental measurement of the DPR in the \nrelevant low invariant-mass region $M\\lesssim200$~MeV is not an easy task\nbecause di-electrons, which are, among dileptons, only available \nin this energy range, are severely contaminated by the Dalitz decays, and\nhigh-precision measurements both of $d\\Gamma\/dM^2$ and hadron spectrum\nare necessary to extract interesting medium effects.\nDespite these challenging requirements, \nit is encouraging\nthat the future HIC programs in GSI-FAIR, NICA-MPD and J-PARC-HI\nare designed to carry out high-precision experiments~\\cite{Galatyuk:2019lcf},\nand also that new technical developments are vigorously being \nmade~\\cite{Adamova:2019vkf}.\n\nFinally, we  remark that such an effort to reveal the significance of the \nenhanced diquark correlations in the hot and dense matter should also \ngive some clue to\nthe modern development \nof hadron physics where \npossible diquark correlations in  hadron structures are\none of the hot topics~\\cite{Barabanov:2020jvn}.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nThe authors thank Naoki Yamamoto for his critical comments. \nT.~N. thanks JST SPRING (Grant No.~JPMJSP2138) and\nMultidisciplinary PhD Program for Pioneering Quantum Beam Application.\nThis work was supported by JSPS KAKENHI \n(Grants No.~JP19K03872, No.~JP90250977, and No.~JP10323263).\n\n\n\n\n\n\n\\bibliographystyle{elsarticle-num} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe recent measurement of the \nmuon\nanomalous \nmagnetic moment,  \n$a_\\mu \\equiv (g_\\mu -2) \/ 2$, \nby the \nE989 experiment at Fermilab \\cite{Abi:2021gix}, in agreement with the previous \nBNL E821 result \\cite{Bennett:2006fi}, implies a 4.2$\\sigma$ discrepancy from the \nStandard Model (SM) \n\\begin{equation} \n\\label{eq:Deltaamu} \n\\Delta a_\\mu \\equiv a_\\mu(\\text{Exp}) - a_\\mu(\\text{SM}) = ( 251 \\pm 59 ) \\times 10^{-11} \\, ,\n\\end{equation}\nfollowing the Muon $g-2$ Theory Initiative recommended value \nfor the SM theory prediction \\cite{Aoyama:2020ynm}. \nAlthough a recent lattice determination of the \nSM hadron vacuum polarization contribution to \n$a_\\mu$ claims no sizeable deviation from the SM \\cite{Borsanyi:2020mff}, \nwe will \nwork here under \nthe hypothesis that $\\Delta a_\\mu$ is due to new physics. \nIn particular, we will focus on the case in which new physics states are so heavy that \ntheir effects can be parameterized \nvia the so-called SM Effective Field Theory (SMEFT) and ask the following question: \n\\emph{What is the scale of new physics behind $\\Delta a_\\mu$?} \n\nThis question is of practical relevance, given the futuristic possibility of resolving the new physics origin of \n$\\Delta a_\\mu$ via direct searches at high-energy particle colliders. \nAs explored recently in \n\\cite{Capdevilla:2020qel,Buttazzo:2020eyl,Capdevilla:2021rwo}, \na muon collider seems to be the best option for this goal. \nHowever, while \nthe very existence of the SMEFT operators contributing to $\\Delta a_\\mu$ \ncould be tested via processes like \n$\\mu^+ \\mu^- \\to Z(\\gamma) h$ or $\\mu^+ \\mu^- \\to t \\bar t$ at a multi-TeV-scale muon collider \\cite{Buttazzo:2020eyl}, \nit is less clear whether the origin of the muon $g-2$ SMEFT operators can be resolved \nvia the direct production of new on-shell states \nresponsible for $\\Delta a_\\mu$. \nThis is the question that we want to address in the present work, using the \ntools of perturbative unitarity. Unitarity bounds on the new physics interpretation of $\\Delta a_\\mu$  \nwere previously considered in \\cite{Capdevilla:2020qel,Capdevilla:2021rwo} focusing however \non a specific class of renormalizable models. Here, \nwe will consider instead the most conservative case in which unitarity limits \nare obtained within the SMEFT \nand \nreach a more pessimistic conclusion about the possibility of \nestablishing a no-lose theorem for testing the origin of $\\Delta a_\\mu$ \nat a future high-energy particle collider. \n\nGenerally speaking, \ngiven a low-energy determination of an EFT coefficient,  \nunitarity methods \ncan be used either within an EFT approach,  \nin order to infer an upper bound on the scale of new physics \nunitarizing EFT scattering amplitudes,  \nor within explicit new physics (renormalizable) models. \nIn the latter case, one obtains a perturbativity bound on certain renormalizable couplings \nthat can be translated into an upper bound on the mass of new on-shell degrees of freedom. \nIn the present work we will be interested in both these approaches.  \nFirst, \nwe will consider a SMEFT analysis in \nwhich \n$\\Delta a_\\mu$\nis explained in terms of \na set of Wilson coefficients normalized to some \ncut-off scale$^2$, $C_i \/ \\Lambda^2$, and later deal with renormalizable models \nfeaturing new heavy mediators that can be matched onto the SMEFT. \nSchematically, \n\\begin{equation} \n\\label{eq:Damustrategy}\n\\Delta a _\\mu \\sim \\frac{C_i}{\\Lambda^2} = \\frac{\\text{(loops)} \\times \\text{(couplings)}}{M^2_{\\rm on-shell}} \\, , \n\\end{equation} \nwhere $M_{\\rm on-shell}$ denote the mass of new on-shell states \nand we included possible loop factors in the matching between the new physics model and the SMEFT operators. \nHence, by fixing the value of the SMEFT coefficients $C_i \/ \\Lambda^2$ in terms of $\\Delta a_{\\mu}$, \nwe will \nconsider high-energy scatterings sourced by the associated effective operators,\ndetermine the $\\sqrt{s}$ that saturates perturbative unitarity \n(according to a standard \ncriterium to be specified in \\sect{sec:unitarity}) and interpret the latter as an \nupper bound on the \nscale of new physics responsible for the muon $g-2$ anomaly. Analogously, in the case of \nnew physics models, we will use the unitarity tool in order to set \nperturbativity bounds on the new physics couplings and in turn (given \\eq{eq:Damustrategy})\nan upper limit \non $M_{\\rm on-shell}$. \nWhile the first approach is model-independent (barring possible degeneracies in the choice of the effective operators) \nand yields the most conservative bound on the scale of new physics, \nthe second approach relies on further assumptions, but it directly connects \nto new on-shell degrees of freedom which are the prime targets of direct searches \nat high-energy particle colliders. \n\nThe paper is structured as follows. We start in \\sect{sec:unitarity} with a brief review \nof partial wave unitarity, in order to set notations and clarify the physical interpretation \nof unitarity bounds. Next, we consider unitarity bounds within a SMEFT approach (\\sect{sec:SMEFT}) \nand within \nrenormalizable models matching onto the SMEFT operators (\\sect{sec:simplmodels}). \nFinally, we comment in \\sect{sec:strongmodels} on non-renormalizable realizations which \ncan saturate the \nunitarity bounds obtained in the SMEFT.   \nOur main findings and implications for the direct resolution of \nthe muon $g-2$ anomaly at high-energy particle colliders \nare summarized in the conclusions (\\sect{sec:conclusions}). \nTechnical aspects of partial wave unitarity calculations, both in the SMEFT and in renormalizable setups, \nare deferred to \\apps{app:unitaritySMEFT}{app:unitarityweakly}. \n\n\n\\section{Partial wave unitarity}\n\\label{sec:unitarity}\n\nWe start with an instant review of \npartial wave unitarity, which will serve to set notations and discuss the physical \nsignificance of unitarity bounds. \n\nThe key point of our analysis is the study of scattering amplitudes  with fixed total angular momentum $J$, the so-called partial waves.\nHere we focus \nonly on the case of $2 \\to 2$ partial waves (while the $2\\to 3$ scattering is discussed in \\app{app:2to3scatterings}) defined as\n\\begin{equation}\n\ta^J_{fi} = \n\t\\frac{1}{32\\pi} \\int_{-1}^1 \\mathrm{d} \\cos\\theta \\, d^J_{\\mu_i\\mu_f}(\\theta) \\, \\mathcal{T}_{fi}(\\sqrt{s},\\cos\\theta) \\,,\n\t\\label{eq:partialwaves}\n\\end{equation}\nwith $\\theta$ the scattering angle in the centre-of-mass frame,\n$(2\\pi)^4\\delta^{(4)}(P_i-P_f)i\\mathcal{T}_{fi}(\\sqrt{s},\\cos\\theta) = \\bra{f} S-1 \\ket{i}$ and $S$ the $S$-matrix. Here, $d^J_{\\mu_i\\mu_f}$ is Wigner's \n$d$-function that arises in the construction of the two-particle incoming (outcoming) state of helicities  $\\mu_{i} $ ($\\mu_{f} $) onto angular momentum $J$ \n\\cite{Jacob:1959at}. \nThe $S$-matrix unitarity condition $S^\\dagger S =1$ then yields the relation\n\\begin{equation}\n\t\\frac{1}{2i}(a^J_{fi} - a^{J^*}_{if}) = \\sum_h a^{J^*}_{hf} a^J_{hi} \\qquad \\Longrightarrow \\qquad \\mbox{Im}\\,(a_{ii}^J) \n\t= \\sum_h | a^J_{hi} |^2 \\geq | a^J_{ii} |^2 \\,,\n\t\\label{eq:Imaii}\n\\end{equation}\nwhere we have restricted ourselves to the elastic channel $h=i=f$. \nThe equation on the right hand side of \\eqref{eq:Imaii} defines a circle in the complex plane inside which the amplitude must lie at all orders,\n\\begin{equation}\n\t\\Big(\\mbox{Re}\\, a^J_{ii} \\Big)^2 + \\Big(\\mbox{Im}\\, a^J_{ii} - \\frac{1}{2} \\Big)^2 \\leq \\frac{1}{4} \\,,\n\\end{equation}\nsuggesting the following bound, under the assumption of real tree-level amplitudes:\n\\begin{equation}\n\t|\\mbox{Re}\\, a^{J}_{ii}| \\leq \\frac{1}{2} \\,. \n\t\\label{eq:partialwavebound}\n\\end{equation} \nHence, \nin order to extract the bound, one needs to fully diagonalize the matrix $a^J$. Once this is achieved, every eigenvalue \nwill give an independent constraint. \nIn the presence of multiple scattering channels, it follows from \n\\eq{eq:partialwavebound} that the strongest bound arises from the largest eigenvalue of $a^J$.\nWhen the latter bound is saturated, \nit basically means that one needs a correction of at least $40$\\% from \nhigher orders to get back inside the unitarity circle, \nthus signaling the breakdown of perturbation theory (see e.g.~\\cite{DiLuzio:2016sur,DiLuzio:2017chi}). \nHere, $a^{J}$ stands for \nthe leading order expansion of the partial wave, \nboth in the coupling constants \nand in external momenta over cut-off scale \nfor the case of an EFT. \n\nAlthough the criterium is somewhat arbitrary, \nand hence \\eq{eq:partialwavebound} should not be understood as a strict bound, \nwe stick to that for historical reasons \\cite{Lee:1977eg}. \nStrictly speaking, a violation of the perturbative unitarity criterium \nin \\eq{eq:partialwavebound} \nshould be conservatively interpreted as the onset of a regime of incalculability \ndue to the breakdown of the perturbative expansion either in couplings or external momenta. \nMore specifically, \nin the case of an EFT \n(where scattering amplitudes grow with energy)\nthe scale of unitarity violation,  \nhereafter denoted as \n\\begin{equation} \n\\label{eq:LambdaU}\n\\sqrt{s} = \\Lambda_U \\qquad \\Longrightarrow \\qquad \n|\\mbox{Re}\\, a^{J}_{ii}| = \\frac{1}{2} \\, , \n\\end{equation}\ncan be associated with the onset of ``new physics'', \nwhere on-shell new degrees of freedom should \nmanifest themselves and be kinematically accessible. \nAlthough one can conceive exotic UV \ncompletions where this is not the case \\cite{Dvali:2010jz}, \nwell-known \nphysical systems \nbehave in this way.\\footnote{Most notably, $\\pi\\pi$ scattering in chiral perturbation theory \nyields $\\Lambda_U = \\sqrt{8\\pi} f_\\pi \\simeq 460$ MeV which is not far from \nthe mass of the $\\rho$ meson resonance.}  \nUnitarity methods can be employed in renormalizable setups as well. In this case, \nthe unitarity limit corresponds to the failure \nof the \ncoupling expansion \nand hence the bound on the renormalizable coupling \ncan be understood as a perturbativity constraint. \n\n\n\n\\section{SMEFT} \n\\label{sec:SMEFT}\n\nIn this section we present the unitarity bounds for the new physics \ninterpretation of \nthe muon $g-2$ anomaly \nwithin a SMEFT approach.   \nThe strategy consists in fixing the Wilson coefficients \n($C_i \/ \\Lambda^2$)\nin terms of the observable $\\Delta a_\\mu$ \nand determine next the energy scale $\\sqrt{s}$ that saturates the unitarity bounds \nderived from the tree-level scattering amplitudes sourced by the effective operator. \nThe shorthand $1\/\\Lambda_i^2 \\equiv C_i \/ \\Lambda^2$ is understood in the following. \n\n\n\n\n\\subsection{SMEFT approach to $\\Delta a_\\mu$}\n\nAssuming a short-distance new physics origin of \n$\\Delta a_\\mu$, \nthe leading SMEFT operators contributing up to one-loop order are \n(see Refs.~\\cite{Buttazzo:2020eyl,Aebischer:2021uvt} for a more systematic discussion)\n\\begin{align} \n\\label{eq:LSMEFTgm2}\n\\mathscr{L}_{g-2}^{\\rm SMEFT} &= \n\\frac{C^\\ell_{eB}}{\\Lambda^2} (\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) H B_{\\mu\\nu} \n+ \\frac{C^\\ell_{eW}}{\\Lambda^2} (\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) \\tau^I H W^I_{\\mu\\nu} \\nonumber \\\\\n&+ \\frac{C^{\\ell q}_{T}}{\\Lambda^2} (\\bar \\ell^a_L \\sigma_{\\mu\\nu} e_R) \n\\varepsilon_{ab} (\\bar Q^b_L \\sigma^{\\mu\\nu} u_R) + \\text{h.c.} \\, , \n\\end{align}\nwhich results in \\cite{Buttazzo:2020eyl} \n\\begin{equation} \n\\label{eq:gm2fromSMEFT}\n\\Delta a_{\\ell} \\simeq \\frac{4 m_\\ell v}{e\\sqrt{2}\\Lambda^2} \n\\( \\mbox{Re}\\, C^\\ell_{e\\gamma} - \\frac{3\\alpha}{2\\pi} \\frac{c^2_W-s^2_W}{s_W c_W} \n\\mbox{Re}\\, C^{\\ell}_{eZ} \\log\\frac{\\Lambda}{m_Z} \\) \n- \\sum_{q=t,c,u} \\frac{4m_\\ell m_q}{\\pi^2} \\frac{\\mbox{Re}\\, C^{\\ell q}_T}{\\Lambda^2} \n\\log\\frac{\\Lambda}{m_q} \\, , \n\\end{equation}\nwhere $C_{e\\gamma} = c_W C_{eB} - s_W C_{eW}$ and \n$C_{eZ} = -s_W C_{eB} - c_W C_{eW}$, \nin terms of the weak mixing angle. \nFor the Wilson coefficients of the dipole operators \nthat contribute at tree level to $\\Delta a_{\\ell}$, one can \nconsistently \ninclude one-loop \nrunning,  \nobtaining \\cite{Degrassi:1998es,Alonso:2013hga}\n\\begin{equation} \nC^{\\ell}_{e\\gamma} (m_\\ell) \\simeq C^{\\ell}_{e\\gamma} (\\Lambda) \n\\( 1 - \\frac{3y^2_t}{16\\pi^2} \\log\\frac{\\Lambda}{m_t} - \\frac{4\\alpha}{\\pi} \\log\\frac{\\Lambda}{m_\\ell} \\) \\, . \n\\end{equation}\nA convenient numerical parameterization reads \n\\begin{equation}\n\\label{eq:DeltaamuSMEFT}\n\\Delta a_{\\mu} \\simeq \n2.5 \\times 10^{-9} \n\\( \\frac{277 \\ \\text{TeV}}{\\Lambda} \\)^2 \n\\( \\mbox{Re}\\, C^\\mu_{e\\gamma} (\\Lambda) - 0.28 \\, \\mbox{Re}\\, C^{\\mu t}_T (\\Lambda)\n- 0.047 \\, \\mbox{Re}\\,  C^\\mu_{eZ} (\\Lambda) \\) \\, , \n\\end{equation}\nwhere we have kept only the leading top-quark contribution for $C_T$ \n(since we are interested on scenarios which maximize the scale of new physics)\nand \nthe logs have been evaluated for $\\Lambda = 277$ TeV.   \nNote, however, that the full log dependence will be retained in the numerical analysis below. \nIn the following, we will drop the scale \ndependence of the Wilson coefficients, \nwhich are understood to be evaluated at the scale $\\Lambda$. \n\n\\subsection{Unitarity bounds} \n\n\nGiven \\eq{eq:LSMEFTgm2}, we can compute the scale of unitarity violation $\\Lambda_U$ \n(defined via \\eq{eq:LambdaU})\nassociated with each of the dimension-6 operators involved. \nTo do so, we consider here only $2\\to 2$ scattering processes,  \nsince the $2 \\to 3$ processes \n(mediated by $\\mathcal{O}_{eW}$) turn out to be suppressed by the weak gauge coupling and the \n3-particle phase space, as shown in Appendix \\ref{app:2to3scatterings}.\nThe results obtained by switching one operator per time are collected in Table \\ref{tab:SMEFTunitaritybounds}, where \nthe bound in correspondence of different initial and final states \n($i \\neq f$)\ncomes from the diagonalization of the scattering matrix (cf.~discussion below \\eq{eq:partialwavebound}). \nIn \\app{app:unitaritySMEFT} we present the full calculation of the unitarity bounds \nstemming from the ${\\rm SU}(2)_L$ dipole operator, which presents several non-trivial aspects, \nlike the presence of higher than $J=0$ partial waves, the multiplicity in ${\\rm SU}(2)_L$ space and the possibility of \n$2 \\to 3$ scatterings. \n\\begin{table}[!t]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|}\n\t\\rowcolor{CGray} \n\t\\hline\n\t\tOperator & $\\Lambda_U$ & $i \\to f$ Channels & $J$ \\\\ \\hline\n\t\t$\\frac{1}{\\Lambda^2_{eB}}(\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) H B_{\\mu\\nu} $ & $2\\sqrt{\\pi}|\\Lambda_{eB}|$ & $B e_R \\to H^\\dagger \\ell_L$ & $1\/2$ \\\\\n\t\t$\\frac{1}{\\Lambda^2_{eW}}(\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) \\tau^I H W^I_{\\mu\\nu}$ & $2\\sqrt{\\pi}\\left(\\frac{2}{3}\\right)^{1\/4}|\\Lambda_{eW}|$ & $W \\bar\\ell_L \\to H^\\dagger \\bar e_R$ & $1\/2$ \\\\ \n\t\t$\\frac{1}{\\Lambda^2_{T,\\ell}}(\\bar \\ell^a_L \\sigma_{\\mu\\nu} e_R) \n\\varepsilon_{ab} (\\bar Q^b_L \\sigma^{\\mu\\nu} u_R)$ & $2\\sqrt{\\frac{\\pi}{3\\sqrt{2}}}|\\Lambda_{T,\\ell}|$ & $e_R u_R \\to Q_L \\ell_L $ & $0$ \\\\\n\\hline\n\t\\end{tabular}\t\n\t\\caption{\\label{tab:SMEFTunitaritybounds} \n\tUnitarity violation scale for the SMEFT operators \n\n\tcontributing \n\tto $\\Delta a_\\mu$.\n\n\n\t}\n\\end{table}\n\n\nWe next make contact with the physical observable $\\Delta a_\\mu$, \nwhose dependence from the Wilson coefficients can be read off \\eq{eq:DeltaamuSMEFT}.  \nTurning on one operator per time, we find the following numerical values for the \nunitarity violation scales:  \n\\begin{itemize}\n\\item $\\mathcal{O}^{\\mu}_{eB} \\equiv (\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) H B_{\\mu\\nu}$\n\\begin{equation}\n\\label{eq:LamUOB}\n\\Lambda_U  \\simeq 277 \\ \\text{TeV} \\ 2\\sqrt{\\pi} \\sqrt{c_W+ 0.047s_W} \\simeq 930 \\ \\text{TeV} \\, .\n\\end{equation}\n\\item $\\mathcal{O}^{\\mu}_{eW} \\equiv (\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) \\tau^I H W^I_{\\mu\\nu}$\n\\begin{equation} \n\\label{eq:LamUOW}\n\\Lambda_U \\simeq 277 \\ \\text{TeV} \\ 2\\sqrt{\\pi} \\left(\\frac23\\right)^{1\/4}\\sqrt{s_W - 0.047c_W} \\simeq 590 \\ \\text{TeV} \\, .\n\\end{equation}\n\\item $\\mathcal{O}^{\\mu t}_{T} \\equiv (\\bar \\ell^a_L \\sigma_{\\mu\\nu} e_R) \n\\varepsilon_{ab} (\\bar Q^b_L \\sigma^{\\mu\\nu} u_R)$\n\\begin{equation}\n\\label{eq:LamUOT}\n\\Lambda_U \\simeq 277 \\ \\text{TeV} \\ 2 \\sqrt{\\frac{\\pi}{3\\sqrt{2}}} \\sqrt{0.28} \\simeq 240 \\ \\text{TeV} \\, .\n\\end{equation}\n\\end{itemize}\nHence, the scale of new physics is maximized if the origin of $\\Delta a_\\mu$ \nstems from a dipole operator oriented in the ${\\rm U}(1)_Y$ direction. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figs\/Leb_LeW_2sigma}\n\\caption{In blue, the region in the ($\\Lambda_{eB}$, $\\Lambda_{eW}$) plane that is \nneeded \nto reproduce the experimental value of $\\Delta a_\\mu$ at the $2\\sigma$ level (with the central line corresponding to the central value of $\\Delta a_\\mu$). The dashed iso-lines represent \nthe unitarity bound $\\Lambda_U$, defined according to Eq.~\\eqref{eq:LU_eBeW}. }\n\\label{fig:Leb_LeW}       \n\\end{figure}\n\nIf more than one operator is switched on, correlations can arise between the Wilson coefficients whenever they couple same sectors of the theory. For instance, in the \ncase in which both the dipole operators \n$\\mathcal{O}^\\mu_{eW}$ and $\\mathcal{O}^\\mu_{eB}$ are present \none can derive a combined bound (see \\eq{eq:LU_eBeW}) \nwhich \nleads to the region displayed in \\fig{fig:Leb_LeW}. \nNote that for $\\Lambda_{eB} \\to \\infty$ ($\\Lambda_{eW} \\to \\infty$) \nwe reproduce the \nbound with $\\mathcal{O}^\\mu_{eW}$ ($\\mathcal{O}^\\mu_{eB}$) only.   \nHowever, if both operators \ncontribute sizeably to $\\Delta a_\\mu$, the unitarity bound can be slightly relaxed above \nthe PeV scale.  \n \n\n\\section{Renormalizable models}\n\\label{sec:simplmodels}\n\nWe next consider simplified models \nfeaturing new \nheavy states, which after being integrated out match onto the \ndipole \nand tensor \nSMEFT operators \ncontributing to $\\Delta a_\\mu$\n(cf.~\\eq{eq:DeltaamuSMEFT}). \nWe will then use unitarity methods to set perturbativity limits on renormalizable couplings \nand in turn set an upper bound on the mass of the new on-shell physics states. \nTo maximize the scale of new physics,  \nwe will focus on two renormalizable setups\nbased scalar-fermion Yukawa theories, allowing for \na left-right chirality flip that is either entirely due to new physics \n(\\sect{sec:oneloopdipole}) or with a top Yukawa insertion (\\sect{sec:treetensor}). \n\n\n\\subsection{One-loop matching onto the dipole operator}\n\\label{sec:oneloopdipole} \n  \nIn order to match onto the dipole operator at one loop\nwe consider a simplified model \nwith a new complex scalar $S =(1,1,Y+1)$ \nand two vector-like fermions $F_\\ell=(1, 2, Y + \\frac{1}{2})$ and \n$F_e = (1, 1,Y)$ allowing for a mixing via the SM Higgs (see e.g.~\\cite{Calibbi:2018rzv,Arnan:2019uhr,Capdevilla:2020qel,Capdevilla:2021rwo})  \n\\begin{align}\n\\label{eq:SFF}\n\\mathscr{L}^{g-2}_{\\rm FFS} &= \n\\lambda_L \\bar  F_\\ell \\ell_L S   + \\lambda_R \\bar  F_e e_R S  + \n\\bar F_\\ell ( y_{L} P_L + y_{R} P_R  ) F_e H + \\rm{h.c.} \\nonumber \\\\\n&-  M_\\ell \\bar F_\\ell  F_\\ell -  M_e \\bar F_e  F_e  - m_ S ^2  |S|^2 \n- \\kappa\\, | H |^2\\,  |S|^2 \n- \\lambda_S |S|^4 \\, . \n\\end{align}\nThe FFS model allows for a chirality flip \nof the external leptons \nvia the product of couplings $\\lambda^*_L y_{L,R} \\lambda_R$ \n(cf.~\\fig{fig:Ceg_macthing}), \nwhich can be used to maximize the scale of new physics. \nFor $v y_{L,R}\\ll M_\\ell,M_e,m_S$, we can integrate out the new physics states and find \nat one loop \n\\begin{align}\n\\label{eq:amu5_gen}\n\\dfrac{C^\\mu_{e\\gamma}}{\\Lambda^2}& \n=-\\dfrac{e \\lambda_L^* \\lambda_R}{32\\pi^2 m_{S}^2}\\dfrac{\\sqrt{x_\\ell x_e}}{(x_\\ell - x_e)}\\,\n\\Bigg\\{\nQ_{S} \n\\left[\ny_R  \\(g_S(x_\\ell)-g_S(x_e)\\)\n+\ny_L \\(\\sqrt{\\dfrac{x_\\ell}{x_e}} g_S(x_\\ell)-\\sqrt{\\dfrac{x_e}{x_\\ell}} g_S(x_e)\\)\n\\right]\n\\nonumber\n\\\\\n&\n+Q_{F} \n\\left[\ny_R  \\(g_F(x_\\ell)-g_F(x_e)\\)\n+\ny_L \\(\\sqrt{\\dfrac{x_\\ell}{x_e}} g_F(x_\\ell)-\\sqrt{\\dfrac{x_e}{x_\\ell}} g_F(x_e)\\)\n\\right]\n\\Bigg\\} \\,,\n\\end{align} \nwhere $Q_{S} = Y +1$, \n$Q_{F} = Y$, \n$x_{\\ell,e}=M^2_{\\ell,e}\/m^2_S$\nand the loop functions \nare given by \n\\begin{equation}\n\\label{eq:FG79}\ng_F(x)=\\dfrac{x^2  - 4x + 3+ 2 \\log x}{2 (x-1)^3}\\,, \\qquad\ng_S(x)=\\frac{x^2-2 x\\log x -1}{2 (x-1)^3}\\,.\n\\end{equation}\nThis result agrees with Ref.~\\cite{Crivellin:2021rbq} \nin which the special case $y_{L}=y_{R}$ was considered. \nNote that in \\eq{eq:amu5_gen} we already matched onto the photon \ndipole operator at the scale $\\Lambda$, \nwhile the connection with the \nlow-energy observable $\\Delta a_\\mu$ is given in \n\\eq{eq:DeltaamuSMEFT}. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/Cegamma_macthing}\n\\caption{Sample diagram of the FFS \nmodel matching onto $C^\\mu_{e\\gamma}$ at the scale $\\Lambda$.}\n\\label{fig:Ceg_macthing}       \n\\end{figure}\n\n\nOur goal is to maximize the mass of the lightest new physics state for a fixed value of the Wilson coefficient.  \nThis is achieved in the degenerate limit \n$m_S = M_\\ell = M_e$, \nyielding  \n\\begin{equation}\n\\label{eq:puta}\n\\dfrac{C^\\mu_{e\\gamma}}{\\Lambda^2}=-\\dfrac{e \\lambda_L^* \\lambda_R}{384 \\pi^2 m_S^2}\\[\\(1+2 Y\\) y_L \n- \\(1+4 Y\\) y_R \\] \\,  \\simeq \n\\dfrac{e Y \\lambda_L^* \\lambda_R}{192 \\pi^2 m_S^2}\\(2 y_R- y_L \\) \\, , \n\\end{equation}\nwhere in the last expression we took $Y  \\gg 1$. \n\nThe unitarity bounds for the FFS model are summarized in \\Table{tab:unitarityboundsFFS}, \nwhere in the case of multiple scattering channels the bound corresponds to \nthe highest eigenvalue of $a^J$.\nWe refer to \\app{app:unitarityweakly} for further details on their derivation.  \nApplying these bounds, the maximum value of the \ncombination $|\\mbox{Re}\\, (\\lambda_L^* \\lambda_R (2 y_R-y_L))|$ \nentering \n\\eq{eq:puta} is $\\approx 121$, while $|eY| \\lesssim 3.5$. Hence, we \nobtain \n\\begin{equation} \n\\Delta a_{\\mu} \\simeq \n2.5 \\times 10^{-9} \n\\( \\frac{131 \\ \\text{TeV}}{m_S} \\)^2 \n\\(\n\\dfrac{e Y}{3.5}\\)\\(\\dfrac{\\mbox{Re}\\,\\(\\lambda_L^* \\lambda_R (2 y_R-y_L )\\)}{121}\n\\)\\,,\n\\label{eq:damuFFS}\n\\end{equation}\nwhich shows that the $\\Delta a_\\mu$ explanation in \nthe FFS model requires an upper bound on the mass \nof the new on-shell states of about 130 TeV. \nOn the other hand, due to the extra loop suppression, \nit is not possible to saturate the unitarity bound that was obtained within the SMEFT \n(see \\eq{eq:LamUOB}). \n\n\n\\begin{table}[th!]\n\t\\centering\n\\renewcommand{\\arraystretch}{1.15}\n\t\\begin{tabular}{|c|c|c|}\n\t\\rowcolor{CGray} \n\t\\hline\n\t\tUnitarity bound & $i\\to f$ Channels & $J$ \\\\ \\hline \n\t\t\t\t$\\left|\\mbox{Re}\\,(\\lambda_L^* \\lambda_R)\\right| < 8\\pi$ & $e_R F_{\\ell_R} \\to e_R F_{\\ell_R}$ & $0$ \\\\\n\t\t$\\left|\\mbox{Re}\\,(y_L^* y_R)\\right| < {8\\pi}\/{\\sqrt{2}}$ & $F_{e_R} \\bar F_{e_L} \\to F_{e_R} \\bar F_{e_L}$ & $0$ \\\\\n\t\t$\\left| \\mbox{Re}\\,(y_L^* y_R) \\pm \\sqrt{4|\\lambda_L|^2|\\lambda_R|^2 + (y_L^*)^2 y_R^2} \\right| < 16\\pi$ & $i, f=F_{\\ell_R}\\bar F_{e_L}, e_R \\bar \\ell_L$ & $0$\\\\\n$2 |\\lambda_L|^2 + |\\lambda_R|^2 < 8\\pi$ & $i, f= F_{\\ell_R} \\bar \\ell_L, F_{e_L} \\bar e_R$ & $0$\\\\\n\t\t$|y_R| < \\sqrt{8\\pi}$ & $H F_{\\ell_L} \\to H F_{\\ell_L}$ & $1\/2$ \\\\\n\t\t$|\\lambda_R|^2 + 2|y_L|^2 < 16\\pi$ & $i, f =  S e_R, H^\\dagger F_{\\ell, R}$ & $1\/2$ \\\\\n\t\t$\\left|\\mbox{Re}\\,(y_L \\lambda_L^*)\\right| < {8\\pi}\/{\\sqrt{2}}$ & $ i, f= F_{e_R} S^\\dagger, e_R H$ & $1\/2$ \\\\\n\t\t$|\\lambda_R|^2 + \\sqrt{32|y_L|^2|\\lambda_R|^2 + |\\lambda_R|^4} < 32\\pi$ & $i, f =S \\bar F_{e_L}, H^\\dagger \\ell_L$ & $1\/2$\\\\\n\t\t$\\left|\\mbox{Re}\\,(\\lambda_L y_L)\\right| < {16\\pi}\/{\\sqrt{2}}$ & $\\ell_L \\bar F_{e_L} \\to S H^\\dagger$ & $1$ \\\\\n\t\t$\\left|\\mbox{Re}\\,(y_L^* \\lambda_R)\\right| < {16\\pi}\/{\\sqrt{2}}$ & $F_{\\ell_R} \\bar e_R \\to H S$ & $1$ \\\\\n\t\t$|\\kappa| < {8\\pi}\/{\\sqrt{2}}$ & $H H^\\dagger \\to S S^\\dagger$ & $0$ \\\\\n\t\t{$|g_Y (Y+1)| < \\sqrt{6\\pi}$} & $S B_\\mu \\to S B_\\mu$ & $1\/2$ \\\\\n\\hline\n\t\\end{tabular}\t\n\t\\caption{\\label{tab:unitarityboundsFFS} \n\tUnitarity bounds for the FFS model.  \n\t}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\\subsection{Tree-level matching onto the tensor operator}\n\\label{sec:treetensor}\n\nWe now consider a simplified \nmodel that matches onto the tensor operator $\\mathcal{O}^{\\mu q}_{T}$. \nThe scalar leptoquarks $R_2= (3,2,\\tfrac{7}{6})$ and $S_1 = (3,1,-\\tfrac{1}{3})$ \nallow for a \ncoupling to the top-quark \nwith both chiralities (see e.g.~\\cite{Dorsner:2016wpm}), \nthus maximizing the effect on $\\Delta a_\\mu$ via a top-mass insertion. \nMassive vectors \ncan also lead to renormalizable extensions, \nbut they result at least into a $m_b \/ m_t$ suppression \ncompared to \nscalar extensions \n(see e.g.~\\cite{Biggio:2016wyy}).    \n\nLet us focus for definiteness on the $R_2$ case (similar conclusions apply to $S_1$). \nThe relevant interaction Lagrangian reads\\footnote{Note that \nthe leptoquark models in \\eq{eq:R2} can be understood as a variant of the FFS model in \\eq{eq:SFF}, \nwhere $F_\\ell$ and $F_e$ are replaced by the SM states $q_L$ and $t_R$, \nwhereas $S$ is the scalar leptoquark (that is the only new physics state).  \nSubstituting instead \n$S$ with the SM Higgs and integrating out the heavy $F_\\ell$ and $F_e$ fermions \ngives contributions to $\\Delta a_\\mu$ through dimension-9 SMEFT operators.}\n\\begin{equation}\n\\label{eq:R2}\n\\mathscr{L}^{g-2}_{R_2} \\supset \n\\lambda_L\\, \\bar t_R \\ell_L^a \\,\\varepsilon_{ab} R_2^b  \n+\n\\lambda_R\\, \\bar q_L^a\\,\\mu_R R_{2a} \n+\n\\text{h.c.} \n\\end{equation}\nwhere $a$ and $b$ are ${\\rm SU}(2)_L$ indices and $\\varepsilon =i \\sigma_2$. \nUpon integrating out the leptoquark with mass $m_{R_2} \\gg v$ (cf.~\\fig{fig:CT_macthing}), \none obtains~\\cite{Feruglio:2018fxo,Aebischer:2021uvt} \n\\begin{equation}\n\\dfrac{C^{\\mu t}_{T}}{\\Lambda^2} =-\\dfrac{ \\lambda^*_L \\lambda_R}{8 m_{R_2}^2 } \\, . \n\\label{eq:R2matching}\n\\end{equation} \nThe unitarity bounds for the $R_2$ model (see \\app{app:unitarityweakly} for details)\nare collected in \\Table{tab:unitarityboundsR2} and they imply $|\\mbox{Re}\\, (\\lambda^*_L \\lambda_R)| \\lesssim 12$. Hence, we can recast the contribution to $\\Delta a_\\mu$ via \\eq{eq:DeltaamuSMEFT} \nas \n\\begin{equation} \n\\Delta a_{\\mu} \\simeq \n2.5 \\times 10^{-9} \n\\( \\frac{180 \\ \\text{TeV}}{m_{R_2} } \\)^2 \n\\(\n\\dfrac{\\mbox{Re}\\,(\\lambda^*_L \\lambda_R)}{12}\\)\\, .\n\\label{eq:damuR2}\n\\end{equation}\nHence, we conclude that in the leptoquark model one expects \n$m_{R_2}\\lesssim 180 \\ \\text{TeV}$ (the same numerical result is obtained for $S_1$), \nthus providing  \nthe largest new physics scale \namong the renormalizable extensions \nresponsible for $\\Delta a_\\mu$. \nMoreover, since the matching with the tensor operator is at tree level, \nthe leptoquark model fairly reproduces the unitarity bound \nfrom the SMEFT operator (see \\eq{eq:LamUOT}).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/CeT_matching}\n\\caption{Sample diagram of the \nleptoquark\nmodel matching onto $C^{\\mu t}_{T}$ at the scale $\\Lambda$. }\n\\label{fig:CT_macthing}       \n\\end{figure}\n\n\\begin{table}[!ht]\n\\renewcommand{\\arraystretch}{1.15}\n\t\\centering\n\t\\begin{tabular}{|c|c|c|}\n\t\\rowcolor{CGray} \n\t\\hline\n\t\tUnitarity bound & $i\\to f$ Channels & $J$ \\\\ \\hline\n\t\t$|\\lambda_L|^2 + |\\lambda_R|^2 < 8\\pi$ & $i, f=t_R \\bar\\ell_L, q_L \\bar\\mu_R$ & $0$\\\\\n\t\t$\\left|\\mbox{Re}\\,(\\lambda_R\\lambda_L^*)\\right| < {8\\pi}\/{\\sqrt{3}}$ & $\\mu_R\\bar\\ell_L \\to q_L\\bar t_R$ & $0$ \\\\\n\t\t$|\\lambda_R|^2 < {8\\pi}\/{3}$ & $q_L R_2^* \\to q_L R_2^*$ & $1\/2$ \\\\\n\t\t$|\\lambda_L|^2 < {16\\pi}\/{3}$ & $t_R R_2^* \\to t_R R_2^*$ & $1\/2$ \\\\\n\t\t\\hline\n\t\\end{tabular}\t\n\t\\caption{\\label{tab:unitarityboundsR2} \n\tUnitarity bounds for the couplings of the leptoquark model defined in \\eq{eq:R2}.}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\\subsection{Raising the scale of new physics via multiplicity?}\n\nNaively, one could be tempted to increase the upper limit on the scale of new physics by adding \n${\\cal N}$ copies of new physics states contributing to $\\Delta a_\\mu$. \nHowever, \nwhile \nboth $C_{e\\gamma}$ and $C_T$ increase by a factor of ${\\cal N}$,  \nthe unitarity bounds on the couplings gets also stronger due to the correlation \nof the scattering channels, so that larger new physics scales cannot be reached. \n\nIn order to see this, consider e.g.~the FFS model with ${\\cal N}$ copies of $F_\\ell$, $F_e$ and $S$.\nThe scaling of the unitarity bounds is most easily seen in processes where the SM states are exchanged in the $s$-channel, for example $S^i F_{\\ell_R}^i \\to S^j F_{\\ell_R}^j$. Since $\\ell_L$ is coupled to all copies in the same way, the $\\mathcal{T}$-matrix can be written as\n\\begin{equation}\n\t\\mathcal{T}^{J=1\/2} = \\frac{1}{32\\pi} |\\lambda_L|^2 J_{\\cal N} \\,,\n\\end{equation}\nwhere $J_{\\cal N}$ is a ${\\cal N} \\times {\\cal N}$ matrix filled with 1. Given that the largest eigenvalue of $J_{\\cal N}$\nis ${\\cal N}$, the unitarity bound on $\\lambda_L$ reads\n\\begin{equation}\n\\label{eq:lamLacsl}\n\t|\\lambda_L| < \\sqrt{\\frac{16\\pi}{{\\cal N}}} \\,.\n\\end{equation} \nSimilar processes can be considered for all the couplings in Eq.~\\eqref{eq:SFF}, leading to a $1\/\\sqrt{{\\cal N}}$ scaling for each Yukawa coupling. Hence, the overall ${\\cal N}$ contribution to \n$\\Delta a _\\mu \\propto \n{\\cal N} \\, \\mbox{Re}\\, (\\lambda_L^* \\lambda_Ry_{L,R}) \/ m^2_S$ is compensated \nby the $1 \/ \\sqrt{{\\cal N}}$ \nscaling of the \nunitarity bounds on the \ncouplings and,   \nfor fixed $\\Delta a _\\mu$, the mass of extra states gets even lowered at large ${\\cal N}$. \nIn this respect, we reach a different conclusion from the analysis in Ref.~\\cite{Capdevilla:2021rwo}.\n\nThe same considerations apply if we consider just one new scalar and ${\\cal N}$ new fermions. The situation is different with ${\\cal N}$ scalars and just one family of fermions, since $S$ does not couple directly to the Higgs (barring the portal coupling $\\kappa$ in \\eq{eq:SFF}, which however does not contribute \nto $\\Delta a_\\mu$). This implies that only $\\lambda_L$ and $\\lambda_R$ will scale as $1\/\\sqrt{{\\cal N}}$, which in turn means that $\\Delta a_\\mu$ does not change. \nSimilar arguments apply when considering larger ${\\rm SU}(2)_L$ representations, thus implying that the minimal choice we made for the FFS model ensures that $m_S$ is maximized.\nThe case of the leptoquark $R_2$ is analogous to what we have just described for ${\\cal N}$ new scalars, with the new fermions of the FFS model replaced by SM fields. Given that $\\lambda_L$ and \n$\\lambda_R$ would scale as $1\/\\sqrt{{\\cal N}}$, there is no gain in taking ${\\cal N}$ \ncopies of leptoquarks. \n\n\n\\section{Non-renormalizable models}\n\\label{sec:strongmodels}\n\n\nTill now we focused on renormalizable extensions of the SM \naddressing $\\Delta a_\\mu$\nand showed that they predict on-shell new physics states well below the \nunitarity bound obtained from the SMEFT dipole operators, \nsuggesting instead that new physics can hide up to the PeV scale. \nNonetheless, the SMEFT bound should be understood as the most conservative \none and applies if the origin of $\\Delta a_\\mu$ can be \nfor instance \ntraced back to a \nstrongly-coupled dynamics. \nWhile such a scenario could have calculability issues, we want to \nprovide here an intermediate step in which the SMEFT dipole operators \nare generated via a tree-level exchange of a new vector \nresonance from \na strongly-coupled sector taking inspiration from the case of the $\\rho$ meson \nin QCD, but whose UV origin we leave \nunspecified.  \n\nSpin-1 vector resonances are conveniently described via \nthe two-index anti-symmetric tensor field \n${\\cal V}_{\\mu\\nu}$, following the formalism of Ref.~\\cite{Ecker:1988te}. \nIn particular, we consider a composite spin-1 state \nfeaturing the same gauge quantum numbers of the SM Higgs doublet  \nand described via the \neffective Lagrangian \n\\begin{align}\n\\label{eq:KRLag}\n\\mathcal{L}_{{\\cal V}} &= \n- {\\cal D}^\\mu {\\cal V}_{\\mu\\nu}^\\dagger {\\cal D}_\\rho {\\cal V}^{\\rho\\nu} \n+ \\dfrac{1}{2} m^2_{{\\cal V}} {\\cal V}_{\\mu\\nu}^\\dagger {\\cal V}^{\\mu\\nu} \\nonumber \\\\ \n& +  c_{HB} {\\cal V}_{\\mu\\nu}^\\dagger  H B^{\\mu\\nu} + {c}_{HW} {\\cal V}_{\\mu\\nu}^\\dagger  \n \\tau^I H W^{I,\\mu\\nu}    +  c_{\\ell e} {\\cal V}_{\\mu\\nu}   (\\bar \\ell_L \\sigma^{\\mu\\nu} e_R) + \\ldots \\, ,  \n\\end{align}\nwhere we neglected ${\\cal V}_{\\mu\\nu}$ self-interactions as well as other higher-dimensional operators. \nIn fact, \\eq{eq:KRLag} should be understood as the leading term of an effective non-renormalizable Lagrangian, with cut-off scale $\\Lambda_{{\\cal V}}$ above $m_{{\\cal V}}$. \nThe free Lagrangian of \\eq{eq:KRLag} propagates three degrees of freedom \ndescribing a free spin-1 particle of mass $m_{{\\cal V}}$, with propagator \\cite{Ecker:1988te,Cata:2014fna,Pich:2016lew} \n\\begin{align} \n\\label{eq:prop} \ni\\Delta_{\\mu\\nu;\\rho\\sigma}(q) &= \n\\frac{2i}{m^2_{\\cal V}-q^2}\\[\n{\\cal I}_{\\mu\\nu;\\rho\\sigma}(q)\n-\\dfrac{q^2}{m^2_{\\cal V}}{\\cal P}_{\\mu\\nu;\\rho\\sigma}(q)\n\\]\n\\, , \n\\end{align}\nwhere ${\\cal I}_{\\mu\\nu;\\rho\\sigma}=\\(g_{\\mu\\rho} g_{\\nu \\sigma}- g_{\\mu\\sigma} g_{\\nu \\rho}\\) \/2$ and \n ${\\cal P}^{\\mu\\nu;\\rho\\sigma}=\\(P_T^{\\mu\\rho} P_T^{\\nu\\sigma}-P_T^{\\mu\\sigma} P_T^{\\nu\\rho}\\)\/2$ with $P_T^{\\mu\\nu}=g^{\\mu\\nu}-q^\\mu q^\\nu\/q^2$. \nAssuming that there is a calculable regime where one can parametrically keep \n$m_{{\\cal V}} \\lesssim \\Lambda_{{\\cal V}}$ \n(in analogy to the chiral approach to the $\\rho$ meson in QCD, for which \n$m_\\rho \\lesssim \\Lambda_{\\chi} \\sim 1$ GeV) we \ncan integrate ${\\cal V}_{\\mu\\nu}$ out and get \nthe \nfollowing\ntree-level matching contribution with the  \nphoton dipole operator (cf.~also \\fig{fig:KRmatch}) \n\\begin{align}\n\\label{eq:KRLag2}\n\\frac{C^\\mu_{e\\gamma}}{\\Lambda^2} =  -\\frac{2  \\(c_W c_{HB}-s_W {c}_{HW}\\) c_{\\ell e} }{m^2_{\\cal V}} \\, .\n\\end{align}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/Vector_Cegamma_macthing}\n\\caption{Tree-level matching \nonto \nthe photon dipole operator \nvia the exchange of \na spin-1 vector resonance.}\n\\label{fig:KRmatch}       \n\\end{figure}\nHence, we obtain\n\\begin{equation} \n\\label{eq:DamuKR}\n\\Delta a_{\\mu} \\simeq \n2.5 \\times 10^{-9} \n\\( \\frac{1 \\ \\text{PeV}}{m_{\\cal V}}\\)^2 \\( \\frac{\\mbox{Re}\\,\\(\\(-c_W c_{HB}+s_W {c}_{HW}\\) c_{\\ell e}\\)}{7.5} \\) \\, ,  \n\\end{equation}\nwhere we normalized $m_{\\cal V}$ at the PeV scale, that is in the ballpark of the unitarity bound \nobtained from the \nSMEFT dipole operators. \nIt should be noted that although the operators in the second line of \\eq{eq:KRLag} have canonical \ndimension equal to 4, scattering \namplitudes involving the $c_{HB,HW,\\ell e}$ couplings, \nas e.g.~$H B \\to e_R \\bar\\ell_L$, \ngrow like $s \/ m^2_{\\cal V}$ due to the \nhigh-energy behaviour of the propagator in \\eq{eq:prop}. \nHence, the effective description of the vector resonance breaks down not far \nabove $m_{\\cal V}$, being the theory non-renormalizable.\\footnote{Another way to generate the dipole operators \nrelevant for \n$\\Delta a_\\mu$ \nat tree level \nis to consider non-renormalizable \nmodels, involving for example \na new vector-like fermion $\\mathcal{F} = (1,2,-\\tfrac{1}{2})$ \\cite{deBlas:2017xtg}.}\n  \n  \n  \n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n \nUnitarity bounds are a useful tool in order to infer the \nregime of validity of a given physical description. \nIn EFT approaches, the energy scale at which unitarity is violated in tree-level \nscattering amplitudes can be often associated to the onset of the new physics \ncompleting the effective description. Instead, within \nrenormalizable setups unitarity bounds are a synonym of \nperturbativity bounds on the size of the adimensional couplings.\nIn this work we have investigated unitarity constraints on the new physics \ninterpretation of the muon $g-2$ anomaly. Assuming a short-distance \nSMEFT origin of the latter, \nwe have first computed unitarity bounds considering a set of leading \n(dipole and tensor) \noperators contributing to $\\Delta a_\\mu$. It turns out that the scale \nof tree-level unitarity violation is maximized in the case of dipole operators \nand reaches the PeV scale \nwhen both ${\\rm U}(1)_Y$ and ${\\rm SU}(2)_L$ dipoles \nare switched on  \n(cf.~\\fig{fig:Leb_LeW}). \nHence, most conservatively, in order to resolve the new physics origin of the \nSMEFT operators behind $\\Delta a_\\mu$ one would need to probe \nhigh-energy scales up to the PeV. \nThis most pessimistic scenario, \nclearly outside from the direct reach of next-generation high-energy particle colliders, \ncan be understood as a  \n\\emph{no-lose theorem} for the muon $g-2$ puzzle. \nOf course, the new physics origin \nof $\\Delta a_\\mu$ might reside well below the PeV scale, as it is indeed suggested \nby simplified models based on renormalizable scalar-Yukawa theories. \nIn the latter case we have considered a couple of well-known scenarios \nmatching either on the tensor (at tree level) or the dipole (at one loop) operators of the SMEFT analysis.  \nIn both cases, we have computed unitarity bounds on renormalizable couplings, \nthus allowing the mass of the new on-shell states to be maximized.  \nThe latter are found to be $M_{\\rm on-shell} \\lesssim 130$ TeV and $\\lesssim 180$ TeV, \nrespectively for the dipole and the tensor operators. \nMoreover, we have shown that multiplicity does not help to relax those bound because \nunitarity limits scale as well with the number of species. \n\nSince the bound obtained within renormalizable models is well below the \nSMEFT bound, it is fair to ask which UV completions could lead to a \nnew physics resolution of the muon $g-2$ puzzle hidden at the PeV scale. \nIn fact, one could imagine a strongly-coupled dynamics at the PeV scale that is \nequivalent to writing the SMEFT Lagrangian.  \nHere, we have \nprovided instead an intermediate step in which the SMEFT dipole operators \nare generated via the tree-level exchange of a new spin-1 vector resonance \ndescribed by a two-index anti-symmetric tensor field \n${\\cal V}_{\\mu\\nu}$\nwith the same quantum numbers of the SM Higgs and whose origin \nshould be traced back \nto the dynamics of a strongly-coupled sector. \nThis effective scenario provides a non-trivial example in which the \ndipole effective operators are generated via \ntree-level matching, thus suggesting that the SMEFT unitarity bound \ncan be saturated with new on-shell states hidden at the PeV scale. \nIt would be interesting to investigate whether a UV dynamics leading to \nsuch effective scenario can be explicitly realized.  \n\n\n\\section*{Acknowledgments} \nWe thank Paride Paradisi and Bartolomeu Fiol for useful discussions. \nFM acknowledges financial support from the State Agency for Research of the Spanish Ministry of Science and Innovation through the \"Unit of Excellence Mar\\'ia de Maeztu 2020-2023\" award to the Institute of Cosmos Sciences (CEX2019-000918-M) and  from PID2019-105614GB-C21 and  2017-SGR-929 grants.  LA acknowledges support from the Swiss National Science Foundation (SNF) under contract 200021-175940.\nThe work of MF is supported by the project C3b of the DFG-funded Collaborative Research Center TRR 257, ``Particle Physics Phenomenology after the Higgs Discovery''. The work of MN was supported in part by MIUR under contract PRIN 2017L5W2PT, and by the INFN grant 'SESAMO'.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The purpose of this paper is to exhibit some peculiar features of  exact\nsupersymmetric solutions of the heterotic string theory. We will consider the\nmost unusual properties of these solutions which all saturate the\nsupersymmetric positivity bound in the limit when the mass of such\nconfigurations tends to zero,\n\\begin{equation}\nM^2 = Z^A {\\cal R } _{AB}(\\Phi_0) Z^B \\rightarrow 0 \\ .\n\\end{equation}\nHere ${\\cal R} $ is a    continuous function of the asymptotic values\nof the  scalar fields $\\Phi_0$ and $Z$ are electric and magnetic charges.\nIt has been pointed out recently by Hull and Townsend \\cite{HullT} and  Witten\n\\cite{W} and Strominger \\cite{Strom} that  since the matrix ${\\cal R} $ is a\ncontinuous function of  $\\Phi_0$,\nthe masses of the Bogomolny states are also continuous functions of  $\\Phi_0$.\nIn particular, for some values of  $\\Phi_0$  massless states may exist\nwhich saturate the bound.\n\nThe first explicit example of such a state was given by Behrndt \\cite{Klaus2}\nand it was interpreted as a massless black hole. This was a significant\nprogress. A wider class of similar solutions was obtained in\n\\cite{BHGAUGE}. The solutions were interpreted as $N_L=0$ states of the\ntoroidally compactified heterotic string. However, all these solutions did not\ninclude non-Abelian fields which are necessary to cancel anomalies of\nsupersymmetry. Therefore it was not quite clear whether these solutions survive\nand remain massless with an account taken of $\\alpha'$ corrections. The\nimportance of quantum corrections to supersymmetry transformations\nincreases  in the situation when it is known that the massless states  may\npresent the\npoints of enhanced gauge symmetry.   Anomalies of supersymmetry result from the\nLorentz anomaly in the effective heterotic string action and may spoil all\nconjectures about the exactness of the BPS bound. The anomalies can be cured\nwhen\nYang-Mills fields  are included according to the Green-Schwarz mechanism of\ncancellation of anomalies.\n\n\n In the present paper we will consider a\ngeneral class of supersymmetric solutions including  non-Abelian fields. We\nwill\nfind massless states which may correspond either to  black holes  or to waves,\nand\nwhich remain exact solutions of equations of motion due to the presence of\nnon-Abelian fields even with an account taken of\n$\\alpha'$ corrections. In other words,\nwe will find massless BPS states  which   saturate\nthe supersymmetric positivity bound and which are free of\nanomalies of supersymmetry.\n\n\nWe will find also a  class of anomaly free massive configurations closely\nrelated to the massless black holes. A rather unusual property of all these\nconfigurations\n(either massive or massless) is that instead of the  usual black hole horizon,\nwhich\n{\\it absorbs} all particles falling into the black hole, they  have a repulsive\n(i.e. antigravitating) naked singularity\nwhich {\\it reflects} all test particles. Since the totally reflecting surface\nis not\nblack but white, it is more proper to call these new singular configurations\nsupersymmetric  white\nholes,  or repulsons.\\footnote{The name ``white holes'' referring to the\ncomplete reflection  (as opposite to the complete absorbtion by black holes)\nseems to be most adequate. Unfortunately,   many years ago this name  was used\nfor   hypothetical\nobjects  associated with the   time reversal of the gravitational\ncollapse. Physical relevance of such objects is rather doubtful\n\\cite{NovFrol}. Therefore we believe that that the name ``white holes'' is\nessentially vacant and can be used    for the  repulsive singular\nconfigurations   discussed in our paper. However, in order to completely avoid\ncollisions with  old terminology, we will often call our  massive and massless\nwhite hole configurations ``repulsons.''  }\n The solutions can be obtained either directly in four-dimensional space or by\ndimensional reduction of ten-dimensional gravitational waves. The  repulsive\nsingularity discussed above   is not present in the\nten-dimensional non-compactified version of the solution. It appears after the\ncompactification in those places of the four-dimensional space  where   the\nvolume of the six-dimensional compactified space shrinks to zero.\n\nOur basic strategy in looking for massless states   is the following.\nIf a configuration  has one-half of unbroken\n$N=4$ supersymmetry and describes a solutions of $N=4$ supergravity interacting\nwith some Abelian and non-Abelian vector multiplets,\nthe vanishing ADM mass simultaneously means the vanishing dilaton\ncharge\\footnote{For solutions without the fundamental axion charge this can be\nshown using the supersymmetry rules.}:\n\\begin{equation}\nM \\rightarrow 0 \\hskip 1 cm  \\Longleftrightarrow   \\hskip 1 cm  \\Sigma\n\\rightarrow 0 \\ .\n\\end{equation}\n If we know any   ten-dimensional solution with one-half of unbroken  $N=1$\nsupersymmetry, we may use the fact that the corresponding four-dimensional\ndilaton $e^{2 \\phi}$ is related to the ten-dimensional dilaton $e^{2\\hat \\phi}$\nas\nfollows:\n\\begin{equation}\ne^{- 2 \\phi} =  e^{- 2\\hat \\phi} \\sqrt {\\det G} \\ ,\n\\end{equation}\nwhere the matrix $ G $ describes the geometry of the internal six-dimensional\nspace.  Knowledge of the ten-dimensional solutions means that both $e^{- 2\\hat\n\\phi}$  and\n${\\det G}$ are known.\nThe massless four-dimensional configurations saturating the BPS bound   are the\nones\nin which the ten-dimensional dilaton charge $\\hat \\Sigma$ is  compensated by\nthe\nmodulus field  charge\n$\\sigma$, where  we define\n\\begin{eqnarray}\ne^{- 2 \\phi} &=& e^{- 2 \\phi_0} + {\\Sigma\\over r} +\\dots ,\\nonumber\\\\\ne^{- 2 \\hat \\phi} &=& e^{- 2\\hat  \\phi_0} + {\\hat\\Sigma\\over r} +\\dots ,\n\\nonumber\\\\\n\\sqrt {\\det G} &=&( \\sqrt {\\det G})_0 +  {\\sigma\\over r} +\\dots\n\\end{eqnarray}\nThe BPS-state is massless  when\n\\begin{equation}\n( \\sqrt {\\det G})_0 \\,\\hat\\Sigma + e^{- 2\\hat  \\phi_0} \\,  \\sigma   =0 \\ .\n\\end{equation}\nIt is clear that for the flat six-dimensional solution with $\\sqrt {\\det G}=\n\\sqrt\n{\\det G}_0$ the massless supersymmetric state of pure $N=4$ supergravity is a\ntrivial flat space. The non-trivial solutions exist only when there are matter\nmultiplets. We will find that for our class of solutions not only $N=4$\nnon-gravitational gauge  multiplets must be present but also  some of them\nhave to be  non-Abelian to keep supersymmetry preserved with an account taken\nof quantum corrections.\n\nThe class of massive supersymmetric solutions which we are going to study will\nbe\nclosely related to the massless configurations. They will also have the\nproperty that\nthe mass of the configuration is proportional to the charge of the\nfour-dimensional\ndilaton. Therefore the new mass formulas for the white holes which will be\nobtained in this paper will simultaneously  give the  dilaton charge formulas.\n\n\n\n\n\n\\section{Supersymmetric string waves and generalized fundamental strings}\n\n We have found various examples of  configurations with vanishing\nfour-dimensional dilaton charge by using some known solutions of the  equations\nof motions of effective action of the heterotic string.   Some of them belong\nto the\nclass of exact supersymmetric heterotic string backgrounds and require the\nnon-Abelian gauge fields to be part of the solution, some other do not seem to\npreserve the unbroken supersymmetry when $\\alpha'$ corrections are taken into\naccount. The first class of exact solutions with $SO(8)$ special holonomy is\ngiven  by\nsupersymmetric string waves (SSW)  \\cite{BKO1} and their T-dual\npartners, generalized fundamental strings (GFS) \\cite{BEK}.    Both the\npp-waves\nand fundamental strings  admit a null Killing vector and  belong to the class\nof  supersymmetric gravitational waves. The Killing spinor for these solutions\nsatisfies a null constraint,  and the dimensionally reduced form of these\nsolutions\nalways gives electrically charged configurations.\n\nWe will consider here the  $u,v$-independent  part of these solutions, which is\ndescribed in terms of  {\\it nine harmonic functions}, satisfying the flat space\neight-dimensional equation $\\sum_{i=1}^8 \\partial_i \\partial_i \\, h (x^i) =0$.\nThese configurations solve the cohomology constraint ${tr} R^2 - {tr} F^2=0$.\n\nThe second class of supersymmetric solutions,  associated with the chiral null\nmodel \\cite{HT}, is\ndescribed by {\\it ten harmonic functions}.  When one of the harmonic functions\nis\ntaken to be a constant, the chiral null model is reduced either to SSW\nsolutions or\nto GFS solutions.  Therefore only for these solutions the embedding of the spin\nconnection into the gauge group is possible. This leads to the preservation of\nunbroken space-time supersymmetry with an account taken of\n$\\alpha'$-corrections, as\nwell as to the left-right  world-sheet supersymmetry. However, when all ten\nfunctions are present in the solutions the status of unbroken supersymmetry in\npresence of  $\\alpha'$-corrections is not clear.  The holonomy of torsionful\nspin\nconnections of this theory, related to the properties of $\\alpha'$-corrections,\nis a\nsubgroup  of the non-compact SO(1.9) Lorentz group. This was established in\n\\cite{KOexact}  for uplifted electrically charged $a=1$ dilaton black holes\n(which\nform a particular case of the chiral null model) and for the complete chiral\nnull\nmodel in  \\cite{HT}. The spin embedding into the gauge group  is the only known\nway to  preserve unbroken supersymmetry. It does not work for generic chiral\nnull\nmodel since the gauge group of the heterotic string is compact.\n\nAll solutions described above admit the null Killing vector and therefore may\nbe\ncalled gravitational waves. Dimensional reduction of these solutions was\nperformed\nin \\cite{KB}.\n\n\nHere we would like to describe first the solutions which solve the cohomology\nconstraint $dH=0$ and which   remain supersymmetric even with an  account taken\nof\n$\\alpha'$-corrections.  After this is done we will look for the massive\nfour-dimensional supersymmetric black holes and study how they approach the\nmassless states. For this\npurpose we start with anomaly free  ten-dimensional solutions of $N=1$\nsupergravity coupled to supersymmetric Yang-Mills theory. The exact SSW as well\nas exact GFS solutions admit  a null Killing vector $l^\\mu$ with\n$l^2=0$.\nThis Killing vector generates an isometry in the $v$ direction where\nwe use light-cone coordinates $x^\\mu = (u,v,x^i)$,\\  $i=1,\\dots ,8$.\nThe solution consists of the ten-dimensional metric, dilaton, two-form and the\nnon-Abelian gauge fields.\n The metric and the 2-form field  are both described in terms of the dilaton\n$e^{-2\\hat \\phi (x^i)}$ and one vector function $A_\\mu(x^i)$\nof the transverse coordinates $x^i$,\n\\begin{equation}\nA_\\mu(x^i) = \\Bigl\\{A_u(x^i) \\equiv  - {K(x^i)\\over 2},\\, A_v=0,\\,\nA_i(x^i)\\Bigr\\}     \\ .\n\\end{equation}\n\n\nFor SSW the dilaton has to be taken constant,  $e^{-2\\hat \\phi(x^i)}  =\ne^{-2\\hat\n\\phi_o} $, for GFS the function $K(x^i)$   has  to be a constant,\n$K(x^i) = K_0$. Under these conditions both solutions can be described as\nfollows\\footnote{ In the chiral null model \\cite{HT}   $e^{2\\hat \\phi} =F$.}:\n\\begin{equation}\nds^2 =  2 e^{2\\hat \\phi} du (dv  + A_{\\mu} dx^\\mu )- \\sum_1^8dx^i dx^i,\n\\qquad A_v =0 \\ ,\n\\label{wave}\\end{equation}\n\\begin{equation}\nB = 2 e^{2\\hat \\phi} du \\wedge (dv + A_{\\mu} dx^\\mu) \\ .\n\\end{equation}\nThe non-Abelian gauge field  $V_\\mu{}^{IJ}$ is obtained by  embedding of the\ntorsionful spin connection into the gauge group of the heterotic string.\n\\begin{equation}\n\\label{eq:embedding}\nV_\\mu{}^{IJ}\n=  l_\\mu V^{IJ}\\hskip .5truecm \\equiv \\hskip .5truecm\n\\Omega_{\\mu-}{}^{ab} =e^{2\\hat \\phi}  l_\\mu {\\cal A}^{ab} ,\n\\ \\ \\ \\ \\ a,b,I,J=1,\\dots ,8\\  .\n\\end{equation}\nThe Yang-Mills indices are in the adjoint representation of\n$SO(8)$.\nThe equations that the dilaton $e^{-2\\hat \\phi (x^i)}$, $K(x^i)$ and $A_i(x^i)$\nhave to satisfy for the configuration to be supersymmetric and solve equations\nof\nmotion are:\n\\begin{equation}\n\\label{eq:Lapl}\n\\triangle e^{-2\\hat \\phi} =  \\triangle K  = 0\\ , \\qquad\n\\triangle\\partial^{[i}A^{j]}=0 \\ , \\qquad \\partial^{[i}A^{j]} \\equiv   {\\cal\nA}^{ij} \\ ,\n\\end{equation}\nwhere the Laplacian is taken over the transverse directions only ($\\triangle\n\\equiv\n\\sum_1^8\\partial_i \\partial_i$). This solution has   $SO(8)$ symmetry.\nObviously, SSW with constant dilaton and GFS with constant $K$ are special\nsolutions of this system of equations.\\footnote{This solution with constant $K\n=\nK_0$ is equivalent to the one presented in \\cite{BEK} after the shift in the\nisometry direction $v' = v -{ K_0 u\\over 2} $.} However, for any other\nsolutions when both\nthe dilaton and the function $K$ are non-constant, the\nproof of unbroken supersymmetry is not available. The holonomy of the\ngeneralized connections in this case includes the non-compact subgroup of the\nLorentz group, since  $\\Omega_{\\mu-}{}^{0i} $   is not vanishing\\footnote{There\nis\nan apparent discrepancy between the statement about the holonomy of generalized\nfundamental string solutions in \\cite{BEK} and \\cite{HT}.   Using\nT-duality rotation from the waves, we have found  that the holonomy  of the SSW\nas well as of GFS is\n$SO(8)$. Meanwhile in \\cite{HT} only the holonomy  of SSW is qualified as that\nof\n$SO(8)$ and the case of GFS is considered as not special.  The analysis shows\nthat\nthere is no discrepancy, however. Spin connections are frame dependent: in the\nframe used in \\cite{HT} one can still find that the curvatures are such that\nthe\nspin embedding into the $SO(8)$ gauge group solves the problem with\n$\\alpha'$-corrections for GFS configurations.}.  This spin connection cannot be\nembedded into $SO(32)$ or $E_8\\times E_8$ gauge group of the heterotic string.\nHowever, without such spin embedding the preservation of supersymmetry at the\nquantum level is questionable. Therefore in what follows we will consider\nonly SSW and GFS with non-Abelian fields included as exact\nsupersymmetric solutions in our class.\n\nWe will limit ourselves to the solutions which in four dimensions correspond\nonly\nto static configurations and not to the stationary ones. For this purpose we\ntake\n$A_1=A_2=A_3 =0 $. Our next specification will be to consider  the solutions\nwhich depend only on $x^1, x^2, x^3$. Thus  the $SO(8)$ symmetry is broken. The\nonly non-vanishing components of the non-Abelian vector field $V_\\mu{}^{IJ}$ in\nthis\ncase are     $V_\\mu{}^{im} $ with $i=1,2,3$ and $m=4,5,6,7,8$.\n\n\n\\section{ Zero mass configurations with asymptotically flat internal  geometry}\n\nThe effective action describing the dynamics of the massless fields of\ntoroidally compactified heterotic string is described by the bosonic part of\nthe action of $N=4$\nsupergravity interacting with 22 abelian $N=4$ vector multiplets.\nTen-dimensional supergravity, dimensionally reduced to four dimensions provides\n6 of them. The additional 16  are coming from the ten-dimensional vector\nmultiplets, or from the gauge sector of the heterotic string. The action in the\nform given by\n Maharana-Schwarz \\cite{MS}\nand  Sen \\cite{Sen} is\n\\begin{eqnarray}\\label{action}\nS &=& {1\\over16\\pi } \\int d^4 x \\sqrt{-\\det g} \\,  \\, \\Big[ R -2  g^{\\mu\\nu}\n\\partial_\\mu \\phi \\partial _\\nu\\phi +{1\\over 8} g^{\\mu\\nu} Tr(\\partial _\\mu\n{\\cal M} L\\partial_\\nu  {\\cal M} L)\n\\nonumber \\\\\n&& -{1\\over 12} e^{-2 \\phi} (H_{\\mu\\nu\\rho})^2\n -  e^{-2 \\phi} g^{\\mu\\mu'} g^{\\nu\\nu'} F^a_{\\mu\\nu} \\, (L {\\cal M} L)_{ab}\n\\, F^b_{\\mu'\\nu'} \\Big] \\, ,\n\\end{eqnarray}\nwhere $\\phi$ is the four-dimensional dilaton, and  ${\\cal M}$ is the 28$\\times\n$28 matrix valued scalar field, describing the moduli. Vector fields include\ngraviphoton as well as vectors from the gauge multiplets. The  28$\\times$28\nsymmetric matrix $L$   with 22 eigenvalues $-1$ and 6 eigenvalues $+1$ defines\nthe metric in the O$(6, 22)$ space. We use the units where $G_N=1$,  which\nshould be taken into account when comparing with $G_N=2$ units often used for\nidentification of string states.\n\n\nDimensional reduction of the chiral null model (without Yang-Mills fields) was\nperformed in \\cite{KB}. The supersymmetric four-dimensional solutions of the\nfield equations following from the action\n(\\ref{action}) have\n the following metric (in the canonical frame):\n\\begin{equation}\nds_{\\rm can}^2 = e^{2\\phi}dt ^2 - e^{-2\\phi} d\\vec x^2 \\ ,\n\\label{spher}\\end{equation}\nand the four -dimensional dilaton is\n\\begin{equation}\ne^{-2\\phi} = \\Bigl(e^{-2\\hat \\phi} \\, K - \\sum_{n= 4}^8 (A_n )^2 \\Bigr)^{1\\over\n2}\\ .\n\\label{4dilaton}\\end{equation}\nOther fields can be also deduced from the ten-dimensional solution by\ndimensional reduction.\n\nThe massless solution was found by Behrndt \\cite{Klaus2} in the framework of\ntoroidally compactified heterotic string theory, and it was further generalized\nin \\cite{BHGAUGE}. It has vanishing dilaton\ncharge and is obtained when the  functions defining the solutions are taken in\nthe\nform \\cite{KB}\n\\begin{equation}\ne^{-2\\hat \\phi}= 1+  {2\\tilde m\\over r}\\ , \\qquad K = 1+\n\\sum_1^s {2\\hat m\\over r}\\ , \\qquad A_n =  {2 q_n\\over r}\\ ,\n\\qquad n=4, \\dots 8, \\qquad r ^2 \\equiv \\vec x^2.\n\\label{klaus}\\end{equation}\nThis choice corresponds to the asymptotically flat internal space.\nThe four-dimensional dilaton is given by\n\\begin{equation}\ne^{-2\\phi}= \\left( 1+ {2(\\tilde m + \\hat m ) \\over r}    - {4(q^2 - \\tilde m\n\\hat\nm)\\over r^2}\n\\right)^{1\\over\n2},\n\\end{equation}\nwhere   $q^2 \\equiv  \\sum_{n = 4}^8 (q_{n})^2$.\n\nIn terms of the right- and left-handed charges the dilaton is given by\n\\begin{equation}\ne^{-2\\phi}= \\left( 1+ {2\\sqrt 2 |Q_R| \\over r}    - {2 (|Q_L|^2 - |Q_R|^2)\\over\nr^2}\n\\right)^{1\\over\n2}.\n\\end{equation}\nRight-handed charge is the charge   corresponding to  the gravi-photon, and\nleft-handed charge is the\ncharge   corresponding to   the vector fields of the matter multiplets. There\nis only one\npossibility to make these solutions massless: to take\n $\\sqrt 2 |Q_R| = \\hat m + \\tilde m=0 $. The dilaton becomes\n\\begin{equation}\\label{DILATON}\ne^{-2\\phi}= \\left(1-  {4 (\\tilde m ^2 +   q^2)  \\over\nr^2}\n\\right)^{1\\over 2} =   \\left(1-  {2|Q_L|^2  \\over\nr^2}\n\\right)^{1\\over 2} .\n\\end{equation}\nUsing the fact that for toroidally compactified string supersymmetric\nconfigurations  $|Q_L|^2 - |Q_R|^2 = - 2 (N_L -1)$, one can see that for\nvanishing $Q_R$ the state is characterized by $N_L=0$. Here $N_L$ is a\nnon-negative integer, describing the total oscillator contribution to the\nsquared mass\nof a state from the left moving oscillators of the string.\nRescaling this solution\nfor arbitrary value of the dilaton at infinity, $e^{2\\phi_0} \\equiv g^2$, we\nget\nfinally  the canonical four-dimensional metric in the form \\cite{BHGAUGE}\n \\begin{equation}\\label{DILATON2}\nds_{\\rm can}^{ 2} ({\\rm el})= \\left(1-  {4 g^2   \\over\nr^2}\n\\right)^{-{1\\over 2}} dt ^2 - \\left(1-  {4  g^2   \\over\nr^2}\n\\right)^{1\\over 2}  d\\vec x ^2\n \\ .\n\\end{equation}\nThe singularity of this configuration at $r=2g$ was found in \\cite{BHGAUGE} to\nbe\na true singularity since the scalar curvature is given by\n\\begin{equation}\nR_{\\rm can}^{\\rm el} = {4g^2 (2 g^2 + r^2) \\over  r (r^2 - 4 g^2)^{5\\over 2}} \\\n{}.\n\\end{equation}\nThe origin of this singularity can be traced back to the fact that the volume\nof the compactified six-dimensional space shrinks to zero for $r=2g$ since\n\\begin{equation}\n\\det G = e^{-4\\phi} e^{ 4\\hat \\phi} ={ r^2 -  4 g^2  \\over (r+  2\\tilde m)^2} \\\n{}.\n\\end{equation}\nIn the magnetic case   $g\\rightarrow {1\\over g}$,  and the volume of the\ncompactified space is\n\\begin{equation}\n\\det G = e^{-4\\phi} e^{ 4\\hat \\phi} ={ r^2 g^2-  4   \\over g^2(r+  2\\tilde\nm)^2}\n\\ .\n\\end{equation}\nIt  shrinks to zero at the point $r= {2\\over g}$ where the four-dimensional\nmagnetic solution is singular,\n \\begin{equation}\\label{DILATON3}\nds^2_{\\rm can} ({\\rm m}) = \\left(1-  {4  \\over\ng^2 r^2}\n\\right)^{-{1\\over 2}} dt ^2 - \\left(1-  {4  \\over\n g^2 r^2}\n\\right)^{1\\over 2}    d\\vec x^2\n \\ .\n\\end{equation}\n\nAll this analysis is valid in the framework of the toroidally compactified\nheterotic string with only Abelian vector fields in the solutions,  and   when\none ignores the issue of $\\alpha'$-corrections to supersymmetry. From now on we\nwill\nconsider only exactly supersymmetric SSW and GFS supplemented by the\nnon-Abelian fields  and described in Sec. 2.\n\nThis means that when both $\\tilde m$ and $\\hat m$ are non-vanishing,\nsupersymmetry is anomalous. If only one of these two numbers  vanishes,\nwhich is acceptable from the point of view of the exactness of the solution, we\ndo\nnot approach the massless state. Only for $\\hat m = \\tilde m=0 $ we  can   have\na massless solution  without anomalies. This special\nconfiguration has the dilaton field given by\n\\begin{equation}\\label{DILATON)}\ne^{-2\\phi}= \\left(1-  {4   q^2 \\over\nr^2}\n\\right)^{1\\over 2}  .\n\\end{equation}\n\nThe presence of the Yang-Mills vector fields in the ten-dimensional solution\nwill add\nsome non-Abelian vectors as well as scalars to the four-dimensional solutions.\nIt\nis quite remarkable that for this solution   to be non-trivial the presence of\nthe\nnon-Abelian vector field is necessary. In fact, using eq. (\\ref{eq:embedding})\nwe\nwill find that for the configuration given in (\\ref{klaus}) the non-Abelian\npart of\nthe solution in ten  dimensions is given by \\cite{BEK}\n\\begin{equation}\n\\label{YM}\nV_\\mu{}^{[in]}\n = l_\\mu \\, \\Bigl(1+  {2\\tilde m\\over r}\\Bigr)^{-1} \\;  {x^i q^n \\over |x|^3},\n\\ \\ \\ \\ \\ \\ i=1,2,3,\\ \\ \\;   n=4,5,6,7,8.\n\\end{equation}\nFor the massless configuration presented above with $\\hat m = \\tilde m=0$ the\nYang-Mills field is\n\\begin{equation}\n\\label{YMI}\nV_\\mu{}^{[in]}\n = l_\\mu  \\;  {x^i q^n \\over |x|^3},\n\\ \\ \\ \\ \\ \\ i=1,2,3,\\ \\ \\;   n=4,5,6,7,8.\n\\end{equation}\n\nThe ten-dimensional manifold  in the process of compactification is split into\nthe four-dimensional manifold $M^4$ with coordinates  $ v=t, x^1,x^2,x^3 $ and\na six-dimensional  manifold $M^6$ with coordinates $ x^4, \\dots , x^8$, $u=x^9\n$\n\\cite{KB}.\nThe moduli space  of our configuration is defined by the six-dimensional metric\n$G_{rs}, \\; r=4, \\dots , 9$,  and by the six-dimensional matrix $B_{rs}$:\n\\begin{equation}\nG_{rs} = \\pmatrix{\n-  \\delta_{mn} &  & e^{2\\hat\\phi} A_n  \\cr\n &   &  \\cr\ne^{2\\hat\\phi} A_m & &  -e^{2\\hat\\phi}  K &  \\cr\n},   \\qquad\nB_{rs} = \\pmatrix{\n0 &  & e^{2\\hat\\phi} A_n  \\cr\n &   &  \\cr\n- e^{2\\hat\\phi} A_m  & & 0&  \\cr\n} .\n\\label{mod}\\end{equation}\nIn addition, there are non-Abelian vectors and scalars which will come from the\nten-dimensional Yang-Mills field (\\ref{YM}). The two matrices $G$ and $B$\ntogether  form the $O(6,6)$ matrix ${\\cal M}$ \\cite{MS} which appears in the\nBogomolny bound \\cite{Sen}.  The ansatz (\\ref{klaus}) used in eq. (\\ref{mod})\nhas\nthe following properties:\n\n\\noindent i) The $O (6,6)$ matrix ${\\cal M}$ build out of $G$ and $B$ at\ninfinity\n($r\\rightarrow\n\\infty$) is\n\\begin{equation}\n{\\cal M}_0 = \\pmatrix{\n-I & 0 \\cr\n0 & -I \\cr\n} .\n\\label{diag}\\end{equation}\n ii) Given this asymptotic value of the matrix ${\\cal M}$, there is only one\nsolution for the massless state which is not changed by quantum corrections:\n\\begin{equation}\nG_{rs} = \\pmatrix{\n-  \\delta_{mn} &  &  {2q_n\\over r}  \\cr\n &   &  \\cr\n{2q_m\\over r}  & &  -1 &  \\cr\n},   \\qquad\nB_{rs} = \\pmatrix{\n0 &  & {2q_n\\over r}   \\cr\n &   &  \\cr\n- {2q_n\\over r}   & & 0&  \\cr\n},  \\qquad e^{-2\\phi}= \\left(1-   {4   q^2 \\over\nr^2}\n\\right)^{1\\over 2}.\n\\label{mod1}\\end{equation}\n The metric of this configuration   coincides with the metric (\\ref{DILATON2}).\nBesides, there are Abelian vector fields and non-Abelian vectors and scalars.\nThe most unusual property of this solution is that the massless state is\ndescribed by a static configuration.  We will return to this issue later after\nwe will find more general exact  massless and massive solutions.\n\n\n\n\n\\section{Special  points in the moduli space}\n\nOne can find a  solution describing a more general family of BPS-states with a\nvanishing ADM mass. For this purpose we may use the fact that in gravitational\nwave\nsolutions in $d=10$ one can use  more general harmonic functions. For the\none-black-hole case one can take\n\\begin{equation}\ne^{-2\\hat \\phi}=e^{-2\\hat \\phi_0} +  {2\\tilde m \\over r}\\ ,\n\\qquad K = K_0 +\n {2\\hat m\\over r}\\ , \\qquad A_n = (A_n)_0 + {2 q_n \\over r}\\ ,\n\\qquad n=4, \\dots 9.\n\\end{equation}\nHowever, for the solution to be exact we have one constraint\n\\begin{equation}\n\\hat m \\tilde m =0 \\ .\n\\end{equation}\nIndeed, this means that either  $\\tilde m  =0$ and the dilaton is constant or $\n\\hat m =0$ and $K$ is constant, which are the conditions for exactness.\n The four-dimensional dilaton is now given by\n\\begin{equation}\ne^{-2\\phi}= \\left(\\Bigl(e^{-2\\hat \\phi_0} +  {2\\tilde m\\over r}\\Bigr)\\Bigl(K_0\n+\n {2\\hat m \\over r}\\Bigr) -\\left[(A_n)_0 + {2  q_n \\over r}\\right ]^2\n\\right)^{1\\over\n2} \\ .\n\\end{equation}\nThis expression can be reorganized as follows:\n\\begin{equation}\ne^{-2\\phi}=e^{-2\\phi_0}\\left(1+   {4M\\over r}    - {4  g^2 q^2 \\over r^2}\n\\right)^{1\\over\n2} ,\n\\end{equation}\nwhere\n\\begin{equation}\ne^{-2\\phi_0}= \\left[e^{-2\\hat \\phi_0}  K_0 - (A_n)_0^2\\right]^{1\/2}  \\equiv\n{1\\over g^2},\n\\end{equation}\nThe mass formula for the  exact SSW case ($\\tilde m=0$) is\n\\begin{equation}\nM = {g^2   \\over 2}  \\left[   e^{-2\\hat \\phi_0}\\, \\hat m  - 2 (A_n)_0\\, q_n\n\\right]\n\\geq 0 \\ ,\n\\label{MSSW}\\end{equation}\n whereas  the mass formula for the case of the exact GFS ($\\hat m=0$) is\n\\begin{equation}\nM = {g^2 \\over 2} \\left[{ K_0\\, \\tilde m   - 2 (A_n)_0\\, q_n }\\right]  \\geq 0 \\\n{}.\n\\label{MGFS}\\end{equation}\n\nThe mass has to be non-negative due to supersymmetric positivity bound, but the\nvanishing value of the mass $M$ is not forbidden by supersymmetry.\n\nThus the metric of the exact non-Abelian electrically charged black hole is\ngiven\nby\n\\begin{equation}\nds_{\\rm can}^2 = \\left(1+   {4M\\over r}    - {4   g^2q^2 \\over r^2}\n\\right)^{-{1\\over\n2} } dt ^2 - \\left(1+   {4M\\over r}    - {4  g^2 q^2 \\over r^2}\n\\right)^{1\\over\n2}   d\\vec x ^2 \\ .\n\\label{II}\\end{equation}\nThe moduli space is presented by the matrix ${\\cal M}$ which asymptotically\n(in the limit $r \\to \\infty$) is\ndescribed in terms of asymptotic values of the matrices $G$ and $B$,\n\\begin{equation}\n(G_{rs})_0 = \\pmatrix{\n-  \\delta_{mn} &  & (A_n)_0\\ \\cr\n &   &  \\cr\n (A_n)_0  & &  -e^{2\\hat \\phi_0}  K_0 &  \\cr\n},   \\qquad\n(B_{rs})_0 = \\pmatrix{\n0 &  &  (A_n)_0  \\cr\n &   &  \\cr\n-  (A_n)_0   & & 0&  \\cr }   .\n\\label{mod2}\\end{equation}\nThus the asymptotic value of the matrix ${\\cal M}$ is very different from the\nsimple diagonal form  in eq. (\\ref{diag}). A complete expression for these\nmatrices\nis\n\\begin{equation}\nG_{rs} = \\pmatrix{\n-  \\delta_{mn} &  & (A_n)_0+  {2q_n\\over r}  \\cr\n &   &  \\cr\n (A_n)_0+ {2q_m\\over r}  & &  -e^{2\\hat \\phi}  K &  \\cr\n},  \\qquad\nB_{rs} = \\pmatrix{\n0 &  &  (A_n)_0+{2q_n\\over r}   \\cr\n &   &  \\cr\n- ( (A_n)_0+{2q_n\\over r})   & & 0&  \\cr\n},\n\\label{mod2a}\\end{equation}\nwhere for SSW and for GFS we have respectively\n\\begin{eqnarray}\ne^{2\\hat \\phi}  K&=& e^{2\\hat \\phi_0} \\Bigl(K_0 +\n {2\\hat m\\over r}\\Bigr) \\ ,  ~~~~ \\tilde m = 0,  \\nonumber\\\\\n \\nonumber\\\\\ne^{-2\\hat \\phi}  K&=&\\Bigl( e^{-2\\hat \\phi_0} + {2\\tilde m \\over r}\\Bigr)^{-1}\nK_0\n\\ ,  ~~~~ \\hat m = 0.\n\\end{eqnarray}\nThe non-Abelian fields for both configurations in the four-dimensional form can\nbe deduced from the ten-dimensional form (\\ref{YM}).\n\nThe moduli space is rather involved and allows to approach the critical points\nof\nthe  massless configuration continuously when the right-hand side in equations\n(\\ref{MSSW}) and (\\ref{MGFS}) tends to zero.\n\nIn all cases considered the four-dimensional configuration has a new\nsingularity\nwhich was not present in the ten-dimensional case. This singularity is present\nin the\ncompactified solution when the volume of the compactified space shrinks to\nzero.\nFor the solutions described above we have\n\\begin{equation}\\label{orbit0}\n\\det G =e^{-4 \\phi}  e^{4 \\hat \\phi}   = g^4  \\left(1+   {4M\\over r}   - {4\ng^2q^2\n\\over r^2}\n\\right) \\left(e^{-2\\hat \\phi_0} +  {2\\tilde m \\over r}\\right)^{-2}  \\ .\n\\end{equation}\nAt the singularity point\n\\begin{equation}\\label{orbit01}\nr_0 =  2(\\sqrt{M^2+g^2 q^2}- M) \\\n\\end{equation}\nthe determinant of the metric of the compactified six-dimensional space\nvanishes,\n\\begin{equation}\\label{orbit02}\n\\det G(r_0) =0 \\ .\n\\end{equation}\nIn stringy frame the geometry of the electric configuration (\\ref{II}) is given\nby\n\\begin{equation}\nds_{\\rm str}^2=  \\left(1+   {4M\\over r}    - {4   g^2q^2 \\over r^2}\\right)^{-1\n} dt ^2 -\n  d\\vec\nx ^2 \\ .\n\\label{stringyelectric}\\end{equation}\nNote that metric in stringy frame is well defined even in the region $r < r_0$,\nwhere $1+   {4M\\over r}    - {4   g^2q^2 \\over r^2} < 0$. It may seem\nmeaningless to continue metric to the region $r< r_0$, since the singularity at\n$r= r_0$\nis a real curvature singularity. However,\nit may be important to have such a continuation in order to investigate the\npossibility of\ntunneling through the singularity, see Sec. 5. One  may suggest  the following\ncontinuation of the canonical metric (\\ref{II}):\n\\begin{equation}\ng_{\\mu\\nu}^{\\rm can} = g_{\\mu\\nu}^{\\rm str}\\ \\sqrt{\\Bigl\\vert1+\n{4M\\over r}    - {4   g^2q^2\n\\over r^2}\\Bigr\\vert}\n\\ ,\n\\label{stringycanonical}\\end{equation}\nwhich gives the following generalization of the canonical metric (\\ref{II}):\n\\begin{equation}\nds_{\\rm can}^2 = \\left(1+   {4M\\over r}    - {4   g^2q^2 \\over r^2}\n\\right)^{-{1} } \\left(\\Bigl\\vert 1+   {4M\\over r}    - {4  g^2 q^2 \\over\nr^2}\\Bigr\\vert\n\\right)^{1\\over\n2}dt ^2 - \\left(\\Bigl\\vert 1+   {4M\\over r}    - {4  g^2 q^2 \\over\nr^2}\\Bigr\\vert\\right)^{1\\over 2}   d\\vec x ^2\n\\ .\n\\label{III}\\end{equation}\nThis continuation preserves an important property of metric\n(\\ref{stringyelectric}): the determinant of metric changes its sign at $r =\nr_0$.\n\n\n\n \\section{Exact supersymmetric black holes are white}\n\nIt is very tempting to associate singular spherically symmetric\nconfigurations    (\\ref{II}), (\\ref{III})  with black holes.\nHowever, it would not be   quite correct. Black holes got their name for the\nreason\nthat they strongly attract all particles,  so that even light cannot escape\nfrom a black hole. Gravitational\nattraction can be described by the Newtonian potential\n$\\Phi  = {1\\over 2}(g_{00} - 1)$. This yields the usual Newtonian attractive\npotential  $\\Phi = -{ M\\over r}\n$ at a large distance from a   massive  Schwarzschild black hole. Meanwhile,\nthe potential corresponding to the metric $g_{00} =\ng^{-1}_{rr} = \\Bigl(1 +{4M\\over r} - {4q^2\\over r^2}\\Bigr)^{-1\/2}$  at\nlarge $r$ is given by  $\\Phi = -{\nM\\over r} + {q^2\\over  r^2}$, and the strength of the gravitational field  is\nproportional to\n$\\Phi' =  { M\\over r^2} - {2 q^2\\over r^3}$. (For notational simplicity we take\nhere the coupling constant $g^2 = 1$.) Thus, in the limit $r \\to \\infty$ we\nstill have gravitational attraction, but only for the configurations with\npositive\nADM mass $M$. However, there is a stable equilibrium for test particles at\n$r_c = {2q^2\\over M}$, and there is a gravitational {\\it repulsion}\n(antigravity) for $r < r_c$. (For massless states the gravitational force is\nrepulsive at all $r$.) This repulsion becomes infinitely strong near\nthe singularity, which appears at\n$r_0 =  2(\\sqrt{M^2+q^2}\\,- M)$.\nInfalling particles cannot touch the singularity at $r= r_0$ and\nbecome totally reflected.\n\n\nIndeed, one can write an equation of motion for a test particle of a small\nmass $m$ in an  external spherically symmetric background  (\\ref{II}), see\n\\cite{Landau}:\n\\begin{equation}\\label{orbit1}\nt = E \\int dr {g_{rr}\\over \\sqrt{g_{00}}}\\left({g_{rr}}\\, E ^2 - g_{00}\\,\n{L^2\\over r^2} - g_{00}\ng_{rr}\\, m^2 \\right)^{-1\/2} \\ .\n\\end{equation}\nHere $E $ is the test particle energy at $r \\to \\infty$, and $L$ is its\nangular momentum with respect to the center of our configuration. For $g_{00} =\ng^{-1}_{rr} = \\Bigl(1 +{4M\\over r} - {4q^2\\over r^2}\\Bigr)^{-1\/2}$  eq.\n(\\ref{orbit1}) reads:\n\\begin{equation}\\label{orbit2}\nt = E { \\int} {dr\\, \\Bigl(1 +{4M\\over r} - {4q^2\\over r^2}\\Bigr)\n\\left({E^2\\Bigl(1 +{4M\\over r} - {4q^2\\over r^2}\\Bigr)   -\n{L^2\\over r^2} - m^2\\sqrt{1 +{4M\\over r} - {4q^2\\over r^2}}}\\right)^{-1\/2} }\\ .\n\\end{equation}\n\nIt is clear from this  equation that test particles with any initial energy $E\n$\ncannot reach the singularity at $r = r_0$. One can easily show that each  test\nparticle  within a finite time reaches  some minimal radius    $r_{\\rm min}  >\nr_0$,\nand becomes reflected.  For example, in the case $M = 0$, the singularity is at\n$r_0 = 2 |q|$, and all massless   test particles with $L \\not = 0$ become\nreflected at\n$r_{\\rm min} =   2 |q|\\, \\sqrt{1+ {L^2\\over 4q^2 E ^2}}\\, > r_0$. (For\ncomparison,   all massless particles with $L < 2ME $ are swallowed by the usual\nSchwarzschild black hole.) In the case $M = 0$, $L = 0$ massive test particles\nare\nreflected at  $r_{\\rm min} =  {2 |q|\\, \\Bigl(1 - {m^4\\over E ^4}\\Bigr)^{-1\/2}}>\nr_0$.\nThe only possible exception is the behavior of massless test particles in the\nS-state ($m = L =\n0$). In this case one should use quantum mechanical treatment similar to the\none\ndeveloped in \\cite{HW,HM}.  An investigation of this question indicates\nthat even in this special case particles are totally reflected.  In this sense\nour\nsolutions describe  white holes  rather than  the black ones.\n\nTo verify the last statement and to get an additional insight into the nature\nof the\nrepulsive singularity at $r = r_0$ we will study the wave equation for a\nmassive scalar field, taking the metric of an electric white hole (repulson) in\nthe form\nwhich allows continuation to $r < r_0$, see eq. (\\ref{III}). For the S-wave,\nthe scalar field equation $\\partial_{\\mu}(\\sqrt{|g|} g^{\\mu\\nu}\n\\partial_{\\nu}\\phi) = - m^2\\phi$ in this metric reads:\n\\begin{equation}\\label{orbit11}\n\\left(1+{4M\\over r} -{4q^2\\over r^{2}}\\right)\\ddot\\phi - \\phi'' -{2\\phi'\\over\nr} =\n-m^2\\phi\\ .\n\\end{equation}\nThe solution of this\nequation in the WKB approximation for $m \\not = 0$ reproduces our previous\nresults about total reflection, being strongly suppressed at  $r < r_{\\rm\nmin}$.\nHowever,   exact solutions of this equation both for $m \\not = 0$ and for $m =\n0$ do not vanish and do not exhibit\nany kind of singular behavior at the  point $r = r_0$.  To  give a particular\nexample, one may consider this equation for $M = 0$, $m = 0$.   In this case\nequation (\\ref{orbit11}) has a stationary solution\n$\\phi = e^{-iEt}\\chi$ in terms of Bessel functions,\n\\begin{equation}\\label{orbit111}\n\\chi = r^{-1\/2}\\, J_\\nu(Er), \\ \\ \\ \\nu_\\pm = \\pm {1\\over 2}\\sqrt{1 +4q^2E^2} \\\n{}.\n\\end{equation}\nThese solutions behave in a regular way at $r = r_0$. This suggests that the\nsingularity at\n$r= r_0$ at the quantum level is transparent for massless particles in the\nS-wave.\nOnly one of these two functions, the Bessel function with\n$\\nu_+ = +\n{1\\over 2}\\sqrt{1 + 4q^2 E^2}$, is normalizable.  It decreases near the\nsingularity at\n$r = 0$ as $r^{\\nu_+-1\/2}$.  In such a situation the probability flux near the\nsingularity $r = 0$ vanishes, which shows that even the massless particles in\nthe\nS-wave ($m = 0$, $L = 0$) are totally reflected, though not by the singularity\nat $r= r_0$ but by\nthe singularity at $r = 0$.\n\n\n\nHere one should make a cautionary   note. In general,  test particles may\ninfluence\nthe background. For $M \\not = 0$ this does not lead to any problems: one may\nconsider test particles with energy\n$E \\ll M$, in which case their influence on the white hole\nbackground can be neglected. Therefore massive states described in our paper do\nexhibit the antigravity regime and can be called white holes, or repulsons.\n\nOn the other hand, in the limit $M = 0$ our semiclassical\nconsiderations may become somewhat misleading.\nIndeed, gravitational repulsion  changes  momentum of a test particle.\nThis may happen only if the white hole itself  acquires the same momentum with\nan opposite sign, which would imply that the massless white hole should start\nmoving with the speed of light. This may suggest that the  state corresponding\nto\na massless white hole  at rest is unstable with respect to infinitesimally\nsmall\nexternal fluctuations, and therefore generically such  states  should be\ndescribed as particles (waves) moving with the speed of light\n\\cite{HullT,Susskind}.\n\nHowever, it might be impossible to give any boost to a massless white hole\nwithout either forming a bound state with it or making it massive. Indeed,\nthese\nstates can be considered massless only at an infinitely large distance from\nthem, but in this case they do not interact at all. Repulsive force\n$-{2q^2\\over\nr^3}$, which appears at a finite distance from the  center of a massless white\nhole, may be interpreted as a gravitational interaction with its massive core.\nThus, gravitational interaction  occurs only with an internal  part of the\nmassless white hole, which leads to its deformation. Such a deformation may\nchange energy and the effective mass of the white hole, and then it will be\nable to carry\nfinite momentum without being accelerated to the speed of light.\n\nAnother problem appears when one tries to understand the nature  of the\nrepulsive gravitational field.  The formal  reason of the repulsion is the\nexistence\nof the non-diagonal terms in the metric of six-dimensional compactified space,\nsee\n(\\ref{mod1}). However, it would be nice to have a simple intuitive\n4-dimensional picture describing the repulsive force from the phenomenological\npoint of view.  One way of thinking\nabout it is that the singularity acts on test particles as a body with a\nnegative\nmass. This mass becomes ``screened'' by positive energy density of physical\nfields.\nTherefore the absolute value of the effective gravitating mass of a sphere of a\nradius $r$ decreases at large\n$r$ as $-{M^2\\over r}$, and finally the total mass vanishes in the limit $r\n\\to \\infty$.\n\nThis intuitive picture  is, in fact,  rather counterintuitive. The\nstates with negative energy do exist in general relativity. For example, the\ntotal\nenergy of a closed universe is equal to zero as a result of exact cancellation\nof positive energy of matter and negative energy of gravitational field. Still\nit is hard to imagine how massive or  massless white holes with a repulsive\ncore could be created\nin the process of gravitational collapse of normal matter with positive energy\ndensity.  This could make such solutions very suspicious.  One should note,\nhowever, that the same is\ntrue  for the usual charged stringy black   holes as well: Typically such black\nholes cannot be formed in the process of gravitational collapse of   charged\nelementary particles. Indeed, in most cases there are no such charged particles\nin the\nunderlying Lagrangian. The description of charged stringy black holes is\nsomewhat\nunconventional as compared with the ordinary Reissner-Nordstrom black holes\ncontaining charged elementary particles. One may consider a sourceless flux of\nelectric or magnetic field, and then imagine a situation where the\ngravitational\nforce squeezes the flux into a singularity. Then the singularity will look like\nan\nelectrically or magnetically charged particle. To describe such a situation at\na\nmore formal level, one should  find  the flux of electric, magnetic and\ngravitational fields at infinity, and then find an extremum of action with\nthese\nboundary conditions, but without imposing any   boundary conditions\nand  solving Lagrange equations   at   the singularity. In particular, there is\nno\nrequirement that the effective charge of the singularity is\ncarried by an elementary particle, or that the  singularity looks like a normal\nparticle\nwith a positive mass. Such requirements appear only if one imposes an\nadditional\ncondition that the black hole is formed as a result of gravitational collapse\nof\nelementary particles. For the reason discussed above, this condition does not\nnecessarily apply to charged stringy black holes. The best constraint which one\ncan obtain on\nthe black hole mass is the supersymmetric Bogomolny positivity bound. This\nconstraint  applies not to the ``effective mass'' of the singularity, but to\nthe total\nADM mass, and it is satisfied by the massless and massive white holes\nconsidered\nin this paper.\n\n\n\n\n\\section{Discussion}\n\nIn the previous papers \\cite{Klaus2} and \\cite{BHGAUGE} the massless $N_L\n=0$ states of the toroidally compactified heterotic string have been found.\nThose states saturate the supersymmetric positivity bound.\nIn this paper we have found exact supersymmetric electrically charged\nfour-dimensional  configurations  whose ADM mass  can  vanish\nwithout the solutions being trivial. One of the features  of these solutions is\nthe\nnecessary presence of non-Abelian vectors  and scalars besides the metric,\nAbelian vectors and scalars. These   solutions have been obtained as  classical\nsolutions of the effective ten-dimensional action of the heterotic string\ntheory.\n\n\n\nThe\nconfigurations which we have discussed here cannot be associated with the\ntoroidally compactified string with $O(6,22)$ duality symmetry. The presence of\nYang-Mills fields  required by preservation of supersymmetry at the\nquantum level means that  we have only\n$O(6,6)$ symmetry with 6 Abelian gravi-photons and 6 Abelian vector multiplets.\nBut instead of the 16 additional Abelian  vector multiplets, which are\nextending\nthe symmetry of the toroidally compactified string from 6 to 22, we have\nnon-Abelian vector multiplets.\n\n\nTherefore\nthe interpretation of these new configurations  as the states of the properly\nquantized string   still has to be investigated.  The quantization conditions\nused\nfor toroidally compactified string should be generalized for the presence of\nnon-Abelian vector multiplets.\n\nThe metric of  the exact supersymmetric configurations which we have studied\nhas the following general form:\n \\begin{equation}\nds_{\\rm can}^2 =  \\left( 1+ {2\\sqrt 2 |Q_R| \\over r}    - {2 (|Q_L|^2 -\n|Q_R|^2)\\over\nr^2}\n\\right)^{-{1\\over\n2}} dt^2 -  \\left( 1+ {2\\sqrt 2 |Q_R| \\over r}    - {2 (|Q_L|^2 - |Q_R|^2)\\over\nr^2}\n\\right)^{1\\over\n2} dx^2 \\ .\n\\end{equation}\nHere $2 (|Q_L|^2 - |Q_R|^2)=q_n^2$,\nand the Yang-Mills field is a necessary part of the solutions when $q_n \\neq\n0$.\nThe  massive\n$a= \\sqrt 3$ electrically charged black hole  of Gibbons and Perry \\cite{GP} is\nincluded into this class. In fact it is the only case with $ |Q_L|^2 -\n|Q_R|^2=2 q_n^2=q_n = 0, \\; V^{YM} =0$\nfor which the configuration does not need the presence of non-Abelian vector\nfields   to be exact:  quantum  corrections vanish due to null properties of\nthe curvature of\nthe pp-waves with $K= 1+ {M\\over r}$ and $e^{-2\\hat \\phi}=1, A_n =0$\n\\cite{BKO1}.\nThis is a\none-parameter extreme black hole solution:\n \\begin{equation}\n ds^2 = \\left( 1+ {4M \\over r}\n\\right)^{-{1\\over\n2}} dt^2 -  \\left( 1+ {4M \\over r}\n\\right)^{1\\over\n2} dx^2 \\ .\n\\end{equation}\nWhen the mass of this solution tends to zero, it becomes  trivial, and one half\nof unbroken supersymmetry gets restored to the\n completely unbroken supersymmetry of the flat space.\nApart from this massive black hole solution, every other one has\n \\begin{equation}\n |Q_L|^2 - |Q_R|^2=2 q_n^2 > 0 \\ .\n\\label{repul} \\end{equation}\nThis means that any solution in this group can become massless ($Q_R =0$)  and\nstill the metric and the right-handed Abelian vectors (from the gravitational\nsupermultiplet)\nwill have some non-vanishing $1\/r^2$ terms.  The left-handed Abelian vectors\n(from the  non-gravitational vector supermultiplet) as well as  the Yang-Mills\nfields  are also present since $ |Q_L|^2 = 2 q_n^2>0$.\n\n\nThus  the non-trivial supersymmetric massless configurations described in this\npaper    do not exist without the non-Abelian multiplets. Even in the limit of\nthe vanishing mass one half of the supersymmetry is unbroken  and the\nother half is broken and serves to form the supercharge of the ultra-short\nmultiplet.\n\n\n\nA very unusual property of the new set of exact supersymmetric solutions\ndescribed in this paper  is the presence of a repulsive singularity when the\ncondition  $|Q_L|^2 - |Q_R|^2=2 q_n^2 > 0$ is valid. This singularity\nappears both for massless ($Q_R = 0$) and for massive ($Q_R  \\not = 0$)\nsolutions. As a result, these solutions can be better classified as  white\nholes rather than  the black ones.\n\nNote that the {\\it gravitational} repulsion which we are discussing here is\nquite different from the repulsive component of interaction between   extreme\nblack holes, which appears due to the {\\it non-gravitational}   interaction  of\ntheir electric, magnetic and dilaton charges \\cite{US}. White holes (repulsons)\nconsidered\nin this paper repel all particles, either charged or not, with the strength\nproportional to their mass. This repulsion, unlike the non-gravitational\nrepulsion considered in \\cite{US}, does not violate the equivalence principle.\nIf  existence of\nrepulsons is confirmed, we will have the first realization of  the universal\ngravitational  force which repels all  particles  and  therefore can be\nassociated with antigravity.\n\nOne should note, however,\nthat the interpretation of our solutions as white holes (repulsons) is rather\nstraightforward for\nmassive states, but, as we  already mentioned, interaction of particles\nwith massless states requires a more detailed investigation, and the classical\nconcept of gravitational repulsion in this case may become inapplicable.\n Formally white holes with vanishing ADM mass are still described by a static\nfour-dimensional geometry. The limit to the massless state has to be considered\nwith a special care since normally one would expect that a massless\nstate has to be described by a wave configuration which admits a null\nKilling vector.   However, to have a link to extreme white hole\nsolutions it is natural to consider those white holes which do not become\ntrivial\nwhen the mass equals zero. One may try to boost this solution to get the\nwave-type\nconfiguration.\n\nAlternatively, after we have found the special points in the moduli space where\nthe four-dimensional white holes become massless, we may return to the original\nform of the ten-dimensional configuration, which from the beginning admitted a\nnull Killing vector. The simplest one, whose four-dimensional metric is given\nin eq. (\\ref{DILATON2}), is indeed a supersymmetric pp-wave \\cite{BKO1}\ndescribed by the\nmetric, the constant dilaton, the two-form and the Yang-Mills field:\n\\begin{eqnarray}\nds^2 &=&  2  du  dv  -du^2 + \\sum_{n=4} ^8  {2q_{n}\\over r} dy^n du -\n\\sum_{i=1}^3\ndx^i dx^i -\\sum_{n=4} ^8 dy^ndy^n,\n\\qquad  e^{2\\hat \\phi}=1, \\nonumber \\\\\nB &=& 2  du \\wedge (dv + {2q_{n}\\over r}  dx^n),   \\qquad\nV_u {}^{in}\n= {x^i q^n \\over  r^3},\n\\ \\ \\ \\ \\ i=1,2,3, \\ \\ n=4,\\dots ,8 .\n\\end{eqnarray}\nThe remarkable feature of this solution is the fact that only if the Yang-Mills\nfield $V$   does not vanish, i.e. $q_{n} \\neq 0$, the geometry and the two-form\nare\nnot trivial.  At $q_n =0$ the metric becomes that of the flat space, $ds^2  =\ndt^2 -\n\\sum_{i=1}^9 dx^i dx^i$, and  the three-form $H$  vanishes. The supersymmetry\nis not broken at all, it is that of the flat space.\nHowever, as long as $q_{n}\\neq 0$, one half of the supersymmetries\nis broken, the\ncondition which the Killing spinor satisfy is  $\\gamma^u \\epsilon =0$\n\\cite{BKO1}.\n\n\nThis solution, as well as the more general ones presented in eqs.\n(\\ref{wave}),\n(\\ref{eq:embedding}) and described in  Sec. 4,   are exact  solutions with\none-half\nof unbroken supersymmetry with an account taken of  perturbative quantum\ncorrections in\n$\\alpha'$. The generalization consists in allowing more general asymptotic\nvalues of the ten-dimensional geometry, which is equivalent to allowing the\nfour-dimensional scalars to have  more general vacuum expectation\nvalues. The new mass formulas for exact supersymmetric non-Abelian white holes\nare\npresented  in\neqs. (\\ref{MSSW}) and (\\ref{MGFS}).\n\nThus we have described the exact supersymmetric non-Abelian configurations\neither as\nten-dimensional gravitational waves or as  electrically charged\nfour-dimensional  white holes, which may be  also called repulsons.\nOur exact supersymmetric massless configurations do not exist\nwithout   the  Yang-Mills fields which form a part of the   white hole\nconfiguration.\n One may expect various\n nonperturbative effects including confinement\/condensation of\nelectric\/magnetic white holes near the special points of the moduli space where\nthese solutions become massless. It would be most appropriate to study these\neffects, but it is outside of the scope of the present paper. Our main purpose\nhere was to demonstrate the possibility of the existence of  a new class of\nsupersymmetric configurations with very unusual properties, and to prepare a\nframework\nfor their subsequent investigation.\n\n\n\n\\section*{Acknowledgements}\n\nWe are grateful  to K. Behrndt, G. Horowitz, A. Sen, L. Susskind, S. Theisen\nand A. Tseytlin   for\nextremely useful discussions and to the referee for the suggestion to call the\nnew objects ``repulsons.\" This\nwork was  supported\nby  NSF grant PHY-8612280.\n\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Classical Frankel Theorem} \\label{intro}\n\nAn $S^1$-action on  a symplectic manifold $(M,\\omega)$ is \\emph{Hamiltonian} if there exists a smooth map, the \\emph{momentum  map}, \n$$\\mu \\colon M \\to (\\mathfrak{s}^1)^* \\simeq \\mathbb{R}$$ into the dual \n$(\\mathfrak{s}^1)^*$ of the Lie algebra $\\mathfrak{s}^1 \\cong \\mathbb{R}$ \nof $S^1$,  such that $$\\mathbf{i}_{\\xi_M}\\omega : = \\omega(\\xi_M, \\cdot) = \n\\mathbf{d} \\mu,$$ for some generator $\\xi$ of $\\mathfrak{s}^1$, that is, the $1$-form $\\mathbf{i}_{\\xi_M}\\omega$ is exact. \nHere $\\xi_M$ is the vector field on $M$ whose flow is given by $\\mathbb{R} \\times M \\ni (t,m) \\mapsto e^{\\rm it \\xi} \n\\cdot m \\in M$, where the dot denotes the $S^1$-action on $M$. If $(M,\\omega)$ is connected, compact and \n\\emph{K\\\"ahler}, the following result of T. Frankel is well-known: \n\\\\\n\\vspace{-2.5mm}\n\\\\\n{\\bf Frankel's Theorem} (\\cite{Frankel1959}).\n \\emph{Let $M$ be a compact connected K\\\"ahler\nmanifold admitting an $S^1$-action  preserving the K\\\"ahler\nstructure. If the $S^1$-action has fixed points, then the\naction is Hamiltonian.}\n\n\\medskip\n\nThis theorem generalizes in various ways; for example, the $S^1$-action may be replaced by a $G$-action, where $G$\nis any compact Lie group and the K\\\"ahler structure may be weakened to a symplectic structure. The purpose of this \npaper is to generalize Frankel's theorem to certain noncompact complete Riemannian manifolds. More specifically, \nwe describe a set of hypotheses under which the proof in the compact case can be generalized. This relies on\nthe existence of a Hodge decomposition on $1$-forms. \n\n\n\\section{Hodge Decomposition implies Frankel's Theorem} \\label{fs}\nLet $(M,\\omega)$ be a symplectic manifold. The triple $(\\omega,g,\\mathbf{J})$ is a \\emph{compatible triple} on $(M,\\omega)$ \nif $g$ is a Riemannian metric and $\\mathbf{J}$ is an almost complex structure such that $g(\\cdot,\\cdot)=\\omega(\\cdot,\\mathbf{J}\\cdot)$. \nDenote by ${\\rm d}V_g$ the measure associated to the Riemannian volume.\n\nLet $G$ be a connected Lie group with Lie algebra $\\mathfrak{g}$ acting on $M$ by symplectomorphisms, i.e., diffeomorphisms \nwhich preserve the symplectic form. We refer to $(M,\\, \\omega)$ as a \\emph{symplectic $G$\\--manifold}. Any element \n$\\xi \\in \\mathfrak{g}$ generates a vector field $\\xi_M$ on $M$, called the \\emph{infinitesimal generator}, given by \n$$\\xi_M(x):= \\left.\\frac{{\\rm d}}{{\\rm d}t}\\right|_{t=0}  \\op{exp}(t\\xi)\\cdot x.$$ \n\nThe  $G$-action on $(M,\\omega)$  is said to be \\emph{Hamiltonian} if there exists a smooth \nequivariant map $\\mu  \\colon M \\to \\mathfrak{g}^*$, called the \\emph{momentum map}, such that for\nall $\\xi \\in \\mathfrak{g}$ we have  $$\\mathbf{i}_{\\xi_M} \\omega : = \\omega(\\xi_M, \\cdot) = \\mathbf{d} \\langle \\mu, \n\\xi \\rangle,$$ where $\\left\\langle \\cdot , \\cdot \\right\\rangle : \\mathfrak{g}^\\ast \\times \\mathfrak{g} \n\\rightarrow \\mathbb{R}$ is the duality pairing. For example, if $G\\simeq (S^1)^k$, $k \\in \\mathbb{N}$, is a torus, the existence\nof such a map $\\mu$ is equivalent to the exactness of the one-forms $\\mathbf{i}_{\\xi_M}\\omega$ for all $\\xi \\in \\mathfrak{g}$. \n\nIn this case the obstruction of the action to being Hamiltonian lies in the first de Rham cohomology\ngroup of $M$. The simplest example of a $S^1$-Hamiltonian action is rotation of the sphere $S^2$ about the polar axis. \nThe flow lines of the infinitesimal generator defining this action are the latitude circles. \n\nDenote by ${\\rm L}^2_\\lambda$ the Hilbert space of square integrable functions relative \nto a given measure ${\\rm d}\\lambda$ on $M$, and write the associated norm either on functions or $1$-forms as \n$\\|\\cdot \\|_{{\\rm L}^2_\\lambda}$. This measure determines a formal \nadjoint $\\delta_\\lambda$ of the de Rham differential.\nA ${\\rm L}^2_\\lambda$ $1$-form $\\omega$ is called \\emph{$\\lambda$\\--harmonic} if it is in the common null space\nof $\\mathbf{d}$ and $\\delta_\\lambda$. \n\n\\begin{theorem}\\label{thm_gen} \nLet $G$ be a compact connected Lie group acting on the symplectic \nmanifold $(M,\\omega)$, with $(\\omega, g, \\mathbf{J})$ a $G$-invariant \ncompatible triple. Suppose, in addition, that \n${\\rm d}\\lambda = f {\\rm d}V_g$ \nis a $G$-invariant measure on $M$, where $f$ is \nsmooth and bounded. Suppose that $\\|\\xi_M\\|_{{\\rm L}^2_\\lambda} < \\infty$ for all $\\xi\\in \\mathfrak{g}$. Assume that every smooth \nclosed $1$-form $\\omega$ in ${\\rm L}^2_ \\lambda$ decomposes as an \n${\\rm L}^2_ \\lambda$-orthogonal sum \n$\\mathbf{d} f + \\chi$, where $f, \\mathbf{d}f \\in {\\rm L}^2_\\lambda$, \n $\\chi \\in {\\rm L}^2_ \\lambda$ is $\\lambda$-harmonic, \nand that each cohomology class in $H^1(M)$ has a unique \n$\\lambda$-harmonic  representative\nin ${\\rm L}^2_ \\lambda$. If $\\mathbf{J}$ preserves the space of ${\\rm L}^2_\\lambda$ harmonic one-forms and the $G$-action has fixed points \non every connected component, then the action is Hamiltonian. \n\\end{theorem}\n\\begin{proof}\nThe proof extends Frankel's method \\cite{Frankel1959}.  For clarity, we divide the proof into several steps.\n\\medskip\n\n\\noindent \\textbf{Step 1} (Infinitesimal invariance of $\\lambda$-harmonic $1$-forms). We show first that if $\\alpha \\in \n\\Omega^1(M)$ is harmonic and $\\| \\alpha\\|_{{\\rm L}^2_\\lambda} < \\infty$, then $\\mathcal L_{\\xi_M} \\alpha = 0$.\nThis is standard in the usual setting, but requires checking here since we have that $\\delta_\\lambda \\alpha = 0$ rather than \n$\\delta \\alpha = 0$. \n\nIf $\\varphi$ is an isometry of $(M,g)$ and preserves the measure $\\operatorname{d}\\!\\lambda$, then \n\\begin{equation}\n\\label{useful_formula}\n\\varphi^\\ast \\left(\\left\\langle \\! \\left\\langle \\nu, \\rho \\right\\rangle \\! \\right\\rangle \\operatorname{d}\\!\\lambda\\right) = \n\\left\\langle \\! \\left\\langle \\varphi ^\\ast \\nu, \\phi^\\ast \\rho \\right\\rangle \\! \\right\\rangle\n\\operatorname{d}\\!\\lambda \\nonumber\n\\end{equation}\nfor any $\\nu, \\rho \\in \\Omega^1(M)$, where $\\left\\langle \\! \\left\\langle \\nu, \\rho \\right\\rangle \\! \\right\\rangle $ dentes\nthe pointwise inner product of $\\nu$ and $\\rho$ on $M$. \n\nNext, denote by $\\Phi: G \\times M \\rightarrow M$ the $G $-action and $F_t := \\Phi_{\\exp(t \\xi)} $ the flow of $\\xi_M$. \nSince $\\mathbf{d} \\alpha = 0 $ it follows that $$\\mathbf{d}F  _t^\\ast \\alpha = F _t^\\ast \\mathbf{d}\\alpha = 0.$$\nIn addition, since $F_t^\\ast$ commutes with $\\lambda$, we also have \n$$\\delta_\\lambda F_t^\\ast \\alpha = F_t^\\ast \\delta_\\lambda \\alpha = 0.$$\nHence if $\\alpha$ is harmonic, then so is $F _t^\\ast \\alpha$.\n\nHowever, because $F_t$ is isotopic to the identity, \n$$\n[F_t^\\ast \\alpha] = F_t^\\star[ \\alpha] = [\\alpha]\n$$ \nin $\\op{H}^1(M, \\mathbb{R})$, where $F_t^\\star$ is the map on cohomology\ninduced by $F_t$.\nThis implies that $F _t^\\ast \\alpha = \\alpha$ since this cohomology class contains only one harmonic representative.\nTaking the $t $-derivative yields \n$$\n{\\mathcal L}_{ \\xi_M} \\alpha =0,\n$$ \nas required.\n\\smallskip\n\n\\noindent\\textbf{Step 2} (Using the existence of fixed points). Define \n$$\\xi_M^\\flat: = g( \\xi_M, \\cdot ) \\in \\Omega^1(M).$$ \nIf $\\alpha\\in \\Omega^1(M)$ is harmonic and $\\| \\alpha\\|_{ {\\rm L}^2_{ \\lambda}} < \\infty$,\nit follows from Step 1 that $$0 = {\\mathcal L}_{\\xi_M} \\alpha = \\mathbf{i}_{\\xi_M} \\mathbf{d} \\alpha + \n\\mathbf{d}\\mathbf{i}_{\\xi_M} \\alpha = \\mathbf{d}\\mathbf{i}_{\\xi_M} \\alpha.$$ Thus $\\alpha(\\xi_M)$ is constant on \neach connected component of $M$. Now, $\\xi_M$ vanishes on the fixed point set of $G$, and each component\nof $M$ contains at least one such point. Thus $\\alpha(\\xi_M) \\equiv 0$ on $M$, whence \n\\begin{equation}\n\\label{vanishing}\n\\left\\langle \\xi_M^\\flat, \\alpha \\right\\rangle_{\\op{L}^2_\\lambda}\n= \\varint_M \\alpha(\\xi_M) \\operatorname{d}\\! \\lambda = 0 \\nonumber\n\\end{equation}\nfor any harmonic one-form $\\alpha$ satisfying $\\| \\alpha\\|_{{\\rm L}^2_{ \\lambda}} < \\infty$. \n\\smallskip\n\n\\noindent\\textbf{Step 3} (Hodge decomposition). \nSince $$\\mathbf{d}\\mathbf{i}_{\\xi_M} \\omega= {\\mathcal L}_{\\xi_M}\\omega = 0$$ and \n$\\|\\mathbf{i}_{\\xi_M} \\omega\\|_{{\\rm L}^2_{\\lambda}} < \\infty$, our hypothesis implies that  \n\\begin{equation}\n\\label{Hodge_in_this_case}\n\\mathbf{i}_{\\xi_M} \\omega = \\mathbf{d} f^ \\xi + \\chi^ \\xi, \\nonumber\n\\end{equation}\nwhere $f^ \\xi \\in \\op{C}^{\\infty}(M) $, $\\chi^ \\xi \\in \\Omega^1(M)$ is $\\lambda$-harmonic and \n$\\|\\mathbf{d} f ^\\xi\\|_{{\\rm L}^2_{ \\lambda}}, \\|\\chi^\\xi\\|_{{\\rm L}^2_{ \\lambda}} < \\infty$. \n\nWe now prove that $\\chi^\\xi=0$. If $ \\alpha \\in \\Omega^1(M) $ is any harmonic one-form with \n$\\|\\alpha\\|_{{\\rm L}^2_{ \\lambda}} < \\infty$, then \n\\begin{equation*}\n\\left\\langle \\mathbf{i}_{\\xi_M} \\omega, \\alpha \\right\\rangle_{{\\rm L}^2_ \\lambda}\n= \\left\\langle \\xi_M^\\flat, \\mathbf{J}\\alpha \\right\\rangle_{\\op{L}^2_\\lambda} = 0\n\\end{equation*}\nby Step 2 since $\\mathbf{J} \\alpha$ is harmonic (by the hypotheses of\nthe theorem). In particular,  since $$\\xi_M^\\flat =  \\mathbf{J}\\mathbf{d}f^\\xi + \n\\mathbf{J}\\chi^\\xi$$ and $\\xi_M^\\flat$ is also orthogonal to the first term on the right, we conclude that $\\chi^\\xi = 0$. \n\nThe conclusion of this step is $$\\mathbf{i}_{\\xi_M} \\omega = \\mathbf{d}f^ \\xi $$ for \nany $\\xi\\in \\mathfrak{g}$; note that both sides of this identity are linear in $\\xi$. \n\\smallskip\n\n\\noindent\\textbf{Step 4} (Equivariant momentum map). Using a basis $\\{e_1, \\ldots, e_r\\}$ of $\\mathfrak{g}$, we \ndefine $\\mu:M \\rightarrow \\mathfrak{g}^\\ast$ by \n$$\n\\mu^ \\xi := \\xi^1f^{e_1}+ \\cdots + \\xi^rf^{e_r}, \\quad \\mbox{where}\\quad \\xi= \\xi^1e_1+ \\cdots + \\xi^re_.\n$$ \nClearly, $$\\mathbf{i}_{\\xi_M} \\omega = \\mathbf{d}\\mu^ \\xi,$$ so $\\mu$  is a momentum map of the $G $-action. \nSince $G$ is compact, one can average $\\mu$ in the standard way (see, e.g., \n\\cite[Theorem 11.5.2]{MaRa2003}) to obtain an equivariant momentum map. \nThis completes the proof of the theorem.\n\\end{proof}\n\n\\section{Applications}\nWe now discuss several different criteria which ensure that the results of the last section can be applied. \nThe first is the classical setting of `unweighted' ${\\rm L}^2$ cohomology, which is the cohomology of\nthe standard Hilbert complex of ${\\rm L}^2$ differential forms on a complete Riemannian manifold. The\nexistence of a strong Kodaira decomposition is known in many instances, and we present a few\nexamples.  We then discuss two other criteria, the first by Ahmed and Stroock and the second by \nGong and Wang, which allow one to prove a similar strong Kodaira decomposition for forms\nwhich are in ${\\rm L}^2$ relative to some weighted measure. We present some examples to which these\ncriteria apply. Finally, we recall some well-known facts\nabout the Hodge theory on spaces with `fibered boundary' geometry; these include asymptotically\nconical spaces, as well as the important classes of ALE\/ALF\/... gravitational instantons. Many\nof these spaces admit circle actions. \n\n\\subsection{Unweighted ${\\rm L}^2$ cohomology} \nThe nature of the Kodaira decomposition and ${\\rm L}^2$ Hodge theory on a complete manifold relative \nto the standard volume form is now classical. An account may be found in de Rham's book \\cite{dR}; \nsee also \\cite{Car2002}. \n\\begin{theorem}\n\\label{GpTr}\nIf $(M^n,g)$ is a complete Riemannian manifold and $0 \\leq k \\leq n$, then the  following conditions are equivalent:\n\\begin{itemize}\n\\item[(i)] ${\\rm Im}(\\mathbf{d} \\delta + \\delta\\mathbf{d})=\n(\\mathcal{H}_2^k(M))^{\\perp}$;\n\\item[(ii)] There is an ${\\rm L}^2$-orthogonal decomposition\n\\[\n{\\rm L}^2(M,\\Lambda^k)={\\rm Im}\\, \\mathbf{d}\\oplus {\\rm Im} \\, \\delta \\oplus \\mathcal{H}_2^k(M);\n\\]\n\\item[(iii)] ${\\rm Im}\\, \\mathbf{d}$ and ${\\rm Im}\\, \\delta$ are closed in ${\\rm L}^2(M,\\Lambda^k)$;\n\\item[(iv)] The quotients\n\\[\n\\overline{ \\mathrm{ran}\\, \\mathbf{d}} \\big\/ \\mathrm{ran}\\, \\mathbf{d} = 0\n\\]\nin ${\\rm L}^2(M, \\Lambda^k)$ and ${\\rm L}^2(M, \\Lambda^{n-k})$. \n\\end{itemize}\nIf the smooth form $\\alpha \\in \\Omega^k(M)$ decomposes as \n$\\mathbf{d}\\beta + \\delta_{\\mu_g} \\gamma + \\chi$, then \n$\\beta\\in\\Omega^{k-1}(M)$, $\\gamma \\in\\Omega^{k+1}(M)$, and \n$\\chi\\in \\Omega^k(M) \\cap \\mathcal{H}^2_k(M)$ are all smooth.\n\\end{theorem}\n\n\n\nTheorems \\ref{thm_gen} and \\ref{GpTr} imply the following result.\n\\begin{cor}\n \\label{nicecor}\nLet $G$ be a compact Lie group which acts isometrically on $(M, \\omega)$, a $2n$-dimensional complete connected \nK\\\"ahler manifold, and suppose that any one of the conditions {\\rm (i) - (iv)} of Theorem \\ref{GpTr} holds. \nIf the infinitesimal generators of the action all lie in $\\op{L}^2_{\\omega^n}$ and if the $G$-action \nhas fixed points, then it is Hamiltonian.\n\\end{cor}\nThe only point to note is that since $M$ is K\\\"ahler, the complex structure $\\mathbf{J}$ \npreserves the space of harmonic forms \\cite[Cor 4.11, Ch. 5]{Wells2008}. \n\n\\subsection{Examples}\nThere are many common geometric settings where the result above applies. We recall a few of these here.\n\n\\medskip\n\n\\noindent{\\bf Conformally compact manifolds:}  A complete manifold $(M,g)$ is called conformally\ncompact if $M$ is the interior of a compact manifold with boundary $\\bar{M}$ and $g$ can be written\nas $\\rho^{-2} \\bar{g}$, where $\\rho$ is a defining function for $\\partial \\bar{M}$ (i.e., $\\partial\\bar{M} = \\{\\rho = 0\\}$\nand ${\\rm d} \\rho \\neq 0$ there) and $\\bar{g}$ is a metric\non $\\bar{M}$ which is non-degenerate and smooth up to the boundary.\nThe sectional curvatures of $g$ become isotropic near any point $p \\in \\partial \\bar{M}$, with common\nvalue $-|{\\rm d}\\rho|^2_{\\bar{g}}$. If this value is constant along the entire boundary, then $(M,g)$ is called\nasymptotically hyperbolic. \n\nAn old well-known result \\cite{Maz-Hodge} states that if $n = \\dim M \\neq 3$ (automatic if $M$\nis symplectic), then the conditions of Theorem \\ref{GpTr} are satisfied when $k = 1$, and hence \nCorollary \\ref{nicecor} holds. There are now much simpler proofs of this result; see \\cite{CarSurvey}. \n\nAs explained in \\cite{CarSurvey}, the conditions of Theorem \\ref{GpTr} are invariant under\nquasi-isometry, which means that we obtain a similar result for any symplectic manifold quasi-isometric \nto a conformally compact space.  This allows us, in particular, to \nsubstantially relax the regularity conditions on $\\rho$ and $\\bar{g}$ in this definition.\n\nThere is an interesting generalization of this to the set of complete edge metrics. The geometry here is\na bit more intricate; as before, $M$ is the interior of a smooth manifold with boundary. Now, however,\nthe boundary $\\partial \\bar{M}$ is assumed to be the total space of a fibration over a compact smooth\nmanifold $Y$ with compact fiber $F$. We can use local coordinates $(x,y,z)$ near a point of the boundary \nwhere $x$ is a boundary defining function, $y$ is a set of coordinates on $Y$ lifted to $\\partial \\bar{M}$\nand then extended inward, and $z$ is a set of functions which restrict to coordinates on each fiber $F$. \nA metric $g$ on $M$ is called a \\textit{complete edge metric} if in each such local coordinate system it takes the form \n\\begin{multline*}\ng = \\frac{\\mathbf{d}x^2 + \\sum a_{0 \\alpha}(x,y,z) \\mathbf{d}x\\mathbf{d}y_\\alpha + \\sum a_{\\alpha \\beta} a_{\\alpha \\beta}(x,y,z) \\mathbf{d}y_\\alpha \\mathbf{d}y_\\beta}{x^2} \\\\\n+ \\sum b_{0\\mu} \\frac{\\mathbf{d}x}{x}\\mathbf{d}z_\\mu + b_{\\alpha \\mu} \\frac{\\mathbf{d}y_\\alpha}{x} \\mathbf{d}z_\\mu + \nb_{\\mu \\nu} \\mathbf{d}z_\\mu \\mathbf{d}z_\\nu. \n\\end{multline*}\nThe prototype is the product $X \\times F$ where $X$ is a conformally compact manifold and $F$ is a compact smooth\nmanifold, or more generally, a manifold which fibers over a neighborhood of infinity in a conformally compact space $X$ \nwith compact smooth fiber $F$.\n\nThe analytic techniques developed in \\cite{Maz-edge} generalize those in \\cite{Maz-Hodge} and show that\nif $(M,g)$ is a space with a complete edge metric, and if $\\dim Y \\neq 2$, then the Hodge Laplace\noperator on $1$-forms is closed. \n\n\\medskip\n\n\\noindent{\\bf Surfaces of revolution:}  \nA Riemannian surface $(M,g)$ which admits an isometric $S^1$ action must be a surface of revolution, hence in\npolar coordinates, \n\\[\ng = \\mathbf{d}r^2 + f(r)^2 \\mathbf{d}\\theta^2,\n\\] \nwhere $\\theta \\in S^1$ and either $f > 0$ on $(0,\\infty)$ and is a function of $r^2$ (i.e., its Taylor expansion\nnear $r=0$ has only even terms) which vanishes at $r=0$, or else $f$ is strictly positive on all of $\\mathbb R$.  \nIn the first case, $M \\cong \\mathbb R^2$, while in the second, $M \\cong S^1 \\times \\mathbb R$. \n\nThe symplectic form is $\\omega = f(r) \\mathbf{d}r \\wedge \\mathbf{d}\\theta$, so the action is generated by the vector field \n$\\partial_\\theta$. Since $\\mathbf{i}_{\\partial\/\\partial\\theta}\\omega=\n-f(r)\\mathbf{d}r$, one of the basic hypotheses becomes\n\\begin{equation}\n\\|\\mathbf{i}_{\\partial_\\theta}\\omega \\|^2_{\\op{L}^2_{\\omega}}  \n= \\varint_{0}^{2\\pi}\\varint_{0}^{\\infty}f(r)^3\\op{d}\\!r\\op{d}\\!\\theta  =2\\pi \\varint_0^\\infty f(r)^3 \\op{d}\\!r < \\infty.\n\\label{normdtheta}\n\\end{equation}\n\\begin{prop}\\cite[Theorem 1.2]{Tr2009} If $M \\cong \\mathbb R^2$ and $f \\leq C r^{-k}$ for some $k > 1\/3$, then the range of the Hodge Laplace\noperator on $1$-forms is closed.\n\\label{Tro}\n\\end{prop}\nWith these hypotheses, we can then apply Corollary \\ref{nicecor} as before.  \n\nIt is worth contrasting Proposition \\ref{Tro} with the well-known criterion of McKean \\cite{McK}. This states that if $(M^2,g)$ \nis simply connected \\emph{and} has Gauss curvature $K_g \\leq -1$, then the ${\\rm L}^2$ spectrum of the Laplacian on functions \nis contained in $[1\/4, \\infty)$. The spectrum of the Laplacian on $2$-forms is the same, and using a standard Hodge-theoretic \nargument, the spectrum of the Laplacian on $1$-forms is contained in $\\{0\\} \\cup [1\/4,\\infty)$. Thus this curvature\nbound would also guarantee the conclusion of Theorem \\ref{GpTr}.  Now, $$K_g = - f''(r)\/f(r) \\leq -1$$ is the same as \n$$f''(r) \\geq f(r).$$ Using this and the initial condition $f(0)=0$, it is not hard to show that $f$ must grow exponentially \nas $r \\to \\infty$, so that \\eqref{normdtheta} cannot hold.  In other words, McKean's condition is useless for our\npurposes. \n\nOf course, if the hypotheses of Proposition \\ref{Tro} hold, then we do not need to apply these Hodge-theoretic\narguments since the momentum map of this circle action is given by any function $\\mu(r)$ satisfying \n$$\\mu'(r) = -f(r).$$ \n\n\\medskip\n\n\\noindent{\\bf Compact stratified spaces:}   Although it is outside the framework of complete manifolds, there is another\nclass of spaces to which these results may be applied. These are the smoothly stratified spaces with iterated edge metrics.\nThese include, at the simplest level, spaces with isolated conic singularities or simple edge singularities. More general\nspaces of this type are obtained recursively, by using spaces such as these as cross-sections of cones, and these\ncones can vary over a smooth base.  Hodge theory on such spaces was first considered by Cheeger \\cite{Cheeger};\nthe recent papers \\cite{ALMP1}, \\cite{ALMP2} provide an alternate approach and generalize the spaces to allow ones\nfor which it is necessary to impose boundary conditions along the strata.  A complete Hodge theory is available, \ncf.\\ the papers just cited.  One important way that such spaces might arise in our setting is if the group $G$ acts symplectically \non a compact smooth manifold $M'$, but \n$G$ commutes with the symplectic action by another group $K$.  Then the action of $G$ descends to \nthe quotient $M = M'\/K$, and this latter space typically has precisely the stratified structure and iterated\nedge metric as described above .\n\n\n\\subsection{Ahmed-Stroock conditions.}  \nUnder certain rather weak requirements on the geometry of $(M,g)$ and an auxiliary measure \n$${\\rm d}\\lambda = e^{-U} {\\rm d}V_g,$$\nAhmed and Stroock \\cite[\\S6]{AhSt2000} have proved a Hodge-type decomposition. In the theorem below and the rest of\nthe paper, $\\Delta f : = \\operatorname{div} \\nabla f = \n- \\delta \\mathbf{d} f$ is \nthe usual Laplacian on functions and $\\nabla^2f : = \\operatorname{Hess} f$\nis the Hessian of $f$, i.e., the second covariant derivative of $f$.\n\n\\begin{theorem}[\\cite{AhSt2000}] Assume that $(M,g)$ is complete and\n\\begin{itemize} \n\\item[$\\bullet$]  $ \\mathrm{Ric}_g \\geq - \\kappa_1$; \n\\item[$\\bullet$] the curvature operator is bounded above, i.e., \n$\\left\\langle \\! \\left\\langle R \\alpha, \\alpha \\right\\rangle \\! \\right\\rangle \n\\leq \\kappa_2 \\| \\alpha\\|_{ {\\rm L}^2}$ for all $\\alpha \\in \\Omega^2(M) $, where $\\kappa_1, \\kappa_2 \\geq 0$.  \n\\end{itemize}\nSuppose further that $U$ is a smooth nonnegative proper function on $M$ which satisfies\n\\begin{itemize}\n\\item $\\Delta U \\leq C(1+U) $ and $\\|\\nabla U\\|^2 \\leq C\\operatorname{e}^{ \\theta U}$ for some $C<\\infty$ and $\\theta\\in (0,1)$;\n\\item $\\varepsilon U^{1+ \\varepsilon} \\leq 1+\\| \\nabla U\\|^2$ for some $\\varepsilon > 0$;\n\\item $\\left\\langle \\! \\left\\langle v, (\\nabla^2 U)(v) \\right\\rangle \\! \\right\\rangle \\geq -B\\|v\\|^2$ for every $x \\in M $ and \n$v \\in \\operatorname{T}_xM$, where $B<\\infty$. \n\\end{itemize} \nWrite $\\delta_\\lambda$ for the adjoint of $\\mathbf{d}$ relative to \n${\\rm d}\\lambda = e^{-U} {\\rm d}V_g$ and ${\\rm L}^2_\\lambda$ for the \nassociated Hilbert space. Note that since $U\\geq 0$, $\\lambda$ is bounded.\n\nThen \n\\begin{enumerate}\n\\item[{\\rm (1)}] \\cite[Theorem 5.1]{AhSt2000} There is a strong Hodge decomposition on $1$-forms. In particular, if \n$\\alpha \\in {\\rm L}^2_\\lambda \\Omega^1 \\cap \\mathcal C^\\infty$ is closed, then $\\alpha = \\mathbf{d}f + \\chi $, where \n$f \\in {\\rm L}^2_\\lambda \\cap \\mathcal C^\\infty$ and $\\chi \\in \\mathcal H_\\lambda^1$.\n\\item[{\\rm (2)}] \\cite[Theorem 6.4]{AhSt2000} Each class $[\\alpha]\\in \\operatorname{H} ^1(M, \\mathbb{R})$ has a unique \nrepresentative in $\\mathcal H_\\lambda^1$. \n\\end{enumerate}\n\\label{AScond}\n\\end{theorem}\n\n\\begin{cor}\\label{Stroock_conditions}\nAssume that $M$ is symplectic and that $(g,\\omega, \\mathbf{J})$ are a $G$-invariant compatible triple,\nand that $U$ is also $G$-invariant. If the hypotheses of Theorem \\ref{AScond} all hold,  \n$\\mathbf{J}\\mathcal{H}^1_\\lambda \\subset \\mathcal{H}^1_\\lambda $,  and if the $G$-action has fixed points, \nthen it is Hamiltonian.\n\\end{cor}\n\n\\subsection{Gong-Wang conditions} There are other conditions, discovered by Gong and Wang,  \nwhich lead to a strong Hodge decomposition. \n\n\\begin{theorem}[\\cite{GoWa2004}]\nLet $G$ act on the noncompact symplectic manifold $(M, \\omega)$, and suppose that $(\\omega, g, \\mathbf{J})$ \nis a $G$-invariant compatible triple. Assume that ${\\rm d}\\lambda = e^V {\\rm d}V_g$ is also $G$-invariant and has finite \ntotal mass. Suppose finally that \n\\begin{itemize}\n\\item $\\mathrm{Ric} - \\operatorname{Hess}(V) \\geq -C \\mathrm{Id}$; \n\\item there exists a positive $G $-invariant proper function $U \\in \\mathcal C^2(M)$ such \nthat $U+V$ is bounded;  \n\\item $\\|\\nabla U\\| \\rightarrow \\infty$ as $U \\rightarrow \\infty $;\n\\item $\\limsup_{U \\rightarrow \\infty}  \\left(\\Delta U\/\\|\\nabla U\\|^2 \\right)< 1$. \n\\end{itemize}\nThen there is a strong Hodge decomposition on \n${\\rm L}^2_\\lambda \\Omega^1(M)$, as before. \n\\end{theorem}\n\n\\begin{cor}\\label{GoWa_conditions}\nWith all notation as above, if $\\mathbf{J}$ preserves $\\mathcal H^1_\\lambda$, and the $G$-action has fixed points, then\nit is Hamiltonian.\n\\end{cor}\n\n\\subsection{Further examples}\nThere are many interesting types of spaces to which the Ahmed-Stroock and Gong-Wang results can be applied, but which \nare not covered by the more classical Theorem \\ref{GpTr}. We describe a few of these here, including spaces with asymptotical \ncylindrical or asymptotically conic ends or with complete fibered boundary geometry. Amongst these are the asymptotically locally \nEuclidean (ALE) spaces, as well as the slightly more complicated ALF, ALG, and ALH spaces which arise in the classification of \ngravitational instantons. (We refer to \\cite{HHM} for a description of the geometry of ALE\/F\/G\/H spaces.) We can also handle Joyce's \nquasi-ALE (QALE) spaces \\cite{Joyce} and their more flexible Riemannian analogues, the quasi-asymptotically conic (QAC) spaces \nof \\cite{DegMaz}. The interest in including all of these spaces is that they seem to be intimately intertwined with symplectic geometry; \nindeed,  many of them arise via hyperK\\\"ahler reduction. \n\nThe obvious idea is to let the function $U$ in Theorem \\ref{AScond} depend only on the radial function $r$ on $M$.  Actually, it \nis clear that Theorem \\ref{AScond} holds on all of $M$ if and only if it holds on each end (with, say, relative boundary conditions \non the compact boundaries), so we can immediately localize to each end. We can also replace $g$ on each end by a perhaps \nsimpler metric which is quasi-isometric to it.   The general feature of all these spaces is that the distance function $r$ from\na suitably chosen inner boundary has ``symbolic decay properties'', i.e., successively higher derivatives of $r$ decay\nincreasingly more quickly.   Writing $U = r^a$, then we require that \n\\begin{align*}\n{\\rm i)} & \\ \\Delta U = a(a-1) r^{a-2} |\\nabla r|^2 + a r^{a-1} \\Delta r \\leq C (1 + r^a) \\\\\n{\\rm ii)} & \\ \\epsilon U^{1+\\epsilon} = \\epsilon r^{a(1+\\epsilon)} \\leq 1 + a^2 r^{2a-2} |\\nabla r|^2 \\\\\n{\\rm iii)} & \\ \\nabla^2 U = a r^{a-1} \\nabla^2 r + a (a-1) r^{a-2} \n\\mathbf{d}r^2  \\geq -B. \n\\end{align*}\nRecalling that $|\\nabla r| = 1$ holds in general, then ii) implies that $a > 2$, while i) shows\nthat $\\Delta r$ must grow slower than $r$, and finally iii) shows that the level sets $\\{r = \\mathrm{const.} \\}$\nhave some sort of convexity.  \n\nRather than trying to determine the most general spaces for which these restrictions hold, we explain why\nthey are true for the various examples listed above. For the reasons we have explained (namely, that it suffices\nto consider a quasi-isometric model), we focus on the simplest models for each of these spaces.  In each\nof the following, we consider one end $E$ of $M$. In general we can apply our results to manifolds $M$\nwhich decompose into some compact piece $K$ and a finite number of ends $$E_1, \\ldots, E_N,$$ each\nof which is of one of the following types. \n \n\\medskip\n\n\\noindent{\\bf Cylindrical ends:}  Here $E = [0,\\infty) \\times Y$ where $(Y,h)$ is a compact\nsmooth Riemannian manifold, and $r$ is the linear variable on the first factor.  The metric is the product\n$\\mathbf{d}r^2 + h$. We obtain conditions i), ii), iii) directly since $\\nabla^2 r = 0$. \n\n\\medskip\n\n\\noindent{\\bf Conic ends:} Now suppose that $E = [1,\\infty) \\times Y$ where $(Y,h)$ is again\na compact smooth manifold and $r \\geq 1$, and the metric is given by \n$g = \\mathbf{d}r^2 + r^2 h$. Then \n$$\\Delta r = (n-1)\/r$$ and $$\\nabla^2 r \\geq 0,$$ so, once again, all three conditions hold.\n\n\\medskip\n\n\\noindent{\\bf Fibered boundary ends:}  This is slightly more complicated.  Suppose that $Z$ is\na compact smooth manifold which is the total space of a fibration $\\pi: Z \\to Y$ with fiber $F$. \nLet $h$ be a metric on $Y$ and suppose that $k$ is a symmetric $2$-tensor on $Z$ which\nrestricts to each fiber $F$ to be positive definite and so that $\\pi^* h + k$ is positive definite on $Z$. \nThen  $$E = [1,\\infty) \\times Z,$$ and \n\\[\ng = \\mathbf{d}r^2 + r^2 \\pi^* h + k.\n\\]\nIn other words, this metric looks conical in the base ($Y$) directions and cylindrical in the fiber ($F$) directions.\nFor the specific cases of such metrics that arise in the gravitational instantons above, $Y$ is the quotient of\nsome $S^k$ by a finite group $\\Gamma$ (typically in $\\mathrm{SU}(k+1)$) and $F$ is a torus $T^\\ell$.\nThe four-dimensional ALF\/ALG\/ALH spaces correspond to the cases $$(k,\\ell) = (2,1),\\ (1,2),\\ (0,3).$$ \nThe pair $(k,\\ell) = (3,0)$ is precisely that of ALE spaces.\n\nFor each of these, it is a simple computation to check that $r$ has all the required properties. \n\n\\medskip\n\n\\noindent{\\bf QALE and QAC ends:}  The geometry of quasi-asymptotically conic spaces\nare considerably more difficult to describe in general, and we defer to \\cite{Joyce} and \\cite{DegMaz} \nfor detailed descriptions of the geometry.  These spaces are slightly more complicated in the sense\nthat while they are essentially conical as $r \\to \\infty$, the cross-sections $\\{r = \\mathrm{const.} \\}$ \nare families of smooth spaces which converge to a compact stratified space. This is consistent with the\nfact that QALE spaces arise as (complex analytic) resolutions of quotients $\\mathbb C^n \/ \\Gamma$. \nThe basic types of estimates for $r$ and its derivatives are almost the same as above, and so conditions \ni), ii), and iii) still hold. We refer to the monograph and paper cited above for full details. \n\n\\medskip\n\n\\noindent{\\bf Bundles over QAC ends:}  The final example consists of ends $E$ which are bundles over \nQAC spaces, and with metrics which do not increase the size of the fibers as $r \\to \\infty$. This is in perfect\nanalogy to how fibered boundary metrics generalize and fiber over conic metrics.  The behavior of the\nfunction $r$ on these spaces is similarly benign and these same three conditions hold.\n\n\\medskip\n\nThese examples have been given with very little detail (in the last two cases, barely any). The reason \nfor including the, here is because they arise frequently.  In particular, the last category, i.e., bundles over\nQAC (or more specifically, QALE) spaces contain the conjectural picture for the important family of \nmoduli spaces of monopoles on $\\mathbb R^3$.   On none of these spaces is the range of the\nLaplacian on unweighted $1$-forms usually closed, but the Ahmed-Stroock conditions provide\nan easily applicable way to obtain Hodge decompositions on these spaces. \n\n\n\n\\section{History of the problem: Frankel's Theorem and further results} \\label{HH}\n\nThe first result concerning the relationship between the existence of fixed points and the Hamiltonian character of the action is\nFrankel's celebrated theorem \\cite{Frankel1959} stating that if the manifold\nis compact, connected, and K\\\"ahler, $G=S^1$, and\nthe symplectic action has fixed points, then it must be Hamiltonian (note that $\\mathbf{J}\n\\mathcal{H} \\subset \\mathcal{H}$ holds, see \\cite[Cor 4.11, Ch. 5]{Wells2008}). Frankel's work has been very influential: \nfor example, Ono \\cite{Ono1984} proved the analogue theorem for compact Lefschetz manifolds and McDuff \\cite[Proposition 2]{McDuff1988} \nhas shown that any symplectic circle action on a compact connected symplectic 4-manifold having fixed points is Hamiltonian. \n\nHowever, this result fails in higher dimensions: McDuff \\cite[Proposition 1]{McDuff1988} gave an example of a compact \nconnected symplectic $6$-manifold with a symplectic circle action which has nontrivial fixed point set (equal to a union\nof tori), which is nevertheless not Hamiltonian. If the $S^1$-action is semi-free (i.e., free off the fixed point set), then \nTolman and Weitsman \\cite[Theorem 1]{ToWe2000} have shown that any symplectic $S^1$-action on a compact connected \nsymplectic manifold having fixed points is Hamiltonian. Feldman \\cite[Theorem 1]{Feldman2001} characterized the obstruction \nfor a symplectic circle action on a compact manifold to be Hamiltonian and deduced the McDuff and Tolman-Weitsman theorems \nby applying his criterion. He showed that the Todd genus of a manifold admitting a symplectic circle action with isolated\nfixed points is equal either to 0, in which case the action is non-Hamiltonian, or to 1, in which\ncase the action is Hamiltonian. In addition, any symplectic circle action on a manifold with positive Todd genus is Hamiltonian. For additional results regarding aspherical\nsymplectic manifolds (i.e.  $\\varint_{S^2} f ^\\ast \\omega = 0$ \nfor any smooth map $f: S^2 \\rightarrow M$) see \\cite[Section 8]{KeRuTr2008} and \\cite{LuOp1995}.\nAs of today, there are no known examples of symplectic $S^1$-actions on compact connected symplectic manifolds that are not Hamiltonian but have at least one isolated fixed point.\n\nLess is known for higher dimensional Lie groups. Giacobbe \\cite[Theorem 3.13]{Giacobbe2005} proved that a symplectic\naction of a $n$-torus on a $2n$-dimensional compact connected symplectic manifold with fixed points is necessarily Hamiltonian;\nsee also \\cite[Corollary 3.9]{DuPe2007}. If $n=2 $ this result can be checked explicitly from the classification \nof symplectic 4-manifolds with symplectic 2-torus actions given in \\cite[Theorem 8.2.1]{Pelayo2010} (since cases 2--5 in the \nstatement of the theorem are shown not to be Hamiltonian; the only non-K\\\"ahler cases are given in items 3 and 4 as proved in \n\\cite[Theorem 1.1]{DuPe2010}). \n\nIf $G$ is a Lie group with Lie algebra $\\mathfrak{g}$ acting symplectically on the symplectic manifold $(M, \\omega) $, \nthe action is said to be \\textit{cohomologically free} if the Lie algebra homomorphism\n$$\\xi \\in \\mathfrak{g} \\mapsto [\\mathbf{i}_{\\xi_M} \\omega] \\in  \\operatorname{H}^1(M, \\mathbb{R})$$ is \ninjective; $\\operatorname{H}^1(M, \\mathbb{R})$ is regarded as an abelian Lie algebra.  Ginzburg \n\\cite[Proposition 4.2]{Ginzburg1992} showed that if a torus $\\mathbb{T}^k = (S^1)^k$, $k \\in \\mathbb{N}$, acts symplectically,\nthen there exist subtori $\\mathbb{T}^{k-r}$, $\\mathbb{T}^r$ such that $\\mathbb{T}^k =\\mathbb{T}^r\\times\\mathbb{T}^{k-r}$,  \nthe $\\mathbb{T}^r$-action is cohomologically free, and the $\\mathbb{T}^{k-r}$-action is Hamiltonian. This homomorphism is the\nobstruction to the existence of a momentum map: it vanishes if and only if the action admits a momentum map. For compact\nLie groups the previous result holds only up to coverings. If $G$ is a compact Lie group, then it is well-known that\nthere is a finite covering $$\\mathbb{T}^k \\times K \\rightarrow G,$$ where $K$ is a semisimple compact Lie group. So\nthere is a symplectic action of $\\mathbb{T}^k \\times K$ on $(M, \\omega)$. The $K$-action is Hamiltonian, since\n$K$ is semisimple. The previous result applied to $\\mathbb{T}^k$ implies that there is a finite covering \n$$\\mathbb{T}^r\\times (\\mathbb{T}^{k-r} \\times K) \\rightarrow G$$ such that the $(\\mathbb{T}^{k-r} \\times K)$-action is\nHamiltonian and the $\\mathbb{T}^r$-action is cohomologically free; this is \\cite[Theorem 4.1]{Ginzburg1992}. The Lie\nalgebra of $\\mathbb{T}^{k-r} \\times K$ is $\\ker\\left(\\xi\\mapsto [\\mathbf{i}_{\\xi_M} \\omega]\\right)$. (It appears that the \nargument in \\cite{Ginzburg1992} implicitly requires $M$ to satisfy the Lefschetz condition or more generally the flux conjecture to hold for $M$. \nThus ultimately it depends on \\cite{Ono2006} where the flux conjecture is established in full generality. We thank\nV. Ginzburg for pointing this out.)\n\\\\\n\\\\\n\\\\\n\\\\\n\\emph{Acknowledgements.} We thank I. Agol, D. Auroux, J. M. Lee, D. Halpern\\--Leistner, X. Tang, and A. Weinstein for many helpful\ndiscussions. We wish to particularly thank A. Weinstein for comments on several preliminary \nversions of the paper. Work on this paper started when the first author was affiliated with the University of California, \nBerkeley (2008\\--2010), and the last two authors were members of MSRI.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}