diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlnmd" "b/data_all_eng_slimpj/shuffled/split2/finalzzlnmd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlnmd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $X$ be a curve, $\\widetilde{X} \\stackrel{\\nu}\\rightarrow X$ its normalization, $\\mathcal{O} = \\mathcal{O}_X$ and $\\widetilde\\mathcal{O} =\\nu_*(\\mathcal{O}_{\\widetilde{X}})$.\n Generalizing an original idea of K\\\"onig \\cite{koe}, we define a sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$ on $X$ called \\emph{K\\\"onig's order} such that the ringed space\n$\\mathbb{X} = (X, \\mathcal A} \\def\\kN{\\mathcal N)$ has the following properties.\n\n\\medskip\n\\noindent\n1.~The non--commutative curve $\\mathbb{X}$ is ``smooth'' in the sense that \\linebreak\n$\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) < \\infty$, where $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$ is the category of coherent $\\mathcal A} \\def\\kN{\\mathcal N$--modules on $X$. In fact, $\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\le 2n$, where $n$ is a certain (purely commutative) invariant of $X$ called \\emph{level}. If the original curve $X$ has only nodes and cusps as singularities, the sheaf $\\mathcal A} \\def\\kN{\\mathcal N$ coincides with \\emph{Auslander's order}\n $$\n \\left(\n \\begin{array}{cc}\n \\mathcal{O} & \\widetilde\\mathcal{O} \\\\\n \\mathcal I} \\def\\kV{\\mathcal V & \\widetilde\\mathcal{O}\n \\end{array}\n \\right)\n $$\n introduced in \\cite{bd}, where $\\mathcal I} \\def\\kV{\\mathcal V$ is the ideal sheaf of the singular locus of $X$.\n\n\\medskip\n\\noindent\n2.~The non--commutative curve $\\mathbb{X}$ is a \\emph{non--commutative} (or \\emph{categorical}) \\emph{resolution of singularities} of $X$, see \\cite{vdb,ku} for the definitions.\nThe category $\\mathop{\\mathsf{Coh}}\\nolimits(X)$ of coherent sheaves on $X$ is a Serre quotient of $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$. Moreover, the triangulated category $\\mathop{\\mathsf{Perf}}\\nolimits(X)$ of perfect complexes on $X$ admits an exact fully faithful embedding $\\mathop{\\mathsf{Perf}}\\nolimits(X) \\hookrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ such that its composition with the Verdier localization $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\rightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ is isomorphic to the canonical inclusion functor.\nIf the curve $X$ is Gorenstein, the constructed categorical resolution of singularities of $X$ turns out to be \\emph{weakly crepant} in the sense of Kuznetsov \\cite{ku}.\n\n\\medskip\n\\noindent\n3.~We show that the triangulated category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$\nis a recollement of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D^b(Q-\\mathrm{mod})$,\nwhere $Q$ is a certain \\emph{quasi-hereditary} artinian ring (in particular, of finite global dimension), determined ``locally'' by the singularity types of the singular points of $X$. In the case of simple curve singularities, we describe the corresponding algebras $Q$\nexplicitly in terms of quivers and relations.\n\n\n\n \\medskip\n\\noindent\n4.~Assume $X$ is projective over some field $\\mathbbm{k}$. According to Orlov \\cite{or}, the Rouquier dimension \\cite{rou} of the triangulated category\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ is equal to \\emph{one}. Let $\\widetilde\\mathcal F} \\def\\kS{\\mathcal S$ be a vector bundle on $\\tilde{X}$ such that\n$\\langle\\widetilde{\\mathcal F} \\def\\kS{\\mathcal S}\\rangle_2 = D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $\\mathcal F} \\def\\kS{\\mathcal S = \\nu_*(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S)$. We show that\n$\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\mathcal{O}_Z\\bigr\\rangle_{n+2}\n$\nwhere $\\mathcal{O}_Z$ is the structure sheaf of the singular locus of $X$ (with respect to the reduced scheme structure) and\n$n$ is the level of $X$.\n\n\n\n\\medskip\n\\noindent\n5.~If our original curve $X$ is moreover rational, then we show that $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ admits a\n\\emph{tilting object} $\\mathcal H} \\def\\kU{\\mathcal U$ such that the finite dimensional $\\mathbbm{k}$--algebra $\\Lambda = \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{D^b(\\mathbb{X})}(\\mathcal H} \\def\\kU{\\mathcal U)\\bigr)^{\\mathrm{op}}$ is quasi--hereditary. In particular, we get an exact fully faithful embedding\n $\\mathop{\\mathsf{Perf}}\\nolimits(X) \\hookrightarrow D^b(\\Lambda-\\mathrm{mod})$,\n giving an affirmative answer on a question posed to the first--named author by Valery Lunts.\n\n\\medskip\n\\noindent\n\\emph{Acknowledgement}. The work on this article has been started during the stay of the second--named author at the Max--Planck--Institut f\\\"ur Mathematik in Bonn. Its final version was prepared during the visit of the second-- and the third--named author to the Institute of Mathematics of the University of Cologne. The first--named author would like to thank Valery Lunts for the invitation and iluminative discussions during his visit to the Indiana University Bloomington. We are thankful to the referees for their helpful comments.\n\n\n\\section{Local description of K\\\"onig's order}\\label{Sec:LocalStory}\n\nLet $(O, \\mathfrak{m})$ be a reduced Noetherian local ring of Krull dimension one, $K$ be its total ring of fractions and\n$\\widetilde{O}$ be the normalization of $O$.\n\n\\begin{proposition}\\label{P:normalizelocal} Consider the ring $O^\\sharp = \\mathop{\\mathrm{End}}\\nolimits_{O}(\\mathfrak{m})$. Then the following properties hold.\n\\begin{itemize}\n\\item $O^\\sharp \\cong \\left\\{x \\in K \\big| x \\mathfrak{m} \\subset \\mathfrak{m} \\right\\}$. Moreover,\n$O \\subseteq O^\\sharp \\subseteq \\widetilde{O}$ and $O = O^\\sharp$ if and only if $O$ is regular.\n\\item Assume that $O$ is not regular. Then the canonical morphisms of $O$--modules\n$$\n\\mathfrak{m} \\stackrel{\\varphi}\\longrightarrow \\Hom_{O}(O^\\sharp, O) \\quad \\mbox{and} \\quad\nO^\\sharp \\stackrel{\\psi}\\longrightarrow \\Hom_{O}(\\mathfrak{m}, O)\n$$\nare isomorphisms.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nFor the first part, see for example \\cite[Proposition 4]{com} or \\cite[Theorem 1.5.13]{deJongPfister}. To show the second part, note that $\\varphi$ assigns\nto an element $a \\in \\mathfrak{m}$ a morphism $O^\\sharp \\stackrel{\\varphi_a}\\longrightarrow O$, where $\\varphi_a(x) = ax$. It is clear that $\\varphi$ is injective. Since $\\Hom_{O}(O^\\sharp, O)$ viewed as a subset of $K$ is a proper ideal in $O$, it is contained in $\\mathfrak{m}$.\nHence, $\\varphi$\n is also surjective, hence bijective.\n\nNext, the canonical morphism $\\Hom_O(\\mathfrak{m}, \\mathfrak{m}) \\stackrel{\\psi}\\longrightarrow \\Hom_O(\\mathfrak{m}, O)$ is injective. On the other hand, there are no surjective morphisms $\\mathfrak{m} \\rightarrow O$ (otherwise, $O$ would be a discrete valuation domain), hence the image of any morphism $\\mathfrak{m} \\rightarrow O$ belongs to $\\mathfrak{m}$ and $\\psi$ is surjective.\n\\end{proof}\n\nFrom now on, let $O$ be an \\emph{excellent} reduced Noetherian ring of Krull dimension one (see for example\n\\cite[Section 8.2]{liu} for the definition and main properties of excellent rings). As before, $K$ denotes its total ring of fractions and\n$\\widetilde{O}$ is the normalization of $O$. Let $X = \\Spec(O)$ and $Z$ be the singular locus of $X$ equipped\nwith the reduced scheme structure. In other words,\n$$Z =\n\\bigl\\{\\mathfrak{m}_1, \\dots, \\mathfrak{m}_t\\bigr\\}\n= \\bigl\\{\\mathfrak{m} \\in \\Spec(O) \\; \\big| \\; O_\\mathfrak{m} \\; \\mbox{is not regular}\\bigr\\}\n$$\n(the condition that $O$ is excellent implies that $Z$ is indeed a finite set).\n\n\\begin{proposition}\\label{P:DetailsKoenigResol} Let\n$\nI = I_Z = \\mathfrak{m}_1 \\cap \\dots \\cap \\mathfrak{m}_t\n$\nbe the vanishing ideal of $Z$ and $O^\\sharp = \\mathop{\\mathrm{End}}\\nolimits_{O}(I)$. Then the following properties are true.\n\\begin{itemize}\n\\item $O^\\sharp \\cong \\left\\{x \\in K \\big| x \\mathfrak{m} \\subset \\mathfrak{m} \\right\\}$. Moreover,\n$O \\subseteq O^\\sharp \\subseteq \\widetilde{O}$ and $O = O^\\sharp$ if and only if $O$ is regular.\n\\item Assume that $O$ is not regular. Then the canonical morphisms of $O$--modules\n$\n\\mathfrak{m} \\stackrel{\\varphi}\\longrightarrow \\Hom_{O}(O^\\sharp, O)$ and\n$O^\\sharp \\stackrel{\\psi}\\longrightarrow \\Hom_{O}(\\mathfrak{m}, O)$\nare isomorphisms.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nFor the first part, see again \\cite[Proposition 4]{com} or \\cite[Theorem 1.5.13]{deJongPfister}. To prove the second, observe that the maps $\\varphi$ and $\\psi$ are well--defined and compatible with localizations with respect to a maximal ideal.\nHence, Proposition \\ref{P:normalizelocal} implies the claim.\n\\end{proof}\n\nWe define a sequence of overrings $O_i$ of the initial ring $O$ by the following recursive procedure:\n\\begin{itemize}\n\\item $O_1 = O$.\n\\item $O_{i+1} = O_i^\\sharp$ for $i \\ge 1$.\n\\end{itemize}\nSince the ring $O$ is excellent, the normalization $\\widetilde{O}$ is finite over $O$, see for example \\cite[Theorem 6.5]{ddone} or \\cite[Section 8.2]{liu}. Hence,\nthere exists $n \\in \\mathbb{N}$ (called the \\emph{level} of $O$) such that we have a finite chain of overrings\n$$\nO_1 \\subset O_2 \\subset \\dots \\subset O_n \\subset O_{n+1}\n$$\nwith $O_1 = O$ and\n$O_{n+1} = \\widetilde{O}$.\n\n\\begin{definition}\\label{D:KoenigsOrder}\nThe ring $A := \\mathop{\\mathrm{End}}\\nolimits_{O}(O_1 \\oplus O_2 \\oplus \\dots \\oplus O_{n+1})^{\\mathrm{op}}$ is called the \\emph{K\\\"onig's order} of $O$.\n\\end{definition}\n\n\n\\begin{proposition}\\label{P:KoenigOrderMatrDescr}\nFor any $1 \\le i, j \\le n+1$ pose $A_{ij} := \\Hom_{O}(O_i, O_j)$. Then the following properties are true.\n\\begin{itemize}\n\\item For $i \\le j$ we have: $A_{ij} \\cong O_j$.\n\\item For $i > j$ we have: $A_{ij} \\cong I_{i,j} := \\Hom_{O_j}(O_i, O_j)$. In particular, $I_{n+1,1} \\cong C := \\Hom_{O}(\\widetilde{O}, O)$\nis the \\emph{conductor ideal}.\n\\item Next, $I_i:= I_{i+1,i}$ is the ideal of the singular locus of $\\Spec(O_i)$ and\nthe ring $\\bar{O}_i := O_i\/I_i$ is semi--simple.\n\\item Moreover, the ideal $I_{n+1, k}$ is projective over $\\mathcal{O}_{n+1}$ for any $1 \\le k \\le n$.\n\\item The ring $A$ admits the following ``matrix description'':\n\\begin{equation}\\label{E:KoenigOrder}\nA \\cong\n\\left(\n\\begin{array}{llccc}\nO_1 & O_2 & \\dots & O_n & O_{n+1} \\\\\nI_1 & O_2 & \\dots & O_n & O_{n+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\nI_{n,1} & I_{n, 2} & \\dots & O_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\end{array}\n\\right)\n\\end{equation}\nand $A \\otimes_O K \\cong \\mathsf{Mat}_{n+1, n+1}(K)$. In other words, $A$ is an \\emph{order} in the semi-simple algebra\n$\\mathsf{Mat}_{n+1, n+1}(K)$.\n\\item For any $2 \\le i \\le n+1$ and $1 \\le j \\le n$ we have inclusions\n\\begin{itemize}\n\\item $I_{i,1} \\subset I_{i,2} \\subset \\dots \\subset I_{i,i-1} \\subset O_i \\subset \\dots \\subset O_{n+1}$\n\\item $I_{n+1,j} \\subset I_{n,j} \\subset \\dots \\subset I_{j+1,j} \\subset O_j$\n\\end{itemize}\ndescribing the ``hierarchy'' between the entries in every row and every column in the matrix description\n(\\ref{E:KoenigOrder}) of the ring $A$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nWe have the following canonical isomorphisms of $O$--modules:\n$$\nO_j \\cong \\Hom_{O_i}(O_i, O_j) \\stackrel{\\cong}\\longrightarrow \\Hom_{O}(O_i, O_j)\n$$\nprovided $i \\le j$ as well as\n$$\nI_{i,j} := \\Hom_{O_j}(O_i, O_j) \\stackrel{\\cong}\\longrightarrow \\Hom_{O}(O_i, O_j)\n$$\nfor $i > j$. Proposition \\ref{P:DetailsKoenigResol} implies that the ideal $I_i = I_ {i+1, i}$ is indeed the ideal of the singular locus of $\\Spec(O_i)$, hence the quotient $\\bar{O}_i = O_i\/I_i$ is semi--simple.\nSince the ring $O_{n+1}$ is regular and the ideal $I_{n+1,k}$ is torsion free as $O_{n+1}$--module, it is projective over $O_{n+1}$.\n\nFinally, for any $1 \\le j \\le n$ and $1 \\le i \\le n+1$ the inclusion $O_j \\subset O_{j+1}$ induces embeddings of $O$--modules\n\\begin{align*}\n \\Hom_{O}(O_{j+1}, O_i)& \\hookrightarrow \\Hom_{O}(O_{j}, O_i)\\\\\n \\intertext{and}\n \\Hom_{O}(O_{i}, O_j)&\\hookrightarrow \\Hom_{O}(O_{i}, O_{j+1}).\n\\end{align*}\n\\vskip-1.5em\n\\end{proof}\n\n\\begin{remark}\nThe idea to study such a ring $A$ is due to K\\\"onig \\cite{koe}, who considered a similar but slightly different construction.\n\\end{remark}\n\n\\smallskip\n\\noindent\nFor any $1 \\le i \\le n+1$ let $e_i = e_{i,i}$ be the $i$-th standard idempotent of $A$ with respect to the presentation (\\ref{E:KoenigOrder}). For $1 \\le k \\le n$ we denote\n\\begin{itemize}\n\\item $\\varepsilon_{k} := \\sum\\limits_{i = k+1}^{n+1} e_i$, $J_k := A \\varepsilon_k A$ and\n$Q_k := A\/J_k$.\n\\item In what follows we write $e = e_{n+1}$, $J = A e A = J_{n+1}$ and $Q := A\/J = Q_n$.\n\\end{itemize}\n\n\\begin{theorem}\\label{T:GlobalDimensionKoenigOrder} The global dimension of $A$ is finite:\n $\\mathrm{gl.dim}(A) \\le 2n$. Moreover, the artinian ring $Q = Q_O$ is quasi--hereditary (hence, its global dimension is finite, too).\n\\end{theorem}\n\n\\begin{proof} A straightforward calculation shows that for every $2 \\le k \\le n+1$ the two--sided ideal $J_{k-1}$ has the following matrix description:\n$$\nJ_{k-1} =\n\\left(\n\\begin{array}{llclllcc}\nI_{k,1} & I_{k,2} & \\dots &I_{k,k-1} & O_k & O_{k+1} & \\dots & O_{n+1} \\\\\n \\vdots& \\vdots & \\vdots &\\vdots & \\vdots& \\vdots & \\vdots& \\vdots\\\\\nI_{k,1} & I_{k,2} & \\dots &I_{k,k-1} & O_k & O_{k+1} & \\dots & O_{n+1} \\\\\nI_{k+1,1} & I_{k+1,2} & \\dots &I_{k+1,k-1} & I_{k+1,k} & O_{k+1} & \\dots & O_{n+1} \\\\\n \\vdots& \\vdots & \\vdots &\\vdots & \\vdots& \\vdots & \\vdots& \\vdots\\\\\n I_{n+1,1} & I_{n+1,2} & \\dots &I_{n+1,k-1} & I_{n+1,k} & I_{n+1,k+1} & \\dots & O_{n+1} \\\\\n\\end{array}\n\\right).\n$$\nIn other words, the $i$-th row of $J_{k-1}$ is the same as for $A$ provided $k \\le i \\le n+1$ and\nin the case $1 \\le i \\le k-1$ the $i$-th and the $k$-th rows of $J_{k-1}$ are the same. In particular, the ideal\n$J = J_n$ has the shape\n\\begin{equation*}\nJ =\n\\left(\n\\begin{array}{llccc}\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\end{array}\n\\right).\n\\end{equation*}\nConsider the projective left $A$--module $P := Ae$. Then we have an adjoint pair\n$$\n\\xymatrix{A-\\mathsf{mod} \\ar@\/^2ex\/[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} & & \\widetilde{O}-\\mathsf{mod} \\ar@\/^2ex\/[ll]|{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}\n}\n$$\n where\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(P, \\,-\\,)$ and $\\tilde\\mathsf F} \\def\\sS{\\mathsf S = P \\otimes_{\\widetilde{O}} \\,-\\,$. The functor $\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ is exact and has the following explicit description: if $M$ is an $\\widetilde{O}$--module then\n$$\n\\tilde\\mathsf F} \\def\\sS{\\mathsf S(M) = M^{\\oplus{(n+1)}} =\n\\left(\n\\begin{array}{c}\nM \\\\\nM\\\\\n\\vdots \\\\\nM\n\\end{array}\n\\right)\n$$\nwhere the left $A$--action on $M^{\\oplus{(n+1)}}$ is given by the matrix multiplication. Since for every\n$1 \\le k \\le n$ the $O$--module $I_{n+1, k}$ is also a projective $\\widetilde{O}$--module, we see that the left $A$--module $J e_k$\nbelongs to the essential image of $\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and is projective over $A$. It is clear that all right $A$--modules\n$e_k J$ are projective, too. Since $P$ is free over $\\widetilde{O} = \\mathop{\\mathrm{End}}\\nolimits_{A}(P)$, \\cite[Lemma 4.9]{bdg} implies that\n$\n\\mathrm{gl.dim}(A) \\le \\mathrm{gl.dim}(Q) + 2.\n$\n\n\\noindent\nNext, observe that for every $1 \\le k \\le n$ the ring $Q_k$ has the following ``matrix description'':\n$$\nQ_k \\cong\n\\left(\n\\begin{array}{llcc}\n\\dfrac{O_1}{I_{k+1,1}} & \\dfrac{O_2}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\dfrac{I_{2,1}}{I_{k+1,1}} & \\dfrac{O_2}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\end{array}\n\\right),\n$$\nwhere $\\dfrac{{O}_k}{I_k} =: \\bar{O}_k$ is semi--simple.\nFor $1 \\le k \\le n$ let $\\bar{e}_k$ be the image\nof the idempotent $e_k \\in A$ in the ring $Q_k = A\/J_k$. Observe that for $2 \\le k \\le n$\n$$L_k := J_{k-1}\/J_k = Q_k \\bar{e}_k Q_k =\nQ_k \\cong\n\\left(\n\\begin{array}{llcc}\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\end{array}\n\\right)\n\\subset Q_k$$\nis projective viewed both as a left and as a right\n$Q_k$--module (via the same argument as for $J$ and $A$). Moreover, $Q_k\/L_k \\cong Q_{k-1}$ and\n$\\bar{e}_k Q_k \\bar{e}_k = \\bar{O}_k$ is semi--simple.\nTherefore, $J_1\/J \\subset J_2\/J \\subset \\dots \\subset J_n\/J$ is a \\emph{heredity chain} in $Q$ and the ring\n$Q$ is \\emph{quasi--hereditary}, see \\cite{CPS,dr} or the appendix of Dlab in \\cite{dk} for the definition and main properties of quasi--hereditary rings. It is well--known that\n$\n\\mathrm{gl.dim}(Q) \\le 2(n-1),\n$\nsee \\cite[Statement 9]{dr}, \\cite[Theorem A.3.4]{dk} (or \\cite[Lemma 4.9]{bdg} for a short proof). The theorem is proven.\n\\end{proof}\n\n\\begin{remark}\nThe bound on the global dimension of $A$ given in Theorem \\ref{T:GlobalDimensionKoenigOrder} is not optimal.\nFor example, if $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{m(n)})$ with $m(n) = 2n$ (respectively $2n+1$) is a simple singularity of type\n$A_{m(n)-1}$, then the level of $O$ is $n$. On the other hand, $O_1 \\oplus \\dots \\oplus O_{n+1}$ is the additive generator of the category of maximal Cohen--Macaulay modules, see \\cite[Section 7]{Bass}, \\cite[Section 5]{lw} or\n\\cite[Section 9]{Yoshino}. Hence, by a result of Auslander and Roggenkamp \\cite{ar}, the global dimension of $A$ is equal to two.\n\n\\smallskip\n\\noindent\nIn the particular cases $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{2})$ (simple node) and $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{3})$ (simple cusp) the K\\\"onig's order $A$ coincides with the Auslander's order\n$\n\\left(\n\\begin{array}{cc}\nO & \\widetilde{O} \\\\\nC & \\widetilde{O}\n\\end{array}\n\\right)\n$\nintroduced in the work \\cite{bd}.\n\\end{remark}\n\n\\begin{remark}\nBasic properties of excellent rings (see \\cite[Section 6]{ddone} or \\cite[Section 8.2]{liu}) imply that\n$$Q_O := Q \\cong Q_{\\widehat{O}_1} \\times \\dots \\times Q_{\\widehat{O}_t},$$ where $\\widehat{O}_i$ is the completion of the local ring\n$O_{\\mathfrak{m}_i}$ for each $\\mathfrak{m}_i \\in \\mathrm{Sing}(O)$. In other words, the quasi--hereditary ring $Q$ depends only on\nthe \\emph{local singularity types} of $\\Spec(O)$.\n\\end{remark}\n\n\\section{K\\\"onig's order as a categorical resolution of singularities}\nFor a (left) Noetherian ring $B$ we denote by $B-\\mathsf{mod}$ the category of all finitely generated left $B$--modules and by $B-\\mathsf{Mod}$ the category of all left $B$--modules.\nAs in the previous section, let $O$ be an excellent reduced Noetherian ring of Krull dimension one and level $n$, $\\widetilde{O}$ be its normalization, $A$ be the K\\\"onig's order of $O$ and $Q$ be the quasi--hereditary artinian algebra attached to $O$. Let $e = e_{n+1}$ and $f = e_1$ be two standard idempotents of $A$, $P = Ae$, $T = Af$ and\n$J = AeA$.\nIt is clear that $\\widetilde{O} \\cong \\mathop{\\mathrm{End}}\\nolimits_A(P)$ and $O \\cong \\mathop{\\mathrm{End}}\\nolimits_A(T)$.\nWe also denote $T^\\vee := \\Hom_A(T, A) \\cong f A$ and $P^\\vee := \\Hom_A(P, A) \\cong e A$.\nThen we have the following diagram of categories and functors:\n\\begin{equation}\n\\xymatrix{O-\\mathsf{mod} \\ar@\/^2ex\/[rr]^{\\,\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\mathsf H} \\def\\sU{\\mathsf U\\,} && A-\\mathsf{mod} \\ar[ll]|{\\,\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && \\widetilde{O}-\\mathsf{mod} \\ar@\/^2ex\/[ll]^{\\,\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nwhere $\\mathsf F} \\def\\sS{\\mathsf S = T \\otimes_O \\,-\\,$, $\\mathsf H} \\def\\sU{\\mathsf U = \\Hom_O(T^\\vee, \\,-\\,)$, $\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(T, \\,-\\,)$ and similarly,\n$\\tilde\\mathsf F} \\def\\sS{\\mathsf S = P \\otimes_{\\widetilde{O}} \\,-\\,$, $\\tilde\\mathsf H} \\def\\sU{\\mathsf U = \\Hom_{\\widetilde{O}}(P^\\vee, \\,-\\,)$,\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(P, \\,-\\,)$. There is the same diagram for the categories of all modules\n$O-\\mathsf{Mod}$, $\\widetilde{O}-\\mathsf{Mod}$ and $A-\\mathsf{Mod}$. The following results are standard, see for example \\cite[Theorem 4.3]{bdg} and references therein.\n\n\\begin{theorem}\nThe pairs of functors $(\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf H} \\def\\sU{\\mathsf U)$ (and respectively $(\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf H} \\def\\sU{\\mathsf U)$) are adjoint and the functors $\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf H} \\def\\sU{\\mathsf U, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful. Both categories\n$O-\\mathsf{mod}$ and $\\widetilde{O}-\\mathsf{mod}$ are Serre quotients of $A-\\mathsf{mod}$:\n$$\nO-\\mathsf{mod} \\cong A-\\mathsf{mod}\/\\mathsf{Ker}(\\mathsf G} \\def\\sT{\\mathsf T) \\quad \\mbox{and} \\quad\n\\widetilde{O}-\\mathsf{mod} \\cong A-\\mathsf{mod}\/\\mathsf{Ker}(\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$$\nMoreover,\n$\n\\mathsf{Ker}(\\tilde\\mathsf G} \\def\\sT{\\mathsf T) = Q-\\mathsf{mod}.\n$\n\\end{theorem}\nThe described picture becomes even better when we pass to (unbounded) derived categories. Observe that\nthe functors $\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are exact. Their derived functors will be denoted\nby $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ respectively, whereas $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ is the left derived functor of $\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ is the right derived functor of $\\mathsf H} \\def\\sU{\\mathsf U$.\n\n\n\\begin{theorem} We have a diagram of categories and functors\n\\begin{equation}\\label{E:descinglobale}\n\\xymatrix{D(O-\\mathsf{Mod}) \\ar@\/^2ex\/[rr]^{\\,\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\sR\\mathsf H} \\def\\sU{\\mathsf U\\,} && D(A-\\mathsf{Mod}) \\ar[ll]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && D(\\widetilde{O}-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nsatisfying the following properties.\n\\begin{itemize}\n\\item The following pairs of functors $(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\sR\\mathsf H} \\def\\sU{\\mathsf U)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)$ and\n$(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs.\n\\item The functors $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$, $\\sR\\mathsf H} \\def\\sU{\\mathsf U$, $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item Both derived categories $D(O-\\mathsf{Mod})$ and $D(\\widetilde{O}-\\mathsf{Mod})$ are Verdier localizations of\n$D(A-\\mathsf{Mod})$:\n\\begin{itemize}\n \\item $\nD(O-\\mathsf{Mod}) \\cong D(A-\\mathsf{Mod})\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$.\n\\item\n$D(\\widetilde{O}-\\mathsf{Mod}) \\cong D(A-\\mathsf{Mod})\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$\n\\end{itemize}\n\\item Moreover,\n$\n\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T) = D_Q(A-\\mathsf{Mod}) \\cong D(Q-\\mathsf{Mod}).\n$\n\\item\nThe derived category $D(A-\\mathsf{Mod})$ is a categorical resolution of singularities of $X = \\Spec(O)$ in the sense of Kuznetsov \\cite[Definition 3.2]{ku}.\n \\item If $O$ is Gorenstein, then the restrictions of $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ on $\\mathop{\\mathsf{Perf}}\\nolimits(O)$ are isomorphic. Hence, the constructed categorical resolution is even \\emph{weakly crepant} in the sense of\n \\cite[Definition 3.4]{ku}.\n\\end{itemize}\nWe have a \\emph{recollement diagram}\n\\begin{equation}\\label{E:recollement}\n\\xymatrix{D(Q-\\mathsf{Mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D(A-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D(\\widetilde{O}-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nand all functors can be restricted on the bounded derived categories $D^b(Q-\\mathsf{mod})$, $D^b(A-\\mathsf{mod})$\nand $D^b(\\widetilde{O}-\\mathsf{mod})$. In particular, we have \\emph{two} semi--orthogonal decompositions\n$$\nD(A-\\mathsf{Mod}) = \\bigl\\langle \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\n\\bigl\\langle \\mathsf{Im}(\\sR\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)\\bigr\\rangle.\n$$\nThe same result is true when we pass to the bounded derived categories.\n\\end{theorem}\n\n\\noindent\n\\emph{Comment on the proof}. The study of various derived functors related with a pair $(B, \\epsilon)$, where $B$ is a ring and $\\epsilon \\in B$ an idempotent (in particular, the recollement diagram (\\ref{E:recollement})) are due to Cline, Parshall and Scott \\cite[Section 2]{CPS}. We also refer to \\cite[Section 4]{bdg} (and references therein) for an exposition focussed on non--commutative resolutions of singularities.\nThe weak crepancy of the categorical resolution $D(A-\\mathsf{Mod})$ of $\\Spec(O)$ follows from \\cite[Theorem 5.10]{bdg}. In particular, the constructed categorical resolution of singularities\nfits into the setting of non--commutative crepant resolutions initiated by van den Bergh in \\cite{vdb}. \\qed\n\n\\section{Survey on the derived stratification of an artinian quasi--hereditary ring}\nThe derived category $D^b(Q-\\mathsf{mod})$ of the quasi--hereditary ring $Q$ introduced in Theorem \\ref{T:GlobalDimensionKoenigOrder} can be further stratified in a usual way \\cite{CPS}, which we briefly describe now adapting the notation for further applications. All details can be found in \\cite{CPS}, \\cite[Appendix]{dr} and \\cite{bdg}.\n\n\n\\medskip\n\\noindent\n1.~Recall that we had started with a reduced excellent Noetherian ring $O$ of Krull dimension one, attaching to it\na certain order $A$. Then we have constructed a heredity chain $J_n \\subset J_{n-1} \\subset \\dots \\subset\nJ_1 \\subset A$ of two--sided ideals and posed $Q_k := A\/J_k$ for $1 \\le k \\le n$. In this notation,\n$Q = Q_n$ is an artinian quasi--hereditary ring we shall study in this section and $Q_1 = \\bar{O}$ is a semi-simple ring (supported on the singular locus of $\\Spec(O)$).\n\n\n\\medskip\n\\noindent\n2.~For any $1 \\le k \\le n$, let\n$\\bar{e}_k$ be the image of the standard idempotent $e_k \\in A$ in $Q_k = A\/J_k$. Then $Q_k\/(Q_k \\bar{e}_k Q_k) \\cong Q_{k-1}$ for all $2 \\le k \\le n$.\n\n\\smallskip\n\\noindent\nLet\n$P_k = Q_k \\bar{e}_k$ be the projective left $Q_k$--module and\n$P_k^\\vee = \\Hom_{Q_k}(P_k, Q_k) = e_k Q_k$ be the projective right $Q_k$--module, corresponding to\nthe idempotent $\\bar{e}_k$. Then we have: $\\mathop{\\mathrm{End}}\\nolimits_{Q_k}(P_k) \\cong \\bar{O}_k$.\nThe functor\n$$\n\\mathsf G} \\def\\sT{\\mathsf T_k = \\Hom_{Q_k}(P_k,\\,-\\,): Q_k-\\mathsf{mod} \\longrightarrow \\bar{O}_k-\\mathsf{mod}\n$$\nis a bilocalization functor: the functors $\\mathsf F} \\def\\sS{\\mathsf S_k = P_k \\otimes_{\\bar{O}_k} \\,-\\,$ and $\\mathsf H} \\def\\sU{\\mathsf U_k = \\Hom_{Q_k}(P_k^\\vee, \\,-\\,)$ are respectively the left and the right adjoints of $\\mathsf G} \\def\\sT{\\mathsf T_k$. Both $\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf H} \\def\\sU{\\mathsf U_k$ are fully faithful. Since\nthe ring $\\bar{O}_k$ is semi--simple, $\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf H} \\def\\sU{\\mathsf U_k$ are also exact. The kernel of $\\mathsf G} \\def\\sT{\\mathsf T_k$ is the category of $Q_{k-1}$--modules.\n\n\n\n\n\\medskip\n\\noindent\n3.~Most remarkably, for any $2 \\le k \\le n$ we have a recollement diagram\n\\begin{equation*}\n\\xymatrix{D^b(Q_{k-1}-\\mathsf{mod}) \\ar[rr]|-{\\,\\mathsf J} \\def\\sW{\\mathsf W_{k}\\,} && D^b(Q_k-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^-{\\,\\mathsf J} \\def\\sW{\\mathsf W_k^*\\,} \\ar@\/_2ex\/[ll]_-{\\,\\mathsf J} \\def\\sW{\\mathsf W_k^{!}\\,} \\ar[rr]|-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T_k\\,}\n && D^b(\\bar{O}_k-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k\\,} \\ar@\/_2ex\/[ll]_-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k\\,}}\n\\end{equation*}\nThis claim in particular includes the following statements.\n\\begin{itemize}\n\\item The functor $\\mathsf J} \\def\\sW{\\mathsf W_k$ (induced by the ring homomorphism $Q_k \\longrightarrow Q_{k-1}$) is fully faithful.\nThe essential image of $\\mathsf J} \\def\\sW{\\mathsf W_k$ coincides with the kernel of $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T_k$ and\n$D^b(Q_k-\\mathsf{mod})\/\\mathsf{Im}(\\mathsf J} \\def\\sW{\\mathsf W_k) \\cong D^b(\\bar{O}_k-\\mathsf{mod})$.\n\\item The functors $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k$ are fully faithful.\n\\end{itemize}\n\n\n\\medskip\n\\noindent\n4.~For all $1 \\le k \\le n$ we have:\n\\begin{itemize}\n\\item $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k(\\bar{O}_k) \\cong \\mathsf F} \\def\\sS{\\mathsf S_k(\\bar{O}_k) \\cong P_k$.\n\\item $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k(\\bar{O}_k) \\cong \\mathsf H} \\def\\sU{\\mathsf U_k(\\bar{O}_k) = \\Hom_{\\bar{O}_k}(P_k^\\vee, \\bar{O}_k) := E_k$ is the injective left $Q_k$--module corresponding to the idempotent $\\bar{e}_k$.\n\\end{itemize}\nThe functor $\\mathsf I} \\def\\sV{\\mathsf V_k: D^b(Q_k-\\mathsf{mod}) \\longrightarrow D^b(Q-\\mathsf{mod})$ induced by the ring epimorphism\n$Q \\longrightarrow Q_k$ is fully faithful. In fact, it admits a factorization $\\mathsf I} \\def\\sV{\\mathsf V_k = \\mathsf J} \\def\\sW{\\mathsf W_n \\dots \\mathsf J} \\def\\sW{\\mathsf W_{k+1}$.\nThe $Q$--module $\\Delta_k := \\mathsf I} \\def\\sV{\\mathsf V_k(P_k)$ (respectively $\\nabla_k := \\mathsf I} \\def\\sV{\\mathsf V_k(E_k)$) is called $k$-th \\emph{standard} (respectively \\emph{costandard}) $Q$--module.\n\n\\medskip\n\\noindent\n5.~The standard and costandard modules have in particular the following properties:\n$$\n\\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\Delta_i, \\Delta_j) = 0 = \\mathop{\\mathrm{Ext}}\\nolimits_Q(\\nabla_j, \\nabla_i) \\; \\mbox{for all} \\; 1 \\le i < j \\le n \\; \\mbox{and} \\; p \\ge 0\n$$\nand\n$$\n\\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\Delta_k, \\Delta_k) = 0 = \\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\nabla_k, \\nabla_k) \\; \\mbox{for all} \\; 1 \\le k \\le n \\; \\mbox{and} \\; p \\ge 1.\n$$\nMoreover, $\\mathop{\\mathrm{End}}\\nolimits_Q(\\Delta_k) \\cong \\mathop{\\mathrm{End}}\\nolimits_Q(\\nabla_k) \\cong \\bar{O}_k$ is semi--simple. The derived category\n$D^b(Q-\\mathsf{mod})$ admits two canonical semi--orthogonal decompositions:\n$$\n\\langle D_1, \\dots, D_n\\rangle = D^b(Q-\\mathsf{mod}) = \\langle D'_n, \\dots, D'_1\\rangle,\n$$\nwhere $D_k$ (respectively $D'_k$) is the triangulated subcategory of $D^b(Q-\\mathsf{mod})$ generated by the object\n$\\Delta_k$ (respectively $\\nabla_k)$. Note that we have the following equivalences of categories:\n$\nD_k \\cong D^b(\\bar{O}_k-\\mathsf{mod}) \\cong D_k'.\n$\n\n\\medskip\n\\noindent\n6.~The stratification of $D^b(Q-\\mathsf{mod})$ by the derived categories $D^b(\\bar{O}_k-\\mathsf{mod})$ can be summarized by the following diagram of categories and functors:\n$$\n\\xymatrix{\nD^b(\\bar{O}_1-\\mathsf{mod}) \\ar[d]_= & D^b(\\bar{O}_2-\\mathsf{mod}) \\ar[d]_{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_2} & & D^b(\\bar{O}_n-\\mathsf{mod}) \\ar[d]^{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_n} \\\\\nD^b(Q_1-\\mathsf{mod}) \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_2} & D^b(Q_2-\\mathsf{mod}) \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_3} & \\dots \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_{n}} & D^b(Q_n-\\mathsf{mod})\\\\\nD^b(\\bar{O}_1-\\mathsf{mod}) \\ar[u]^= & D^b(\\bar{O}_2-\\mathsf{mod}) \\ar[u]^{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_2} & & D^b(\\bar{O}_n-\\mathsf{mod}) \\ar[u]_{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_n}\n}\n$$\n\\section{Derived stratification and curve singularities}\nRecall that we also have the following recollement diagram\n\\begin{equation*}\n\\xymatrix{D^b(Q-\\mathsf{mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D^b(A-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D^b(\\widetilde{O}-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation*}\nwhere $\\mathsf I} \\def\\sV{\\mathsf V$ is induced by the ring epimorphism $A \\rightarrow Q$. Abusing the notation, we shall write\n$\\Delta_k = \\mathsf I} \\def\\sV{\\mathsf V(\\Delta_k)$ for all $1 \\le k \\le n$. This implies the following result.\n\n\\begin{theorem}\nThe derived category $D^b(A-\\mathsf{mod})$ admits two semi--orthogonal decompositions\n$\n\\bigl\\langle\\mathsf{Im}(\\mathsf I} \\def\\sV{\\mathsf V), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigl\\rangle = D^b(A-\\mathsf{mod}) =\n\\bigl\\langle\\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Im}(\\mathsf I} \\def\\sV{\\mathsf V)\\bigl\\rangle.\n$\n\\end{theorem}\n\n\\noindent\nNext, recall that we have a bilocalization functor $$\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T: D^b(A-\\mathsf{mod}) \\longrightarrow D^b(O-\\mathsf{mod}).$$\n\\begin{lemma}\\label{L:calcullocale}\nFor any $1 \\le k \\le n$ we have: $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\bar{O}_k$. Moreover, $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(Q) \\cong\nO_1\/C_1 \\oplus \\dots \\oplus O_n\/C_n$, where $C_k := I_{n+1,k} = \\Hom_O\\bigl(O_{n+1}, O_k\\bigr)$.\n\\end{lemma}\n\n\\begin{proof} The first result follows from the following chain of isomorphisms:\n$$\n\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) = \\Hom_A(Af, \\Delta_k) \\cong f\\cdot \\Delta_k \\cong \\bar{O}_k.\n$$\nThe proof of the second statement is analogous.\n\\end{proof}\n\n\\section{K\\\"onig's resolution in the projective setting}\\label{S:ProjectiveSetting}\n\n\\smallskip\n\\noindent\nLet $X$ be a reduced projective curve over some base field $\\mathbbm{k}$. In this section we shall explain the construction\nof K\\\"onig's sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$, ``globalizing'' the arguments of Section \\ref{Sec:LocalStory}.\n\\begin{itemize}\n\\item Let $\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ be the normalization of $X$ and $Z$ be the singular locus of $X$\n(equipped with the reduced scheme structure).\n\\item In what follows, $\\mathcal{O} = \\mathcal{O}_X$ is the structure sheaf of $X$, $\\mathcal K} \\def\\kX{\\mathcal X$ is the sheaf of rational functions on $X$, $\\widetilde{\\mathcal{O}}: = \\nu_*(\\mathcal{O}_{\\widetilde{X}})$\nand $\\mathcal I} \\def\\kV{\\mathcal V$ is the ideal sheaf of the singular locus $Z$.\n\\item We consider the sheaf of rings $\\mathcal{O}^\\sharp := \\mbox{\\textit{End}}_X(\\mathcal I} \\def\\kV{\\mathcal V)$ on the curve $X$.\n\\end{itemize}\nThe next result follows from the corresponding affine version (Proposition \\ref{P:DetailsKoenigResol}).\n\\begin{proposition}\nWe have inclusions of sheaves $\\mathcal{O} \\subseteq \\mathcal{O}^\\sharp \\subseteq \\widetilde{O}$, and $\\mathcal{O} = \\mathcal{O}^\\sharp$ if and only if $X$ is smooth. Moreover, there are isomorphisms of $\\mathcal{O}$--modules\n$\n\\mathcal I} \\def\\kV{\\mathcal V \\cong \\mbox{\\textit{Hom}}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O})$ and $\\widetilde{\\mathcal{O}} \\cong \\mbox{\\textit{Hom}}_X(\\mathcal I} \\def\\kV{\\mathcal V, \\mathcal{O})$.\n\\end{proposition}\n\n\\noindent\nNow we define a sequence of sheaves of rings $\\mathcal{O} \\subset \\mathcal{O}_k \\subset \\widetilde{\\mathcal{O}}$ by the following recursive procedure.\n\\begin{itemize}\n\\item First we pose: $\\mathcal{O}_1 := \\mathcal{O}$.\n\\item Assume that the sheaf of rings $\\mathcal{O}_k$ has been constructed. Then it defines a projective curve $X_k$ together with a finite birational morphism $\\nu_k: X_k \\longrightarrow X$ (partial normalization of $X$) such that $\\mathcal{O}_k = \\bigl(\\nu_k\\bigr)_*(\\mathcal{O}_{X_k})$.\n\\item Let $Z_k$ be the singular locus of the curve $X_k$ (as usual, with respect to the reduced scheme structure).\nThen we write\n$$\n\\mathcal{O}_{k+1} := \\mathcal{O}_k^\\sharp \\cong (\\nu_k)_*\\bigl(\\mbox{\\textit{End}}_{X_k}(\\mathcal I} \\def\\kV{\\mathcal V_{Z_k})\\bigr).\n$$\n\\end{itemize}\nThen there exists a natural number $n$ (called the \\emph{level} of $X$) such that we have a finite chain of sheaves of rings\n$$\n\\mathcal{O} = \\mathcal{O}_1 \\subset \\mathcal{O}_2 \\subset \\dots \\subset \\mathcal{O}_n \\subset \\mathcal{O}_{n+1} = \\widetilde{\\mathcal{O}}.\n$$\nObviously, the level of $X$ is the maximum of the levels of local rings $\\widehat{\\mathcal{O}}_x$,\nwhere $x$ runs through the set of singular points of $X$.\n\n\\begin{definition}\nThe sheaf of rings\n$\\mathcal A} \\def\\kN{\\mathcal N := \\mbox{\\textit{End}}_X\\bigl(\\mathcal{O}_1 \\oplus \\dots \\oplus \\mathcal{O}_{n+1}\\bigr)$ is called the \\emph{K\\\"onig's sheaf of orders} on $X$.\n\\end{definition}\n\n\\noindent\nIn what follows, we study the ringed space $\\mathbb{X} = (X, \\mathcal A} \\def\\kN{\\mathcal N)$. We denote by $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$ (respectively\n$\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})$) the category of coherent (respectively quasi--coherent) sheaves of $\\mathcal A} \\def\\kN{\\mathcal N$--modules on the curve $X$.\n\n\\begin{theorem} The sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$ admits the following description:\n\\begin{equation}\\label{E:KoenigOrderSheaf}\n\\mathcal A} \\def\\kN{\\mathcal N \\cong\n\\left(\n\\begin{array}{llccc}\n\\mathcal{O}_1 & \\mathcal{O}_2 & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_1 & \\mathcal{O}_2 & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_{n,1} & \\mathcal I} \\def\\kV{\\mathcal V_{n, 2} & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_{n+1,1} & \\mathcal I} \\def\\kV{\\mathcal V_{n+1, 2} & \\dots & \\mathcal I} \\def\\kV{\\mathcal V_n & \\mathcal{O}_{n+1} \\\\\n\\end{array}\n\\right) \\subset \\mbox{\\textit{Mat}}_{n+1, n+1}(\\mathcal K} \\def\\kX{\\mathcal X),\n\\end{equation}\nwhere $\\mathcal I} \\def\\kV{\\mathcal V_{i, j} := \\mbox{\\textit{Hom}}_{X}(\\mathcal{O}_i, \\mathcal{O}_j)$ for all $1 \\le j < i\\le n+1$ and $\\mathcal I} \\def\\kV{\\mathcal V_k = \\mathcal I} \\def\\kV{\\mathcal V_{k+1, k}$ for $1 \\le k \\le n$. Moreover, $\\mathcal A} \\def\\kN{\\mathcal N \\otimes_\\mathcal{O} \\mathcal K} \\def\\kX{\\mathcal X \\cong \\mbox{\\textit{Mat}}_{n+1, n+1}(\\mathcal K} \\def\\kX{\\mathcal X)$. Next, we have:\n$$\n\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) \\le 2n,\n$$\nwhere $n$ is the level of $X$.\n\\end{theorem}\n\n\\begin{proof}\nThe result follows from the corresponding local statements in Proposition \\ref{P:KoenigOrderMatrDescr} and\nTheorem \\ref{T:GlobalDimensionKoenigOrder} and the fact that\n$$\n\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) = \\max\\bigl\\{\\mathrm{gl.dim}(\\widehat{\\mathcal A} \\def\\kN{\\mathcal N}_x) \\, | \\, x \\in X_{\\mathrm{cl}}\\bigr\\},\n$$\nsee for instance \\cite[Corollary 5.5]{bdg}.\n\\end{proof}\n\n\nFor any $1 \\le i \\le n+1$, let $e_i \\in \\Gamma(X, \\mathcal A} \\def\\kN{\\mathcal N)$ be the $i$-th standard idempotent with respect to the matrix\npresentation (\\ref{E:KoenigOrderSheaf}). As in the affine case, we use the following notation.\n\\begin{itemize}\n\\item We write $e = e_{n+1}$ and $f = e_1$. Let $\\kP := \\mathcal A} \\def\\kN{\\mathcal N e$ and $\\mathcal{T} := \\mathcal A} \\def\\kN{\\mathcal N f$ be the corresponding locally projective left $\\mathcal A} \\def\\kN{\\mathcal N$--modules. Then we have the following isomorphisms of sheaves of $\\mathcal{O}$--algebras:\n\\begin{equation}\\label{E:keysheafisom}\n\\mathcal{O} \\cong \\mbox{\\textit{End}}_{\\mathbb{X}}(\\mathcal{T}) := \\mbox{\\textit{End}}_{\\mathcal A} \\def\\kN{\\mathcal N}(\\mathcal{T}) \\ \\mbox{ and } \\ \\widetilde{\\mathcal{O}} \\cong \\mbox{\\textit{End}}_{\\mathbb{X}}(\\kP) :=\n \\mbox{\\textit{End}}_{\\mathcal A} \\def\\kN{\\mathcal N}(\\kP).\n\\end{equation}\nWe shall also use the notation\n$$\n\\kP^\\vee := \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kP, \\mathcal A} \\def\\kN{\\mathcal N) \\cong e\\mathcal A} \\def\\kN{\\mathcal N \\ \\mbox{ and } \\ \\mathcal{T}^\\vee := \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal{T}, \\mathcal A} \\def\\kN{\\mathcal N) \\cong f\\mathcal A} \\def\\kN{\\mathcal N.\n$$\n\\item For any $1 \\le k \\le n$ we set\n$$\\varepsilon_k := \\sum\\limits_{i = k+1}^{n+1} e_i \\in \\Gamma(X, \\mathcal A} \\def\\kN{\\mathcal N).\n$$\nThen $\\mathcal J} \\def\\kW{\\mathcal W_k:= \\mathcal A} \\def\\kN{\\mathcal N \\varepsilon_k\\mathcal A} \\def\\kN{\\mathcal N$ denotes the corresponding sheaf of two--sided ideals in $\\mathcal A} \\def\\kN{\\mathcal N$.\n\\item The sheaves of $\\mathcal{O}$--algebras $\\kQ_k := \\mathcal A} \\def\\kN{\\mathcal N\/\\mathcal J} \\def\\kW{\\mathcal W_k$ are supported on the finite set $Z$ for all $1 \\le k \\le n$. In what follows, we shall identify them with the corresponding finite dimensional\n $\\mathbbm{k}$--algebras of global sections\n $\n Q_k := \\Gamma(X, \\kQ_k),\n $\nwhich have been shown to be quasi--hereditary, see Theorem \\ref{T:GlobalDimensionKoenigOrder}. As before, we shall write $\\mathcal J} \\def\\kW{\\mathcal W = \\mathcal J} \\def\\kW{\\mathcal W_n$ and $Q = Q_n$.\n\\item In a similar way, the torsion sheaf $\\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k$ will be identified with the corresponding ring of global sections\n$\n\\bar{O}_k:= \\Gamma\\bigl(X, \\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k),\n$\nwhich is a semi--simple finite dimensional $\\mathbbm{k}$--algebra, isomorphic to the ring of functions of the singular locus $Z_k$ of the partial normalization $X_k$ of our original curve $X$.\n\\end{itemize}\n\n\\begin{proposition}\\label{P:recollementSheaves} Consider the following diagram of categories and functors\n\\begin{equation}\n\\xymatrix{\\mathop{\\mathsf{Coh}}\\nolimits(X) \\ar@\/^2ex\/[rr]^{\\,\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\mathsf H} \\def\\sU{\\mathsf U\\,} && \\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X}) \\ar[ll]|{\\,\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && \\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X}) \\ar@\/^2ex\/[ll]^{\\,\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nwhere $\\mathsf F} \\def\\sS{\\mathsf S = \\mathcal{T} \\otimes_O \\,-\\,$, $\\mathsf H} \\def\\sU{\\mathsf U = \\mbox{\\textit{Hom}}_X(\\mathcal{T}^\\vee, \\,-\\,)$, $\\mathsf G} \\def\\sT{\\mathsf T = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal{T}, \\,-\\,)$ and similarly,\n$\\tilde\\mathsf F} \\def\\sS{\\mathsf S = \\kP \\otimes_{\\widetilde{O}} \\,-\\,$, $\\tilde\\mathsf H} \\def\\sU{\\mathsf U = \\mbox{\\textit{Hom}}_{\\widetilde{X}}(P^\\vee, \\,-\\,)$,\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kP, \\,-\\,)$. Here we identify (using the functor $\\nu_*$) the category $\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})$ with the category of coherent $\\widetilde{O}$--modules on the curve $X$. Then the following results are true.\n\\begin{itemize}\n\\item The pairs of functors $(\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T), (\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf H} \\def\\sU{\\mathsf U)$ and $(\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T), (\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs. The functors $\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf H} \\def\\sU{\\mathsf U, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item The functors $\\mathsf G} \\def\\sT{\\mathsf T$ and $\\tilde\\mathsf G} \\def\\sT{\\mathsf T$ are bilocalization functors. Moreover, $\\mathsf{Ker}(\\mathsf F} \\def\\sS{\\mathsf S) \\cong\nQ-\\mathsf{mod}$.\n\\item The pairs of functors $(\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ and $(\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf H} \\def\\sU{\\mathsf U)$ between\n$\\mathop{\\mathsf{Coh}}\\nolimits(X)$ and $\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})$ form adjoint pairs, too. Moreover, these functors admit the following ``purely commutative'' descriptions:\n$$\n\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\nu_*,\\ \\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf H} \\def\\sU{\\mathsf U \\simeq \\nu^!,\\ \\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\widetilde\\mathcal{O}} \\nu^*(\\,-\\,)\n\\ \\mbox{\\emph{ and }} \\ \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U \\simeq \\nu_*(\\mathcal C} \\def\\kP{\\mathcal P^\\vee \\otimes_{\\widetilde{\\mathcal{O}}} \\,-\\,),\n$$\nwhere $\\mathcal C} \\def\\kP{\\mathcal P := \\mbox{\\textit{Hom}}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O}) = \\mathcal I} \\def\\kV{\\mathcal V_{n+1,1}$ is the conductor ideal sheaf.\n\\end{itemize}\nThe same results are true when we replace each category of coherent sheaves by the corresponding category of quasi--coherent sheaves.\n\\end{proposition}\n\n\\begin{proof}\nThe proofs of the first two parts follow from standard local computations. Let $\\mathcal F} \\def\\kS{\\mathcal S$ be a coherent $\\mathcal{O}$--module and\n$\\mathcal G} \\def\\kT{\\mathcal T$ a coherent $\\widetilde{O}$--module (identified with the corresponding coherent sheaf on $\\widetilde{X}$).\nThen we have:\n$$\n\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf F} \\def\\sS{\\mathsf S(\\mathcal F} \\def\\kS{\\mathcal S) = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal A} \\def\\kN{\\mathcal N e, \\mathcal A} \\def\\kN{\\mathcal N f \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S) \\cong \\bigl(e \\mathcal A} \\def\\kN{\\mathcal N f\\bigr)\\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S \\cong\n\\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S \\cong \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\mathcal{O}} \\bigl(\\widetilde\\mathcal{O} \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S\\bigr)\n$$\nand\n$$\n\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S(\\mathcal G} \\def\\kT{\\mathcal T) = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal A} \\def\\kN{\\mathcal N f, \\mathcal A} \\def\\kN{\\mathcal N e \\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T) \\cong\n\\bigl(f \\mathcal A} \\def\\kN{\\mathcal N e\\bigr)\\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T \\cong \\widetilde\\mathcal{O} \\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T \\cong \\mathcal G} \\def\\kT{\\mathcal T.\n$$\nThis proves the isomorphisms of functors $\\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\widetilde\\mathcal{O}} \\nu^*(\\,-\\,)$ and\n$\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\nu_*$. Since $\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ and $\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf H} \\def\\sU{\\mathsf U$ are right adjoints of $\\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S$\n respectively, the remaining isomorphisms are true as well.\n\\end{proof}\n\n\\noindent\nThe next statement summarizes the main properties of the K\\\"onig's resolution in the projective framework.\n\\begin{theorem}\\label{T:main} We have a diagram of categories and functors\n\\begin{equation}\\label{E:descinglobale2}\n\\xymatrix{D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr) \\ar@\/^2ex\/[rr]^{\\,\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\sR\\mathsf H} \\def\\sU{\\mathsf U\\,} && D(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X}) \\ar[ll]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nsatisfying the following properties.\n\\begin{itemize}\n\\item The pairs of functors $(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\sR\\mathsf H} \\def\\sU{\\mathsf U)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)$ and\n$(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs.\n\\item The functors $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$, $\\sR\\mathsf H} \\def\\sU{\\mathsf U$, $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item Both derived categories $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr)$ and $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr)$ are Verdier localizations of\n$D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)$:\n\\begin{itemize}\n \\item $\nD\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr) \\cong D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$.\n\\item\n$D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\cong D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$\n\\end{itemize}\n\\item Moreover,\n$\n\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T) \\cong D(Q-\\mathsf{Mod}).\n$\n\\item\nThe derived category $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)$ is a categorical resolution of singularities of $X$ in the sense of Kuznetsov \\cite[Definition 3.2]{ku}.\n \\item If $X$ is Gorenstein, then the restrictions of $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ on $\\mathop{\\mathsf{Perf}}\\nolimits(X)$ are isomorphic. Hence, the constructed categorical resolution is even \\emph{weakly crepant} in the sense of\n \\cite[Definition 3.4]{ku}.\n\\end{itemize}\nWe have a \\emph{recollement diagram}\n\\begin{equation}\\label{E:recollement2}\n\\xymatrix{D(Q-\\mathsf{Mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nand all functors can be restricted on the bounded derived categories $D^b(Q-\\mathsf{mod})$, $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$\nand $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. In particular, we have \\emph{two} semi--orthogonal decompositions\n$$\nD\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\n\\bigl\\langle \\mathsf{Im}(\\sR\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)\\bigr\\rangle.\n$$\nThe same result is true when we pass to the bounded derived categories.\n\\end{theorem}\n\n\\begin{corollary}\nFor each $1 \\le k \\le n$ let $D_k$ (respectively $D'_k$) be the full subcategory of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ generated\nby the $k$--th standard module $\\Delta_k$ (respectively, the $k$--th costandard module $\\nabla_k$). Then we have\nequivalences of categories\n$\nD_k \\cong D^b(\\bar{O}_k-\\mathsf{mod}) \\cong D'_k\n$\nand semi--orthogonal decompositions\n\\begin{equation}\\label{E:semiorthdec}\n\\bigl\\langle D_1, \\dots, D_n, \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U), D'_n, \\dots, D'_1\\bigr\\rangle.\n\\end{equation}\nBoth triangulated categories $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$ and $\\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ are equivalent\nto the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. Note that they are \\emph{different} viewed as subcategories\nof $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$.\n\\end{corollary}\n\n\\begin{remark}\nAs in the setting at the beginning of this section, let $X$ be a reduced excellent curve, $\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ its normalization and\n$\\mathcal C} \\def\\kP{\\mathcal P := \\mathit{Hom}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O})$ the conductor ideal. Then $\\mathcal C} \\def\\kP{\\mathcal P$ is also a sheaf of ideals in $\\widetilde\\mathcal{O}$, hence the scheme $S = V(\\mathcal C} \\def\\kP{\\mathcal P) \\stackrel{\\eta}\\hookrightarrow X$ is a \\emph{non--rational locus of $X$ with respect to $\\nu$} in the sense of Kuznetsov and Lunts \\cite[Definition 6.1]{kl}. Starting with the Cartesian diagram\n$$\n\\xymatrix{\n\\widetilde{S} \\ar[r]^{\\tilde\\eta} \\ar[d]_{\\tilde{\\nu}} & \\widetilde{X} \\ar[d]^{\\nu} \\\\\nS \\ar[r]^\\eta & X\n}\n$$\none can construct a partial categorical resolution of singularities of $X$ obtained by the ``naive gluing'' of the\nderived categories $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(S)\\bigr)$, see \\cite[Section 6.1]{kl}.\nIt would be interesting to compare the obtained triangulated category with the derived category $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widehat\\mathbb{X})\\bigr)$ of the non--commutative curve\n$\n\\widehat\\mathbb{X} = \\left(\nX,\n\\left(\n\\begin{array}{cc}\n\\mathcal{O} & \\widetilde\\mathcal{O} \\\\\n\\mathcal C} \\def\\kP{\\mathcal P & \\widetilde\\mathcal{O}\n\\end{array}\n\\right)\n\\right),\n$\nsee also \\cite[Section 8]{bd}. Next, \\cite[Theorem 6.8]{kl} provides a recipe to construct a categorical resolution of singularities of $X$, which however, involves some non--canonical choices.\nIt is an interesting question to compare these categorical resolutions with K\\\"onig's resolution $\\mathbb{X}$ constructed in our article. Another important problem is to give an ``intrinsic description'' of the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$, i.e.~to provide a list of properties describing it uniquely up to a triangle equivalence.\nWe follow here the analogy with non--commutative crepant resolutions, see \\cite[Conjecture 5.1]{bo} and \\cite[Conjecture 4.6]{vdb}. All such resolutions are known to be derived equivalent in certain cases, see for example \\cite[Theorem 6.6.3]{vdb}. Recall that K\\\"onig's resolution $\\mathbb{X}$ is weakly crepant in the case the curve $X$ is Gorenstein.\n\\end{remark}\n\n\\section{Purely commutative applications}\nResults of the previous sections allow to deduce a number of interesting ``purely commutative'' statements.\nLet $X$ be a reduced projective curve over some base field $\\mathbbm{k}$ and\n$\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ be its normalization. According to Orlov \\cite{or}, the Rouquier dimension of the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ is equal to one. In fact, Orlov constructs an explicit vector bundle\n$\\widetilde\\mathcal F} \\def\\kS{\\mathcal S$ on $\\widetilde{X}$ such that $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr) = \\langle \\widetilde\\mathcal F} \\def\\kS{\\mathcal S\\rangle_2$ (here we follow the notation of Rouquier's seminal article \\cite{rou}).\n\n\\begin{theorem} Let $\\mathcal F} \\def\\kS{\\mathcal S := \\nu_*(\\widetilde{\\mathcal F} \\def\\kS{\\mathcal S})$ be the direct image of the Orlov's generator of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. Then the following results are true.\n\\begin{itemize}\n\\item Let $Z$ be the singular locus of $X$ (with respect to the reduced scheme structure) and $\\mathcal{O}_Z$ be the corresponding structure sheaf. Then\n\\begin{equation}\\label{E:1stEstderDim}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\mathcal{O}_Z\\bigr\\rangle_{n+2},\n\\end{equation}\nwhere $n$ is the level of $X$.\n\\item Let $\\kS = \\mathcal{O}_1\/\\mathcal C} \\def\\kP{\\mathcal P_1 \\oplus \\dots \\oplus \\mathcal{O}_n\/\\mathcal C} \\def\\kP{\\mathcal P_n$, where $\\mathcal C} \\def\\kP{\\mathcal P_k := \\mbox{\\textit{Hom}}_X(\\mathcal{O}_{n+1}, \\mathcal{O}_k)$ is the conductor ideal sheaf of the $k$--th partial normalization of $X$ for $1 \\le k \\le n$. Then we have:\n\\begin{equation}\\label{E:2ndEstderDim}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\kS\\bigr\\rangle_{d+3},\n\\end{equation}\nwhere $d$ is the global dimension of the quasi--hereditary algebra $Q$ associated with $X$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nAccording to Theorem \\ref{T:main}, the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ admits a semi--orthogonal decomposition\n$$\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle D^b(Q-\\mathsf{mod}), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle.\n$$\nMoreover, the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ is the Verdier localization of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ via the functor $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T$. This implies that whenever we have an object $\\kX$ of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ with\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\langle \\kX\\rangle_m$ then $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\langle \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\kX)\\rangle_m$.\nAccording to Proposition \\ref{P:recollementSheaves} we have:\n$$\n(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\cdot\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) \\cong \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) \\cong \\nu_*(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) =: \\mathcal F} \\def\\kS{\\mathcal S.\n$$\nNext, Lemma \\ref{L:calcullocale} implies that for all $1 \\le k \\le n$ we have:\n$$\n\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k.\n$$\nLet $\\nu_k: X_k \\longrightarrow X$ be the $k$--th partial normalization of $X$ and $Z_k = \\left\\{y_1, \\dots, y_p\\right\\}$ be the singular locus of $X_k$ (as usual, equipped with the reduced scheme structure). Then\n$$\\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k \\cong (\\nu_{k})_*\\bigl(\\mathcal{O}_{X_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{Z_k}\\bigr) \\cong\n(\\nu_{k})_*\\bigl(\\mathcal{O}_{Z_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{y_1} \\oplus \\dots \\oplus \\mathcal{O}_{Z_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{y_p}\\bigr).\n$$\nObserve that if $y \\in Z_k$ and $x = \\nu_k(y)$ then $(\\nu_{k})_*(\\mathcal{O}_{X_k}\/\\mathcal I} \\def\\kV{\\mathcal V_y) \\cong (\\mathcal{O}\/\\mathcal I} \\def\\kV{\\mathcal V_x)^{\\oplus l}$,\nwhere $l = \\mathsf{deg}\\bigl[\\mathbbm{k}_y: \\mathbbm{k}_x\\bigr]$. Therefore,\n$$\n\\mathsf{add}\\bigl(\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_1) \\oplus \\dots \\oplus \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_n)\\bigr) = \\mathsf{add}(\\mathcal{O}_Z)\n$$\nand (\\ref{E:1stEstderDim}) is just a consequence of \\cite[Lemma 3.5]{rou}. The equality (\\ref{E:2ndEstderDim}) follows in a similar way from Lemma \\ref{L:calcullocale} and \\cite[Proposition 7.4]{rou}.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{C:boundRouquierDim}\nLet $X$ be a reduced quasi--projective curve over some base field $\\mathbbm{k}$. Then there is the following upper bound\non the Rouquier dimension of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$:\n\\begin{equation}\\label{E:boundRouquierDim}\n\\mathsf{dim}\\Bigl(D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)\\Bigr) \\le \\mathsf{min}(n+1, d +2),\n\\end{equation}\nwhere $n$ is the level of $X$ and $d$ is the global dimension of the quasi--hereditary algebra $Q$ associated with\n$X$.\n\\end{corollary}\n\n\\begin{remark}\nIn the case $X$ is rational with only simple nodes or cusps as singularities, the bound (\\ref{E:boundRouquierDim}) has been obtained in \\cite[Theorem 10]{bd}. Note that $n = 1$ and $d=0$ is this case. We do not know whether\nthe estimates (\\ref{E:1stEstderDim}) and (\\ref{E:2ndEstderDim}) are strict.\n\\end{remark}\n\n\n\\noindent\nThe following result gives an affirmative answer on a question posed to the first--named author by Valery Lunts.\n\\begin{theorem}\\label{T:tilting}\nFor any reduced rational projective curve $X$ over some base field $\\mathbbm{k}$ there exists a finite dimensional\nquasi--hereditary $\\mathbbm{k}$--algebra $\\Lambda$ having the following properties.\n\\begin{itemize}\n\\item There exists a fully faithful exact functor\n$\n\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf I} \\def\\sV{\\mathsf V}\\longrightarrow D^b(\\Lambda-\\mathsf{mod})\n$ and a Verdier localization $D^b(\\Lambda-\\mathsf{mod})\n \\stackrel{\\sP}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr),\n$\nsuch that $\\sP \\mathsf I} \\def\\sV{\\mathsf V \\simeq \\mathsf{Id}_{\\mathop{\\mathsf{Perf}}\\nolimits(X)}$.\n\\item The triangulated category $D^b(\\Lambda-\\mathsf{mod})$ is a recollement of the triangulated categories $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D^b(Q-\\mathsf{mod})$, where $Q$ is the quasi--hereditary algebra associated with $X$.\n\\item We have: $\\mathrm{gl.dim}(\\Lambda) \\le d +2$, where $d = \\mathrm{gl.dim}(Q)$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof} According to Theorem \\ref{T:main}, there exists a fully faithful exact functor\n$\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ and a Verdier localization $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\stackrel{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ such that $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T \\cdot \\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathsf{Id}_{\\mathop{\\mathsf{Perf}}\\nolimits(X)}$.\nIt suffices to show that the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ has a tilting object. Recall that we have constructed a semi--orthogonal decomposition\n\\begin{equation}\\label{sorth}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\langle\\kQ\\rangle,\\,\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle,\n\\end{equation}\nwhere $\\langle \\kQ\\rangle \\cong D^b(Q-\\mathsf{mod})$ is the triangulated subcategory generated by\n$\\kQ = \\mathcal A} \\def\\kN{\\mathcal N\/\\mathcal J} \\def\\kW{\\mathcal W$ and $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S) \\cong D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$.\n\nSince the curve $X$ is rational and projective, we have: $\\widetilde{X} = \\widetilde{X}_1 \\cup \\dots \\cup\n\\widetilde{X}_t$, where $\\widetilde{X}_k \\cong \\mathbbm{P}^1_{\\mathbbm{k}}$ for all $1 \\le k \\le t$.\nThen $$\\widetilde\\mathcal B} \\def\\kO{\\mathcal O := \\bigl(\\mathcal{O}_{\\widetilde{X}_1}(-1) \\oplus \\mathcal{O}_{\\widetilde{X}_1}\\bigr) \\oplus \\dots \\oplus\n\\bigl(\\mathcal{O}_{\\widetilde{X}_t}(-1) \\oplus \\mathcal{O}_{\\widetilde{X}_t}\\bigr)$$ is a tilting bundle on $\\widetilde{X}$ and the algebra $E:= \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{\\widetilde{X}}(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O)\\bigr)^{\\mathrm{op}}$ is isomorphic to the direct product of $t$ copies of the path algebra of the Kronecker quiver $\\kron{\\scriptscriptstyle\\bullet}\\bu$.\nThen $\\mathcal B} \\def\\kO{\\mathcal O:= \\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O) \\cong \\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O)$ is a tilting object in\nthe triangulated category $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$.\n\nThe semi--orthogonal decomposition \\eqref{sorth} implies that $\\Hom_{D^b(\\mathbb{X})}(\\kY, \\kX) = 0$ for any $\\kX \\in \\langle \\kQ\\rangle$ and $\\kY \\in \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$.\n\nIt is clear that $\\mathop{\\mathrm{Ext}}\\nolimits_{\\mathbb{X}}^p(\\kQ, \\kQ) = 0$ for $p \\ge 1$ and $Q \\cong \\mathop{\\mathrm{End}}\\nolimits_{\\mathbb{X}}(\\kQ)^{\\mathrm{op}}$.\nSince the ideal $\\mathcal J} \\def\\kW{\\mathcal W$ is locally projective\nas a left $\\mathcal A} \\def\\kN{\\mathcal N$--module, we have: $\\mbox{\\textit{Ext}}_{\\mathbb{X}}^p(\\kQ,\\,-\\,) = 0$ for $p \\ge 2$. Moreover, since\n$\\mathcal B} \\def\\kO{\\mathcal O$ is locally projective and $\\kQ$ is torsion, we also have vanishing $\\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) = 0$.\nSince $\\Hom_{\\mathbb{X}}(\\kX_1, \\kX_2) \\cong \\Gamma\\bigl(X, \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kX_1, \\kX_2)\\bigr)$ the local--to--global spectral sequence implies that\n$\n\\mathop{\\mathrm{Ext}}\\nolimits^p_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) = 0\n$\nunless $p = 1$ and \\begin{equation}\\label{E:local-to-global}\n\\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) \\cong \\Gamma\\bigl(X, \\mbox{\\textit{Ext}}^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)\\bigr).\n\\end{equation}\nSumming up, the complex $\\mathcal H} \\def\\kU{\\mathcal U := \\kQ[-1] \\oplus \\mathcal B} \\def\\kO{\\mathcal O$ is \\emph{tilting} in the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$. A result of Keller \\cite{kel} implies that the derived categories\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ and $D^b(\\Lambda-\\mathsf{mod})$ are equivalent, where $\\Lambda := \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{D^b(\\mathbb{X})}(\\mathcal H} \\def\\kU{\\mathcal U)\\bigr)^{\\mathrm{op}}$. Finally, observe that\n$\n\\Lambda\n\\cong\n\\left(\n\\begin{array}{cc}\nQ & W \\\\\n0 & E\n\\end{array}\n\\right),\n$\nwhere $W := \\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)$ viewed as a ($Q$--$E$)--bimodule. Since the algebra $Q$ is quasi--hereditary and $E$ is directed, the algebra $\\Lambda$ is quasi--hereditary as well. According to \\cite[Corollary 4']{pr}, we have:\n$\\mathrm{gl.dim}(\\Lambda) \\le \\mathrm{gl.dim}(Q) + 2$.\n\\end{proof}\n\n\\begin{remark} In a recent work \\cite[Theorem 4.10]{Wei}, the following inversion of Theorem \\ref{T:tilting} was obtained. Assume $X$ is a projective curve over an algebraically closed field $\\mathbbm{k}$ and $\\Lambda$ a finite dimensional $\\mathbbm{k}$--algebra of finite global dimension such that there exist functors\n$$\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf I} \\def\\sV{\\mathsf V}\\longrightarrow D^b(\\Lambda-\\mathsf{mod}) \\stackrel{\\sP}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)\n$$\n with $\\mathsf I} \\def\\sV{\\mathsf V$ fully faithful, $\\sP$ essentially surjective and $\\sP \\mathsf I} \\def\\sV{\\mathsf V \\simeq \\mathsf{Id}$. Then $X$ is rational. This result can be shown by examining the Grothendieck groups of the involved triangulated categories.\n\\end{remark}\n\n\\begin{remark}\nIn the case $X$ has only simple nodes or cusps as singularities, Theorem \\ref{T:tilting} has been obtained\nin \\cite[Theorem 9]{bd}. See also \\cite[Definition 3]{bd} for an explicit description of the algebra $\\Lambda$ is this case.\n\\end{remark}\n\n\n\n\\begin{remark}\\label{R:bimodule}\nNow we outline how the $Q$--$E$--bimodule $W = \\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)$ from the proof of Theorem \\ref{T:tilting} can be explicitly determined. The isomorphism\n(\\ref{E:local-to-global}) implies that $W$ can be computed locally and we may assume that $X = \\Spec(O)$ and\n$O$ is a complete local ring. We follow the notation of Section \\ref{Sec:LocalStory}. For any $1 \\le k \\le n$ the\nleft $A$--module $R_k:= Q e_k$ has projective resolution\n$$\n0 \\longrightarrow Je_k \\longrightarrow A e_k \\longrightarrow R_k \\longrightarrow 0.\n$$\nThis yields the following isomorphisms of $\\widetilde{O}$--modules:\n\\begin{equation}\\label{E:localcomputation}\nW_k = \\mathop{\\mathrm{Ext}}\\nolimits^1_A(R_k, P) \\cong \\dfrac{\\Hom_A(J e_k, A e)}{\\Hom_A(A e_k, A e)} \\cong \\dfrac{\\Hom_O(C_k, \\widetilde{O})}{\\Hom_O(O_k, \\widetilde{O})} \\cong \\dfrac{C_k^\\vee}{\\widetilde{O}},\n\\end{equation}\nwhere $P = A e$ and $C_k = \\Hom_{O}(\\widetilde{O}, O_k) = \\Hom_{O_k}(\\widetilde{O}, O_k)$ is the conductor ideal of the\npartial normalization\n$O_k \\subset \\widetilde{O}$. Since $\\widetilde{O}$ is regular, we have a (non--canonical) isomorphism of\n$\\widetilde{O}$--modules $\\dfrac{C_k^\\vee}{\\widetilde{O}} \\cong \\dfrac{\\widetilde{O}}{C_k}$.\nSince $\\widetilde{O} \\cong \\mathop{\\mathrm{End}}\\nolimits_A(P)$, this leads to a description of the right $E$--action on $W$. To say more about the left action of $Q$ on $W$, we need an explicit description of the algebra $Q$.\n\\end{remark}\n\n\\section{Quasi--hereditary algebras associated with simple curve singularities}\nLet $\\mathbbm{k}$ be an algebraically closed field of characteristic zero.\nIn this section we compute the algebra $\\kQ$ for the\n \\emph{simple plane curve singularities} in the sense of Arnold \\cite{avg}. These singularities are in\n one--to--one correspondence with the simply laced Dynkin graphs.\n\n\\begin{proposition}\nThe algebra $Q$ associated with the simple singularity $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{m+1})$\nof type $A_{m}$ is the path algebra of the following quiver\n\\begin{align*}\n &\\xymatrix{ 1 \\ar@\/_\/[r]_{\\beta} \\def\\eps{\\varepsilon_1}& 2\\ar@\/_\/[l]_{\\alpha} \\def\\de{\\delta_1} \\ar@\/_\/[r]_{\\beta} \\def\\eps{\\varepsilon_2} &\n \t3 \\ar@\/_\/[l]_{\\alpha} \\def\\de{\\delta_2} \\ar@{.}[r] & (n-1) & n \\ar@\/_\/[l]+<4ex,.9ex>*{}_-{\\alpha} \\def\\de{\\delta_{n-1}}\n \t\\ar@{<-}@\/^\/[l]+<4ex,-.9ex>*{}^-{\\beta} \\def\\eps{\\varepsilon_{n-1}}\t}\n \\end{align*}\n where $n=\\big[\\frac{m+1}2\\big]$ with the relations\n \\begin{align*}\n \t&\\ \\beta} \\def\\eps{\\varepsilon_k\\alpha} \\def\\de{\\delta_k=\\alpha} \\def\\de{\\delta_{k+1}\\beta} \\def\\eps{\\varepsilon_{k+1}\\ \\text{ if }\\ 1\\le k*{}_{\\alpha} \\def\\de{\\delta_{n-1}}\n \t\\ar@{<-}@\/^\/[l]+<4ex,-.9ex>*{}^{\\beta} \\def\\eps{\\varepsilon_{n-1}} }\n \\]\n with the relations\n \\begin{align*} \t&\\ \\beta} \\def\\eps{\\varepsilon_k\\alpha} \\def\\de{\\delta_k=\\alpha} \\def\\de{\\delta_{k+1}\\beta} \\def\\eps{\\varepsilon_{k+1}\\ \\text{ if }\\ 1\\le k0$. As one can see that if the thermodynamic quantities are evaluated in the time interval between $t=0$ and any time $t$, always the change in entropy, entropy production, and entropy extraction rates become greater than zero revealing such systems are inherently irreversible. Moreover, we show that when a distinct temperature difference is not retained between the hot and cold baths, in absence of load, ${\\dot e}_{p}={\\dot h}_{d}=0$ showing that the system is reversible. In the presence of load and when the viscous friction decreases in time, we show that the entropy $S(t)$ monotonously increases with time and saturates to a constant value as $t$ further steps up. The entropy production rate ${\\dot e}_{p}$ decreases in time and at steady state (in the presence of load), ${\\dot e}_{p}={\\dot h}_{d}>0$ which agrees with the results shown in the works \\cite{muuu177}. On the contrary, when the viscous friction increases in time, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. \n\n\n\n\nMost of the previous studies have also focused on exploring the thermodynamic feature of systems such as Brownian heat engines by assuming temperature invariance viscous friction. In reality, the viscous friction of a medium tends to decrease as the temperature of the medium increases. This is because as the intensity of the background temperature increases, the force of interaction between neighboring molecules decreases. In this paper, considering a spatially varying viscosity, the non-equilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. The potential is also coupled with a spatially varying temperature. In this case, the direction of the particle velocity is dictated by the magnitude of the external load of $f$. As one can note that the steady state velocity of the engine is positive when $f$ is smaller and the engine acts as a heat engine. In this regime ${\\dot e}_{p}={\\dot h}_{d}>0$. When $f$ steps up, the velocity of the particle steps down, and at stall force, we find that ${\\dot e}_{p}={\\dot h}_{d}=0$ revealing that the system is reversible at this particular choice of parameter. For large force, the current is negative and the engine acts as a refrigerator. In this region ${\\dot e}_{p}={\\dot h}_{d}>0$. In the absence of load, ${\\dot e}_{p}={\\dot h}_{d}>0$ as long as a distinct temperature difference is retained between the hot and cold baths. \n\n\n\nAt this point, we want also to stress that most of the previous works have focused on calculating\nthe thermodynamic features of different model systems\nby considering additive noise. On the contrary, most realistic\n systems such as neuron system can be also described by Langevin\nequations with multiplicative noise were in this case, the noise\namplitude varies spatially \\cite{mar12}. In this paper, we\nstudy how thermodynamic features of such systems behave.\n\n\nThe rest of the paper is organized as follows: in Section II, we derive the expression for various thermodynamic relations to a Brownian particle walking in overdamped and underdamped media. In Section III, the role of viscous friction is studied by considering viscous friction that varies spatially and temporally. In section IV, we explore the model system in the presence of multiplicative noise. Section V deals with a summary and conclusion.\n\n\n\n\n\n\n\n\n\n\\section{Free energy, entropy production and entropy extraction rates } \nRecently, the dependence for entropy production, free energy, and entropy extraction rates on the system parameters was explored \\cite{muuu177} by considering a Brownian particle that walks in a medium where its viscous friction is insensitive to time or position. However, in most realistic systems, the viscous friction of the medium varies spatially or temporally. To address this issue, let us consider a Brownian particle that moves in an underdamped medium along the potential $U(x)=U_{s}(x)+fx$ where $U_{s}(x)$ and $f$ are the periodic potential and the external force, respectively. Next, the relation for the entropy production and extraction rates will be derived considering a spatially varying viscous friction.\n\n\n\\subsection{Underdamped case}\n{\\it Derivation for entropy production and entropy extraction rates.\\textemdash} Let us consider a single Brownian particle that is arranged to undergo a random walk in an underdamped medium. Here the single-particle and\nits trajectory are considered to be the system as contrasted with the underdamped medium which provides friction and acts as a heat bath. The dynamics of the system is governed by \n\\begin{equation}\nm{dv\\over dt} = -\\gamma (x,t){dx\\over dt}+ {d U(x) \\over dx} + \\sqrt{2k_{B}\\gamma (x,t) T(x)}\\xi(t).\n\\end{equation}\nFor simplicity, Boltzmann constant $k_{B}$ is assumed to be unity. The random noise $\\xi(t)$ is assumed to be Gaussian white noise satisfying the relations \n$\\left\\langle \\xi(t) \\right\\rangle =0$ and $\\left\\langle \\xi(t) \\xi(t') \\right\\rangle=\\delta(t-t')$. The viscous friction $\\gamma (x,t)$ and $T(x)$ are assumed to vary spatially along the medium.\n\nFor underdamped case, the Fokker-Plank equation is given by\n\\begin{eqnarray}\n{\\partial P\\over \\partial t}&=&-{\\partial (vP) \\over \\partial x}-{1 \\over m}{\\partial(U'(x)P) \\over \\partial v}+ \\nonumber \\\\\n&&{\\gamma(x,t) \\over m}{\\partial (vP) \\over \\partial v}+{\\gamma(x,t) T(x) \\over m^2}{\\partial^2 P \\over \\partial v^2}\n\\end{eqnarray}\nwhere $P(x,v,t)$ is the probability of finding the particle at particular position $x$, velocity $v$ and time $t$.\n\n\n \n \nFor convenience, Eq. (2) can be rearranged as \n\\begin{eqnarray}\n{\\partial P\\over \\partial t}&=& -(k+{\\partial J' \\over \\partial v} )\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nk=v{\\partial P \\over \\partial x} ={\\partial J \\over \\partial x}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nJ'= -{\\gamma (x,t)\\over m}vP+{1\\over m}(U'P) -{\\gamma (x,t) T(x)\\over m^2}{\\partial P\\over \\partial v}.\n\\end{eqnarray}\nFrom Eqs. (4) and (5), one gets \n\\begin{eqnarray}\n{\\partial P \\over \\partial v}= -{m^{2}J'\\over \\gamma (x,t)T(x)}+{mU'P\\over \\gamma (x,t) T(x)}-{m vP\\over T(x)}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\partial P \\over \\partial x}= {k\\over v}.\n\\end{eqnarray}\nNext we derive the expressions for the entropy production by considering two cases. \n\n\n\n{\\it Case1.\\textemdash} Here we want to stress that the approach by Lee. $et$. $al$. \\cite{mar1} and Ge. $et$. $al$. \\cite{mar2} indicate that under time reversal operation, the total entropy production and extraction rate can be obtained. Particularly, the analysis by Ge. $et$. $al$. \\cite{mar2} shows that the entropy production rate is given by\n\\begin{eqnarray}\n{\\dot e}_{p}&=&-\\int {1\\over T(x) \\gamma(x,t)}\\left[F-{T(x) \\gamma(x,t)\\nabla_{v}\\ln(P)\\over m}\\right]^2P dx dv \\nonumber \\\\\n&=&-\\int {1\\over T(x) \\gamma(x,t)}\\left[F-{T(x) \\gamma(x,t){\\partial P\\over \\partial v}\\over m P}\\right]^2P dx dv\n\\end{eqnarray}\nwhere $F=-T(x) \\gamma(x,t)+U'$.\nSubstituting Eq. (6) into Eq. (8), one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv \n\\end{eqnarray}\nOn the other hand, the entropy extraction rate can be found via the method devloped by Ge. $et$. $al$. \\cite{mar2} as\n\\begin{eqnarray}\n{\\dot h}_{d}&=&-\\int {1\\over T(x) }\\left[T(x) \\gamma(x,t)+{T(x) \\gamma(x,t)\\nabla_{v}\\ln(P)\\over m}\\right]P v dxdv \\nonumber \\\\\n&=&-\\int {1\\over T(x) }\\left[T(x) \\gamma(x,t)+{T(x) \\gamma(x,t){\\partial P\\over \\partial v}\\over m P}\\right]P v dxdv.\n\\end{eqnarray}\nSubstituting Eq. (6) into Eq. (10) leads to\n \\begin{eqnarray}\n{\\dot h}_{d} &=& \\int {(U'J-vmJ') \\over T(x) } dx dv.\n\\end{eqnarray} Here ${\\dot e}_{p}={d e_{p}\\over dt} $ and ${\\dot h}_{d}={d h_{d}\\over dt}$ denote the entropy production and extraction rates.\nThe above expressions (for the entropy production and extraction rates) can be also derived at the ensemble level via the approach stated in the work \\cite{mu7}. \n\n{\\it Case2.\\textemdash} One can also rederive the expressions for the entropy production and extraction rates at the ensemble level by first analyzing the entropy of the system at the trajectory level as\n\\begin{eqnarray}\ns(t)=-ln P(x,v,t)\n\\end{eqnarray}\nwhere $x(t)$ denotes the stochastic trajectory. \nThe rate of entropy\nchange at trajectory level is then given by\n\\begin{eqnarray}\n{\\dot s}(t)=-{\\partial_{t} P(x,v,t) \\over P(x,v,t)}-{\\partial_{x} P(x,v,t) \\over P(x,v,t)}{\\dot x}-{\\partial_{v} P(x,v,t) \\over P(x,v,t)}{\\dot v}.\n\\end{eqnarray}\n\nSubstituting Eqs . (6) and (7) into Eq. (13), the entropy production and dissipation\nrates at trajectory level are given as\n\\begin{eqnarray}\n{\\dot e}_{p}^{*}=-{\\partial_{t} P(x,v,t) \\over P(x,v,t)}+{m^{2}J'\\over \\gamma (x,t)T(x)P(x,v,t)}{\\dot v}-{k \\over P(x,v,t)}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d}^{*}={mU'\\over \\gamma (x,t) T(x)}{\\dot v}-{m\\gamma(x,t) v\\over T(x)}{\\dot v}.\n\\end{eqnarray}\nBecause averaging overall trajectories yields $\\left\\langle {\\dot v}|x\\right\\rangle={J'\\over P(x,t,v)}$ and $\\int \\partial_{t} P(x,v,t ) =0$, after some algebra one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv -\\int P v dv\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=& \\int {m(U'-v\\gamma (x,t))J' \\over T(x)\\gamma (x,t) } dx dv,\n\\end{eqnarray} respectively.\nWhen a periodic boundary condtion is imposed, Eqs. (9) and (16) as well as Eqs. (11) and (17) converge to\n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=& -\\int {vmJ' \\over T(x) } dx dv.\n\\end{eqnarray} \n\nThe heat dissipation rate ${\\dot H}_{d}$ can be calculated \\cite{am4,am5} as \n\\begin{eqnarray}\n{\\dot H}_{d} \n&=&-\\left\\langle \\left(-\\gamma(x,t){\\dot x}+ \\sqrt{2k_{B}\\gamma(x,t) T(x)}\\right).{\\dot x}\\right\\rangle \\nonumber \\\\\n&=&-\\left\\langle m{vdv\\over dt} +v U'(x) \\right\\rangle.\n\\end{eqnarray}\nOur previous analysis also suggests \\cite{mu17,muu17,muuu17} that the entropy extraction rate ${\\dot h}_{d}$ can be expressed as \n\\begin{eqnarray}\n{\\dot h}_{d} \n&=&-\\int \\left({m{vdv\\over dt} +v U'(x) \\over T(x)} \\right)P dxdv.\n\\end{eqnarray}\n One should note that Eq. (21) is exact and does not depend on any boundary condition. \nSince ${d S(t)\\over dt}$ and ${\\dot h}_{d} $ are computable, the entropy production rate can be readily obtained as\n\\begin{eqnarray}\n{\\dot e}_{p}&=&{d S(t)\\over dt}+{\\dot h}_{d}. \n\\end{eqnarray}\nIn the long time limit, ${d S(t)\\over dt}=0$ which implies ${\\dot e}_{p}={\\dot h}_{d}>0$ at steady state and ${\\dot e}_{p}={\\dot h}_{d}=0$ at stationary state.\n\n\n\n\nOnce the expressions for \n${\\dot S}(t)$, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ are computed as a function of time $t$, the analytic expressions for the change in entropy production, heat dissipation and total entropy can be found analytically via \n$\n\\Delta h_d(t)= \\int_{0}^{t}{\\dot h}_{d}(t)dt $,\n$\\Delta e_{p}(t)= \\int_{0}^{t} {\\dot e}_{p}(t) dt $ and \n$\\Delta S(t) =\\int_{0}^{t} {\\dot S}(t)dt $\nwhere $\\Delta S(t)=\\Delta e_p(t)-\\Delta h_d(t)$. \n\n \n{\\it Derivation for free energy.\\textemdash} Our next objective is to write the expression for the free energy in terms of ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$ where ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$ are the terms that are associated with ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$. As discussed before, the heat dissipation rate is either given by Eq. (20) (for any cases) or if a periodic boundary condition is imposed, ${\\dot H}_{d}(t)$ is given by \n\\begin{eqnarray}\n{\\dot H}_{d} &=&-\\int m(v)J' dx dv\n\\end{eqnarray}\nwhich is notably different from Eq. (19), due to the term $T(x)$. \nThe term associated to ${\\dot e}_{p}$ is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& -\\int {m^{2} J'^{2} \\over P \\gamma(x,t)} dx dv.\n\\end{eqnarray}\nThe entropy balance equation \n\\begin{eqnarray}\n{d S^T(t)\\over dt}&=&{\\dot E}_{p}-{\\dot H}_{d}\n\\end{eqnarray}\nis associated to Eqs. (11) or (17) except the term $T(x)$.\nOnce again, employing the expressions for \n${\\dot S}^T(t)$, ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$, one can get \n$\\Delta H_d(t)= \\int_{0}^{t}{\\dot H}_{d}(t)dt$, \n$\\Delta E_{p}(t)= \\int_{0}^{t} {\\dot E}_{p}(t) dt $ and \n$\\Delta S(t)^T =\\int_{0}^{t} {\\dot S}(t)^{T}dt$\nwhere $\\Delta S(t)^T=\\Delta E_p(t)-\\Delta H_d(t)$. \n\n\nOn the other hand, the expression for the internal energy has a form \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int ({\\dot K}+ v U'_{s}(x))P(x,v,t)dvdx\n\\end{eqnarray}\nwhere ${\\dot K}=m{vdv\\over dt}$ and $U'_{s}$ denote the rate of kinetic and potential energy, respectively. The network work done by the system \n\\begin{eqnarray}\n{\\dot W}&=& \\int v f P(x,v,t)dvdx\n\\end{eqnarray}\nexplicitly depends on the velocity $V$ and the load $f$. \nIn terms of ${\\dot H}$ and ${\\dot W}$, the rate of the internal energy is given by \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}\n\\end{eqnarray}\nand after some algebra, the first law of thermodynamics can be written as \n\\begin{eqnarray}\n\\Delta E_{in}= -\\int_{0}^{t}( {\\dot H}_{d}(t)+{\\dot W}) dt.\n\\end {eqnarray}\n\nRearranging some terms, one gets \nthe rate of free energy as ${\\dot F}={\\dot E}-T{\\dot S}$ for isothermal case and ${\\dot F}={\\dot E}-{\\dot S}^T$ for nonisothermal case where ${\\dot S}^T={\\dot E}_{p}-{\\dot H}_{d}$. The rate of free energy dissipation \n\\begin{eqnarray}\n{\\dot F}&=&{\\dot E}_{in}- {\\dot S}^T \\nonumber \\\\\n&=&{\\dot E}_{in}-{\\dot E}_{p}+{\\dot H}_{d}\n\\end{eqnarray}\ncan be expressed as a definite integral as\n\\begin{eqnarray}\n\\Delta F(t)&=&-\\int_{0}^{t} \\left( {\\dot W}+ {\\dot E}_{p}(t) \\right)dt.\n\\end{eqnarray} \nFor isothermal case, at quasistatic limit where the velocity approaches zero $ v=0$, ${\\dot E}_{p}(t) =0$ and ${\\dot H}_{d}(t) =0$ and far from quasistatic limit \n$E_{p}={\\dot H}_{d}>0$ which is expected as the particle operates irreversibly. \n\n\\subsection{Overdamped case}\n\n{\\it Derivation for free energy, entropy production and entropy extraction rates.\\textemdash} For overdamped case,\n as discussed by Sancho. $et$ .$at$ \\cite{am3} and Jayannavar $et$ .$at$ \\cite{am33}, \nEq. (1) converges to \n\\begin{eqnarray}\n\\gamma(x,t){dx\\over dt}&=&{-\\partial U(x)\\over \\partial x} -{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)}+\\nonumber \\\\\n&&\\sqrt{2k_{B}\\gamma(x,t) T(x)}\\xi(t)\n\\end{eqnarray}\nwhich corresponds to the Stratonovich interpretation \\cite{am1,am2}.\n The corresponding Fokker Planck equation is given by \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({U'(x)\\over \\gamma(x,t)}+{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)^2}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T(x)\\over \\gamma(x,t)}{\\partial P(x,t)\\over \\partial x}\\right)\n\\end{eqnarray}\nwhich can be rewritten as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&-{\\partial J \\over \\partial x}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nJ&=&-\\left({U'(x)P(x,t)\\over \\gamma(x,t)}+{P(x,t)(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)^2}\\right)-\\nonumber \\\\\n&&\\left({T(x)\\over \\gamma(x,t)}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\n\nThe rate of entropy\nchange at trajectory level is given by\n\\begin{eqnarray}\n{\\dot s}(t)=-{\\partial_{t} P(x,t) \\over P(x,t)}-{\\partial_{x} P(x,t) \\over P(x,t)}{\\dot x}.\n\\end{eqnarray}\nOn the other hand, from Eq. (35) one gets \n\\begin{eqnarray}\n{\\partial P \\over \\partial x}= -{\\gamma (x,t) J\\over T(x)}-{U'(x)P(x,t)\\over T(x)}-{P(x,t)(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}.\n\\end{eqnarray}\n\n\nSubstituting Eq. (37) into Eq. (36), the entropy production and dissipation\nrates at trajectory level are given as\n\\begin{eqnarray}\n{\\dot e}_{p}^{*}=-{\\partial_{t} P(x,t) \\over P(x,t)}+{\\gamma (x,t)J\\over T(x)P(x,t)}{\\dot v}\n\\end{eqnarray}\nand \n \\begin{eqnarray}\n{\\dot h}_{d}^{*}={U'(x)\\over T(x)}+{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}.\n\\end{eqnarray}\nBecause averaging overall trajectories yields $\\left\\langle {\\dot x}|x\\right\\rangle={J\\over P(x,t)}$ and $\\int \\partial_{t} P(x,t ) =0$, after some algebra one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {\\gamma(x,t) J^{2} \\over P T(x)} dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({JU'(x)\\over T(x)}+{J(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}\\right) dx \n\\end{eqnarray} respectively. One should note that Eqs. (18) and (19) (underdamped case) as well as Eqs. (40) and (41) (overdamped case) approach \n\\begin{eqnarray}\n{\\dot h}_{d} ={\\dot e}_{p}\n&=&\\int \\left({J U'(x) \\over T(x)} \\right)dx\n\\end{eqnarray}\nat steady state ($v {dv\\over dt}=0$), \nas long as a \nperiodic boundary condition is imposed.\nFrom Eqs. (40) and (41), it is evident that when detailed balance conditions are satisfied the velocity of equivalently the current $J=0$ and as a result $ {\\dot e}_{p}=0$ and $ {\\dot h}_{d}=0$. Far from equilibrium, $J>0$, and in this case when the viscous friction decreases either spatially or temporally, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ approach to a constant value. When the system is driven out of equilibrium and when viscous friction increases spatially and temporally, $ {\\dot e}_{p}$ and $ {\\dot h}_{d}$ \nmonotonously step up.\nMoreover, from Eq. (41), the heat dissipation rate is derived as \n\\begin{eqnarray}\n{\\dot H}_{d} &=& \\int \\left(JU'(x)+{J(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)} \\right) dx .\n\\end{eqnarray}\nOn the other hand, the term ${\\dot E}_{p}$ is related to ${\\dot e}_{p}$ and it is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& \\int {\\gamma(x,t) J^{2} \\over P } dx.\n\\end{eqnarray}\nThe new entropy balance equation has a simple form \n\\begin{eqnarray}\n{d S^T(t)\\over dt}&=&{\\dot E}_{p}-{\\dot H}_{d}.\n\\end{eqnarray}\nFurthermore, the internal energy \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int J U'_{s}(x) dx\n\\end{eqnarray}\nhas functional dependence on the current $J$ and the potential profile $U_s$. \nThe total work done is then given by \n\\begin{eqnarray}\n{\\dot W}&=& \\int \\left(Jf+{J(\\gamma'(x,t)T(x))\\over 2 \\gamma(x,t)} +{JT'(x)\\over 2 }\\right)dx.\n\\end{eqnarray}\nThe first law of thermodynamics can be written as \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}.\n\\end{eqnarray}\n The change in the internal energy reduces to \n$\n\\Delta E_{in}= -\\int_{0}^{t}( {\\dot H}_{d}(t)+{\\dot W}) dt.\n$\nOnce again \nthe rate of free energy dissipation can be written as \n${\\dot F}={\\dot E}_{in}- {\\dot S}^T \n={\\dot E}_{in}-{\\dot E}_{p}+{\\dot H}_{d}$. The change in the free energy is then given by \n$\\Delta F(t)=-\\int_{0}^{t} \\left( {\\dot W}+ {\\dot E}_{p}(t) \\right)dt.\n$\n\n \n \n\\section{ Time dependent viscous friction }\n\nSome viscous fluids show a change in viscosity when\ntime changes. This is because as the fluid shear stress\nchanges in time, so does the viscosity. Often, the dynamics\nof systems with self-organized criticality also can be explored\nby considering time dependent diffusion (viscous\nfriction) and drift terms \\cite{marr1,marr2, marr3}. Some studies have also\nfocused on calculating the mean first passage time by considering a time dependant diffusion term \\cite{muuuu17}. To explore the non-equilibrium thermodynamic features of a Brownian particle that hops in a medium where its\nviscosity depends on time, we consider a Brownian particle\nthat walks on a periodic isothermal medium (in the\npresence or absence of load) where its viscosity is given\nby \n\\begin{eqnarray}\n\\gamma(t)&=&{1\\over g(1+t^z)}.\n\\end{eqnarray}\nIn this case, the corresponding Fokker Planck equation in overdamped medium is given as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({f\\over \\gamma(t)}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T\\over \\gamma(t)}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nImposing a periodic boundary condition $P(0,t)=P(L,t)$ and let us choose a Fourier cosine series \n\\begin{eqnarray}\nP(x,t)=\\sum_{n=0}^\\infty b_{n}(t)cos\\left({n\\pi \\over L_0}(x+{f\\over \\gamma(t)})\\right)\n\\end{eqnarray}\nas a possible solution. \nAfter some algebra, we get \n the probability distribution as \n\\begin{eqnarray}\nP(x,t)&=&\\sum_{n=0}^\\infty \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right)\\right]\\zeta.\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\zeta&=&e^{-{(n\\pi )^2T\\left(gt+{gt^{z+1}\\over(z+1)}\\right) \\over L^2}}.\n\\end{eqnarray}\nHere $f$ is the external load and $T$ is the temperature of the medium.\nThe current is then given by \n\\begin{eqnarray}\nJ(x,t)&=&-\\left[{f P(x,t)\\over \\gamma(t)} + {T\\over \\gamma(t)}{\\partial P(x,t) \\over \\partial x}\\right].\n\\end{eqnarray}\nThe current $J(x,t)>0$, only when $f \\ne 0$ since $\\gamma(x)$ is not the necessary parameter to keep the system out of equilibrium. \n As stated before, $\n{\\dot e}_{p}= {\\dot h}_{d}+{d S(t)\\over dt}$ where \n$\n{d S(t)\\over dt}=-\\int {J\\over P(x,t)}{\\partial \\over \\partial x} P(x,t) dx$. \nAfter some algebra, we write \n\\begin{eqnarray}\n{d S(t)\\over dt}&=&-\\int J {\\sum_{n=0}^\\infty {{n\\pi\\over 2} \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right) \\right] \\zeta \\over \\sum_{n=0}^\\infty \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right)\\right]\\zeta}} dx.\n\\end{eqnarray}\n The entropy production and entropy extraction rates are given by the relations \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {J^{2} \\over P(x,t) T g(1+t^z) } dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({Jf\\over T}\\right) dx.\n\\end{eqnarray} \nSubstituting Eqs. (52) and (54) into Eqs. (56) and (57), one can explore how ${\\dot e}_{p}$ and ${\\dot h}_{d}$ depend on time. For $z \\ge 0$, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ decrease and approach to a constant value at steady state. When $z<0$, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ step up continuously. \nHereafter, for simplicity, the parameter $g$ is considered to be a constant. \n Furthermore, the heat dissipation rate is given by \n\\begin{eqnarray}\n{\\dot H}_{d} &=& \\int \\left(Jf \\right) dx \n\\end{eqnarray}\nwhile the term ${\\dot E}_{p}$ is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& \\int {J^{2} \\over P(x,t)g(1+t^z) } dx.\n\\end{eqnarray}\nOn the other hand, the internal energy has a form \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int J U'_{s}(x) dx.\n\\end{eqnarray}\nThe total work done is then given by \n\\begin{eqnarray}\n{\\dot W}&=& \\int \\left(Jf \\right)dx.\n\\end{eqnarray}\nThe first law of thermodynamics can be written as \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}\n\\end{eqnarray}\nHereafter, whenever we plot any figures, we use the following dimensionless \nload ${\\bar f}=fL_{0}\/T_{c}$, ${\\bar U}=U\/T_{c}$, temperature ${\\bar \\tau}=T(x) \/T_{c}$ where $T_c$ is the reference temperature.\nWe also introduced dimensionless parameter ${\\bar x}=x\/L_{0}$, ${\\bar v}=vm\/ \\gamma L_{0}$ and ${\\bar t}=t \\gamma \/m$. Hereafter the bar will be dropped.\nFrom now on all the figures will be plotted in terms of the dimensionless parameters.\n\nThe expression for the rate of entropy production as well as entropy extraction rate can be readily calculated via Eqs. (56) and (57). In the absence of load $f=0$, ${\\dot e}_{p} = {\\dot h}_{d}=0$. This is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. When a distinct temperature difference is retained between the hot and cold baths, in absence of load, ${\\dot e}_{p}={\\dot h}_{d}=0$ showing that such a system is inherently reversible. In the presence of load and when the viscous friction increases in time (see Fig. 1), ${\\dot e}_{p}$ and ${\\dot h}_{d}$ decrease in time and as time further\nsteps up ${\\dot e}_{p}$ and ${\\dot h}_{d}$ monotonously increase in time as shown in Fig. 1. Figure 1 is plotted by fixing $\\tau=1.0$, $f=1.0$ and $z=-0.5$. On the contrary, in the presence of load and when the viscous friction decreases in time (see Fig. 2), ${\\dot e}_{p}$ and ${\\dot h}_{d}$ monotonously decrease with time and saturate to a constant value as $t$ further steps up. Fig. 2 is plotted by fixing $\\tau=1.0$, $f=1.0$ and $z=1.0$. \n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n9.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n10.eps}\n}\n\\caption{ (Color online) (a) The entropy extraction rate ${\\dot h}_{d}(t)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) The plot of entropy production rate ${\\dot e}_{p}(t)$ as a function of $t$.\n${\\dot e}_{p}(t)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The two figures exhibit that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ decrease in time and as time further steps up, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase. In the both figures, the parameters are fixed as $f=1.0$, $\\tau=1.0$ and $z=-0.5$. \n} \n\\label{fig:sub}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n7.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n8.eps}\n}\n\\caption{ (Color online) (a) ${\\dot h}_{d}(t)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) ${\\dot e}_{p}(t)$ as a function of $t$.\n${\\dot e}_{p}(t)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The figure exhibits that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ decrease in time and as time further steps up, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ approach a constant value. In the both figures, the parameters are fixed as $f=1.0$, $\\tau=1.0$ and $z=1.0$. \n} \n\\label{fig:sub}\n\\end{figure}\n\n\n\n\n\\section{ Spatially varying viscous friction }\n\n\nMost of the previous studies have focused on exploring the thermodynamic feature of systems such as Brownian heat engines by assuming temperature invariance viscous friction. However, various studies have indicated that the viscous friction of a medium tends to decrease as the temperature of the medium increases \\cite{mar15, mar16, aa15}. Particularly in the liquid medium, the viscosity decreases as the intensity of the background temperature steps up. This is because when the temperature of the medium increases, more molecules start vibrating, and as a result their speed increases. This speedy motion of the molecules creates a reduction in interaction time between neighboring molecules. At the macroscopic level, there will be a reduction in the intermolecular force, and hence reduced viscosity of the fluid. Consequently, when the temperature of the viscous medium decreases, the viscous friction in the medium decreases. \n\n\n\n\n\n\n\nIn this paper, considering a spatially varying viscosity, the non-equilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. The potential is also coupled with a spatially varying temperature. \n\n{\\it The model .\\textemdash} Let us consider a Brownian particle that walks in a piecewise linear potential with an external load $U(x)=U_{s}(x) + fx$, where the ratchet potential\n$U_{s}(x)$ is given by\n\\begin{equation}\n U_{s}(x)=\\left\\{\n \\begin{array}{ll}\n 2U_{0}\\left({x\\over L_{0}}\\right),& \\text{if}~~~ 0 \\le x \\le {L_{0}\\over 2};\\\\\n 2U_{0}\\left(1-{x\\over L_{0}}\\right),& \\text{if} ~~~{L_{0}\\over 2} \\le x \\le L_{0}.\n \\end{array}\n \\right.\n\\end{equation}\nHere $U_{0}$ and $L_{0}$ denote the barrier height and the width of the ratchet potential,\nrespectively. $f$ designates the external force. \nThe potential exhibits its maximum value\n$U_{0}$ at $x={L_{0}\\over 2}$ and its minima at $x=0$ and $x={L_{0}}$.\nThe \nspatially varying temperature is arranged as \n\\begin{equation}\nT(x)=\\left\\{\n\\begin{array}{ll}\nT_{h},& \\text{if} ~~~0 \\le x \\le {L_{0}\\over 2};\\\\\nT_{c},& \\text{if} ~~~ {L_{0}\\over 2} \\le x \\le L_{0}\n\\end{array}\n\\right.\n\\end{equation}\n as shown in Fig. 1. The potential $U_{s}(x)$ and \n$T(x)$ are assumed to be periodic with a period $L_0$, $U_{s}(x+L_{0})=U_{s}(x)$ and $T(x+L_{0})=T(x)$.\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=8cm]{am14.eps}\n}\n\\caption{(Color online) Schematic diagram for a particle that walks in a piecewise linear potential\nin the absence of an external load. }\n\\label{fig:ratchet}\n\\end{figure}\n\nFor a fluid such as blood, it is reasonable to\nassume that when the temperature of the blood sample increases by $1.0$ degree Celsius, its viscosity steps down by $2.0$ degree Celsius \\cite{aa15}. Thus let us consider viscous friction that varies as \n \\begin{equation}\n \\gamma(x)=\\gamma'-C(T(x)-T_c)\n\\end{equation}\nwhere $C$ is a constant which is less than one. In the next two sections, we will explore the model system in the overdamped and underdamped limits.\n\n\\subsection{ Underdamped case in the absence of ratchet potential }\nIn this section, we consider an important model system where a colloidal particle that undergoes a biased random walk in a spatially varying thermal arrangement in the presence of external load $f$ with no potential. \nSolving Eq. (3) at steady state, the general expression for the probability distribution is obtained as\n\\begin{equation} \nP(x,v)=\\frac{e^{-\\frac{m (f-(\\gamma+c T_c) v+c v T[x])^2}{2 T[x] (\\gamma+c T_c-c\nT[x])^2}} \\sqrt{\\frac{m}{T[x]}}}{\\sqrt{2 \\pi }}.\n \\end{equation}\n The average velocity is found to be \n\\begin{equation} \nv={f\\over \\gamma(x) }.\n\\end{equation}\n In the absence of force, the velocity of the particle approaches zero. \n\n \n\nVia Eqs. (9) and (11), the entropy production and extraction rates are calculated as \n \\begin{eqnarray} \n{\\dot h}_{d}(t)&=&{\\dot e}_{p}(t) \\nonumber \\\\\n&=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma T_c}+\\frac{1}{(\\gamma+c (T_c-T_h)) T_h}\\right)\n \\end{eqnarray}\nOne can see that in the limit where the load approaches the stall force, ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. \tExploiting Eq. (62), it is evident that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase when the temperature difference between the hot and cold baths decreases. This is feasible since when the temperature difference between the heat baths steps up, the magnitude of the viscose friction decreases. \n\t\n\tThe rate of heat dissipation \nis calculated employing Eq. (20) and it converges to\n \\begin{eqnarray} \n{\\dot H}_{d}(t)&=&{\\dot E}_{p}(t)\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\n\tIn the limit where the load approach zero,\n\t${\\dot H}_{d}(t)={\\dot E}_{p}(t)=0$ showing that at quasistatic limit the system is reversible.\n\tOn the other hand, the rate of work done is given by \n\\begin{eqnarray} \n{\\dot W}(t)&=&{\\dot E}_{p}(t) \\nonumber \\\\ &=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\n\tFor isothermal case $T_{h}=T_{c}$, one gets $v=f\/\\gamma$, ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=f^2L_{0}\/\\gamma T_{c}$ and ${\\dot H}_{d}(t)={\\dot E}_{p}(t)=f^2L_{0}\/\\gamma$.\n\t\n\n\t\n\n \n\n\n\n\\subsection{ Overdamped case }\n\nIn the presence of ratchet potential, in the overdamped limit, the closed-form expression for the steady-state current can be given as\n\\begin{equation}\nJ= -{\\varsigma_{1}\\over \\varsigma_{2}\\varsigma_{3}+(\\varsigma_{4}+\\varsigma_{5})\\varsigma_{1}}\n\\end{equation}\nwhere the expressions for $\\varsigma_{1}$, $\\varsigma_{2}$, $\\varsigma_{3}$, and $\\varsigma_{4}$\nare given by\n\\begin{eqnarray}\n\\varsigma_{1}& = &-1+e^{{L_0(f-{2U_{0}\\over L_0})\\over 2T_{c}}+{L_0(f+{2U_{0}\\over L_0})\\over 2T_{h}}}\\\\ \\nonumber\n\\varsigma_{2}& = &{e^{-{{fL_{0}(T_{c}+T_{h})\\over T_{c}}+2U_{0}\\over 2T_{h}}}\\left(e^{fL_{0}\\over\n 2T_{c}}-e^{U_{0}\\over T_{c}}\\right)L_0\\over fL_{0}-2U_{0}}\\\\ \\nonumber\n &&-{\\left(e^{-{fL_{0}+2U_{0}\\over 2T_{h}}}-1\\right)L_{0}\\over fL_{0}+2U_{0}} \\\\ \\nonumber\n\\varsigma_{3} & = &{e^{-{U_{0}\\over T_{c}}+{fL_{0}+2U_{0}\\over 2T_{h}}}\n\\left(e^{fL_{0}\\over 2T_{c}}-e^{U_{0}\\over T_{c}}\\right)L_0T_{c} \\gamma\\over fL_{0}-2U_{0}}+\\\\ \\nonumber\n &&{\\left(e^{fL_{0}+2U_{0}\\over 2T_{h}}-1\\right)L_{0}T_{h}(\\gamma +c(T_c-T_h))\\over fL_{0}+2U_{0}}\\\\ \\nonumber\n\\varsigma_{4}& = &(\\gamma +c(T_c-T_h)){L_{0}^2\\left(fL_{0}+2((-1+e^{-{fL_{0}+2U_{0}\\over 2T_{h}}})T_{h}+U_{0})\\over\n 2(fL_{0}+2U_{0})^2\\right)}\n\\end{eqnarray}\nand $\\varsigma_{5}=L_{0}^2(t_1+t_2+t_3t_4)$. Here $t_1$, $t_2$, $t_3$ and $t_4$ are given by \n \\begin{eqnarray}\n t_1&=&{\\gamma \\over 2fL_{0}-4U_{0}}\\\\ \\nonumber\n t_2&=&{\\gamma T_c(-1+e^{-{{fL_{0}-2U_0}\\over 2T_{c}}})T_c\\over (fL_{0}-2U_{0})^2}\\\\ \\nonumber\n t_{3}&=&{(-1+e^{-{{fL_{0}-2U_0}\\over 2T_{h}}})T_h\\over (f^2L_{0}^2-4U_{0}^2)} \\\\ \\nonumber\n t_{4}&=&{{e^{-{fL_{0}(T_{c}+T_{h})\\over 2T_{c}T_{h}}-{U_{0}\\over T_{h}}}(e^{{fL_{0}\\over\n 2T_{c}}}-e^{{U_0\\over T_{c}}})(\\gamma +c(T_c-T_h))}\\over (f^2L_{0}^2-4U_{0}^2)}.\n \\end{eqnarray}\n\tThe steady state current converges to zero (at quasistatic limit) when \n\\begin{eqnarray}\nf'= {2U_{0}(T_{h}-T_{c})\\over (L_{0}(T_{h}+T_{c}))}.\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n1.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n2.eps}\n}\n\\caption{ (Color online) (a) The dependence of $J$ on $U_{0}$ for fixed $\\tau=2.0$, $\\gamma'=1$ and $f=0.5$. The parameter $C$ is also fixed as $0.4$ (solid line), $0.2$ (dashed line) and $0.04$ (dotted line). \n (b) The plot $J$ as a function of $f $ for parameter choice $U_{0}=2.0$ and $\\tau=2.0$. \n The parameter $C$ is fixed as $0.4$, (solid line), $0.2$ (dashed line) and $0.04$ (dotted line). }\n\\label{fig:sub}\n\\end{figure}\nNext, let us explore the dependence for the thermodynamic quantities on the model parameters. In Fig. 4a, the current as a function of potential height is plotted. The current exhibits a maximum value at a particular barrier height. As shown in Fig. 4b, the current monotonously decreases with the load. When $f0$ while when $f>f'$, $J<0$.\n\n\n\n\nOnce the expression for steady-state current is obtained, the values for ${\\dot h}_{d}$ and ${\\dot e}_{p}$ can be readily evaluated via Eq. (42). \nAt steady state ($v {dv\\over dt}=0$), to both underdamped and overdamped cases, one finds \n\\begin{eqnarray}\n{\\dot h}_{d} ={\\dot e}_{p}\n&=&\\int \\left({J U'(x) \\over T(x)} \\right)dx.\n\\end{eqnarray}\nThe rate of heat extraction is given by \n\\begin{eqnarray}\n{\\dot H}_{d}=&=&\\int \\left({J U'(x)} \\right)dx.\n\\end{eqnarray}\nAt quasistatic limit ( $f\\to f'$), ${\\dot h}_{d} ={\\dot e}_{p}=0$ as well as ${\\dot H}_{d}=0$ since at this limit $J=0$.\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n3.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n4.eps}\n}\n\\caption{ (Color online) (a) The plot for ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $U_{0}$ for parameter choice of $\\tau=2.0$, $C=0.04$ and $f=0.5$. \n (b) The plot ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $f$ for parameter choice of $U_{0}=2.0$, $C=0.04$ and $\\tau=2.0$. } \n\\label{fig:sub}\n\\end{figure}\nLet us explore how the rate of entropy production ${\\dot e}_{p}(t)$ and the rate of entropy extraction ${\\dot h}_{d}(t)$ behave. The plot of ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $U_{0}$ is depicted in Fig. 5a. The entropy production and extraction rates take a zero value at the stall force (zero velocity), ${\\dot e}_{p}(t)={\\dot h}_{d}(t)=0$ which implies that at the stall force the system is reversible. The plot ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $f$ is depicted in Fig. 5b. As depicted in the figure, the entropy production and extraction rates decrease as the load increases and attains a zero value at the stall force. As the load further increases, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ step up. \nThe entropy production and extraction rates increase as $C$ and the temperature difference between the two baths decreases. On the other hand, entropy production and extraction rates decrease as $\\tau$ increases and attain a zero value at a particular $\\tau$. As the temperature further increases, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase\n\nIf one considers a periodic boundary condition at steady state in the absence of ratchet potential $U_{0}=0$, the results obtained quantitively agree with the underdamped case (Section IV ) and one gets \n\t\\begin{eqnarray} \n{\\dot h}_{d}(t)&=&{\\dot e}_{p}(t) \\nonumber \\\\\n&=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma T_c}+\\frac{1}{(\\gamma+c (T_c-T_h)) T_h}\\right) \\end{eqnarray}\n\tand \n\\begin{eqnarray} \n{\\dot H}_{d}(t)&=&{\\dot E}_{p}(t) \\nonumber \\\\ &=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\t\n\t\t\t\t\n\nLet now explore the energetics of the model system. When the Brownian particle along the reaction coordinate, the heat that is taken from the hot heat bath $Q_{h}$ is given as \n\\begin{eqnarray}\nQ_{h} &=&U_{0}+{fL_{0}\\over 2}\n \\end{eqnarray}\n\twhile \n\tthe rate of heat flow into the cold heat bath $Q_{h}$ can be found as \n\t\\begin{eqnarray}\nQ_{c} &=&U_{0}-{fL_{0}\\over 2}\n \\end{eqnarray}\n\twhich implies the work done is given by \n\t\\begin{eqnarray}\nW &=& Q_{h}-Q_{c}= fL_{0}.\n \\end{eqnarray}\n\nLet us now explore how the efficiency $\\eta$ and the coefficient of performance of the refrigerator $P_{ref}$ behave. When the engine acts as a heat engine, the efficiency is given by\n\\begin{eqnarray}\n\\eta= {W\\over Q_{h}}={fL_{0}\\over U_{0}+fL_{0}\/2}.\n\\end{eqnarray}\nAt quasistatic limit, plugging Eq. (76) into Eq. (84), one gets \n\\begin{eqnarray}\n\\eta= 1-{T_{c}\\over T_{h}}\n\\end{eqnarray}\nwhich is the efficiency of the Carnot heat engine. When the engine performs as a refrigerator, the coefficient of performance of the refrigerator $P_{ref}$ is given by \n\\begin{eqnarray}\nP_{ref}= { Q_{c}\\over W}={U_{0}-fL_{0}\/2 \\over fL_{0}}\n\\end{eqnarray}\nand at quasistatic limit, plugging Eq. (76) into Eq. (86), $P_{ref}$ approaches Carnot refrigerator\n\\begin{eqnarray}\nP_{ref}= {T_{c}\\over T_{h}-T_{c}}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\n \n\n\\section{ Multiplicative noise }\nMost of the previous works have focused on calculating the thermodynamic features of different model systems by considering additive noise. Most realistic\n systems such as neuron system can be also described by Langevin\nequations with multiplicative noise were in this case, the noise\namplitude varies spatially \\cite{mar12}. Considering multiplicative noise, \nthe intrinsic noise-induced ordering phase transition has been also studied in the work \\cite{mar10}. \nIn this work, we study how entropy, entropy production, and extraction rate depend on the strength of the background noise by solving the model exactly.\n\nFor the case where the temperature is position-dependent $T(x)= \\sqrt{D}|x|^{{-z \\over 2}}$, in the absence any external potential, the corresponding Fokker Planck equation is given as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({T'(x))\\over 2 \\gamma}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T(x)\\over \\gamma}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nThe probability current is given as \n\\begin{eqnarray}\nJ&=&-\\left({P(x,t)T'(x))\\over 2 \\gamma}\\right)-\\nonumber \\\\\n&&\\left({T(x)\\over \\gamma}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nThe solution for the probability distribution is well known \\cite{mulu1} and it is given by \n\\begin{eqnarray}\nP(x,t) = {|x|^{z \\over 2} e^{-{|x|^{z+2}\\over D(z+2)^2 t}}\\over \\sqrt{4\\pi D t}}.\n\\end{eqnarray}\n \nFrom Eqs. (42) and (43), one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int { \\gamma J^{2} \\over P T(x) } dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({J(T'(x))\\over 2 T(x)}\\right) dx. \n\\end{eqnarray}\n\n\n\nThe expression for the entropy production and extraction rates can be found by substituting Eqs. (87) and (88) into Eqs. (89) and (90).\n The plot ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ as a function of $t$ for parameter choice $\\tau=1$, $D=1.0$ and $z=-4.0$ is depicted in Figs. 6a and 6b. The figures depicts that ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ decrease as time increases and in long time limit, it approaches its stationary value ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. Only in the long time limit, $t\\to \\infty$, ${d S(t)\\over dt}=0$ since ${\\dot e}_{p}(t)= {\\dot h}_{d}(t)=0$. This can be intuitively comprehended on physical grounds. For isothermal case, in the long time limit, the system approaches stationary state and only at this particular state, $\\Delta h_d=0$, $\\Delta S=0$ or $\\Delta e_p=0$ (at stationary state). However when the particle operates at finite time, the system operates irreversibility and in this regime, the second law of thermodynamics states that $\\Delta S(t)>0$. As it can be seen from Fig. 6 that if the thermodynamic quantities are evaluated in the time interval between $t=0$ and any time $t$, always the inequality \n$\\Delta h_d(t)=h_d(t)-h_d(0)>0$, $\\Delta S(t)=S(t)-S(0)>0$ or $\\Delta e_p(t)=e_p(t)-e_p(0)>0$ holds true and as time progresses the change in this parameters increases. \nIn fact, in small $t$ regimes, ${\\dot e}_{p}(t)$ becomes much larger than ${\\dot h}_{d}(t)$ (see Figs. 6a and 6b) showing that the entropy production is higher (than entropy extraction) in the first few period of times. When time increases, more entropy will be extracted ${\\dot h}_{d}(t)>{\\dot e}_{p}(t)$. Over all, since the system produces enormous amount of entropy at initial time, in latter time or any time $t$, $\\Delta e_p(t)>\\Delta h _d(t)$ and hence $\\Delta S(t)>0$. \n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n11.eps}} \n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n12.eps}\n}\n\\caption{ (Color online) The plot ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ as a function of $t$ for parameter choice $\\tau=1$, $D=1.0$ and $z=-4.0$ is depicted in Figs. 6a and 6b, respectively. The figures depicts that ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ decrease as time increases and in long time limit, it approaches its stationary value ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. \n} \n\\label{fig:sub}\n\\end{figure}\n\n\n\n\\section{Summary and conclusion}\n\n\n\nThe influence of viscous friction on the thermodynamic properties of a Brownian particle that walks in overdamped and underdamped media is studied. The viscous friction is considered to vary either spatially or temporally. \nBy extending Seifert stochastic approach to underdamped and overdamped media, the general expressions for entropy production, free energy, and entropy extraction rates are derived. To explore \nthe non-equilibrium thermodynamic features of a Brownian particle that hops in medium where its viscosity depends on time, a Brownian particle that walks on a periodic isothermal medium (in the presence or absence of load) is considered. The analytical results depict that in the absence of load, the entropy production rate balances the entropy extraction rate which is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. It is shown that when a distinct temperature difference is not retained between the hot and cold baths, in absence of load, the entropy production still balances the entropy extraction rate revealing the system is reversible. When the external load is zero and \nwhen the viscous friction decreases in time, the entropy monotonously increases with time and saturates to a constant value as $t$ further steps up. The entropy production rate decreases in time and at steady state (in the presence of load), ${\\dot e}_{p}={\\dot h}_{d}>0$ which agrees with the results shown in the works \\cite{muuu177}. On the contrary, when the viscous friction increases in time, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. \n\n\n\nFor a system where the viscous friction of a medium tends to decrease as the temperature of the medium increases, the non-equilibrium thermodynamic features of the model system are explored. In this case, the load $f$ dictates the direction of the particle velocity. The steady-state velocity of the engine is positive when $f$ is smaller and the engine acts as a heat engine. In this regime the entropy production and extraction rates become nonzero. When $f$ steps up, the velocity of the particle steps down and at stall force, the entropy production rate balances the entropy extraction rate revealing the system is reversible at this particular choice of parameter. For large loads, the current is negative and the engine acts as a refrigerator. In this region the entropy production and extraction rates become nonzero. In the absence of load, the entropy production and extraction rates become larger than zero as long as a distinct temperature difference is retained between the hot and cold baths. We further explore the thermodynamic features of such systems by considering a multiplicative noise wherein\ncase the noise amplitude varies spatially. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn conclusion, in this work, we derive several thermodynamic relations to a Brownian particle moving in underdamped and overdamped media by considering viscous friction that varies temporally and spatially. We believe that the present theoretical work serves as a basic tool to understand the nonequilibrium thermodynamics.\n\n\n \n\\section*{Acknowledgment}\nI would like to thank Blaynesh Bezabih and Mulu Zebene for their\nconstant encouragement. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nTwo of the mainstreams of Noncommutative Geometry \nconcentrate around the notions of a\nprojective module \\cite{conbook,coq} and \nof a quantum group \\cite{frt,cmp}. Quite\nrecently (see \\cite{bmq,md1,mp}), the concept of a quantum principal \nbundle was systematically developed with quantum groups (Hopf algebras) in the\nrole of structure groups.\nHence, since both projective modules and quantum principal bundles \nserve as starting\npoints for quantum geometric considerations, \nthe conceptual framework provided by the notion of a \nquantum principal bundle has a good chance of unifying \nthose two branches of Noncommutative Geometry. \n\nIn the classical differential geometry,\nit is hard to overestimate the interplay between Lie groups and $K$-theory.\nTherefore, it is natural to expect that establishing a similar\ninteraction in the noncommutative case is necessary for better understanding of\nquantum geometry. \nIt is already known that \nthe classification of quantum principal bundles over manifolds depends only on\n the classical subgroups of quantum \nstructure groups \\cite{md2}. This leads to \nthe following questions: Is the classification of (general)\nquantum principal bundles over\nnoncommutative spaces richer then the classification of classical-group bundles\nover noncommutative spaces? Is there \na bimodule that can be obtained \nas the bimodule of intertwiners (noncommutative analogue of\nequivariant vector valued functions on a total space) only from a bundle with a \nnoncommutative structure Hopf algebra? More generally, when does a \ndeformation of a group into a quantum group entail essential consequences in the\ngeometry (e.g., in the classification of bundles or in the Yang--Mills theory)?\n\nSince this article is to a great extent a\nfollow-up of \\cite {bmq}, much of the mathematical\nand physical motivations listed there can also\nbe considered as motivations for this work, and so\nwill not be repeated here. Let us just emphasize\nthat our main purpose is to specify and analyze a class of\nconnections on quantum principal\nbundles (called strong connections) that enjoy\nsome additional properties making\nthem more like their classical counterparts, and\n(taking advantage of the notion of a strong connection) to discuss a link\nbetween the two approaches to noncommutative\ndifferential geometry based on quantum principal bundles and\nprojective modules. The study of the precise relationship between those two\napproaches is thought of as a move towards answering the questions \nmentioned above.\n\nWe begin this article by fixing the notation and recalling the fundamentals of\n quantum bundles and Yang--Mills theory on projective modules. \nIn the first section, in addition\nto this vocabulary review, we also study the definition\nof a quantum principal bundle using snake diagrams (see Remark~\\ref{snake} \nand Proposition~\\ref{galois}). Taking advantage of Remark~\\ref{snake}, we prove \n(Corollary~\\ref{dcor}) that the fundamental vector field compatibility condition\n(see Point~3 in Definition~\\ref{qpbdef}) implies its stronger version. \n(The latter version of\nthe fundamental vector field compatibility condition was assumed in \nExample~4.11\\bmq .)\n\nThe formalism \nused in this paper is a generalization of the\ncorresponding formalism used in classical\ndifferential geometry. The calculations showing this, \nthough often very instructive, are straightforward\n and we will not fully elaborate on\nthat fact later on. Differential geometry on\nquantum principal bundles is still in the process\nof being born --- the umbilical cord has hardly\nbeen cut yet --- and it seems premature at\nthis point to make precise categorical\nstatements establishing the relationship\nbetween classical and quantum differential\ngeometries. As Yu.I.Manin mentioned in a\nsimilar context (see p.86 in \\cite{ma}), ``Here,\none should not act too hastily since even in\nsupergeometry this program was started only\nrecently and revealed both rich content and\nsome puzzling new phenomena.\"\n\nIn Section~2, \nwe define and provide examples of strong connections.\nProposition~\\ref{scprop}\nallows one to interpret strong connections on trivial quantum bundles\nas those induced from the base space, and to produce examples of strong\nconnections in the case of trivial quantum bundles with the universal calculus. \nIn the Introduction to the preliminary version of \\bmq ,\none can read regarding inducing connection forms from the\nbase space that ``...in the general non-commutative or quantum case there\nwould appear to be slightly more possibilities...\" than in the classical case.\nExamples of connections that are not strong (and thus realize the\n just mentioned ``quantum possibility\") are supplied as well. More precisely,\nwe construct both strong and non-strong connections on a very simple (yet\n rich enough) example of a `discrete bundle' and on a quantum version of \nthe Hopf fibration $S\\sp2\\ra RP\\sp2$.\n(As a byproduct of our considerations we obtain a \\mbox{$q$-deformation}\n of~$RP\\sp2$.)\nIt might be worthwhile to note that the latter construction does not employ\nthe trivial-bundle or the quantum-group-quotient techniques. We end this section\nwith presenting an example of a non-strong connection in the set-up of a strict\nmonoidal category dual to the category of sets with Cartesian product.\n\nIn the subsequent section, we describe the action of (global) \ngauge transformations on the space of connections on a bundle with\nthe universal differential calculus, and show that this action \npreserves the strongness of a connection.\n\nIn Section~4, we use the notion of a strong connection to justify the\ndefinition of a global curvature form. (We need to assume that a connection \nis strong if we want to show that its curvature form has\ncertain properties that classical curvature\nforms possess automatically; e.g., the usual relation to the square of the exterior\ncovariant derivative.) \n\nIn Section~5, we present a link between gauge theory on a\nquantum principal bundle and Yang--Mills theory\non a projective module. First we show how, in the case of a free module,\nto incorporate the Yang--Mills action constructed in \\cite{conri,ri}\ninto the quantum bundle picture. Then, to obtain a\nHermitian metric compatible connection\non a free module from a strong $U\\sb q(2)$-connection on a\ntrivial quantum principal bundle, we mimic the\n classical geometry formula which permits one to\ndetermine the values of a connection form on a Hopf\nalgebra of smooth functions on a matrix Lie group\nby knowing its values on the matrix of\ngenerators. (Note that usually one thinks of a\nconnection form as a map sending smooth\n vector fields to elements of a Lie algebra, but\nwe can also view it as a map from the Hopf\nalgebra of smooth functions on a Lie group into\nthe space of smooth \\mbox{1-forms}.) It turns out that the connections\ncompatible with a particular $q$-dependent Hermitian structure can be \nidentified with the strong $U\\sb q(2)$-connections that satisfy certain condition.\nWe close this section by concluding that, in the setting under consideration,\nthe moduli space of critical points of $U\\sb q(2)$ and $U(2)$--Yang--Mills\ntheory coincide.\nThus, at least in this case, the $q$-deformation of the structure group alone\nhas no essential\nbearing on the Yang--Mills theory. This seems to bring us a step closer \nto answering the question posed at the end of the first paragraph: \nOne should expect geometrically interesting effects of the noncommutativity \nof a Hopf algebra in Yang--Mills theory\nonly for non-trivial bundles (comodule algebras that are not crossed product \nalgebras) or non-strong connections. \n\nFinally, in the Appendix, we examine the\nadvantages of adding a twist to the definition\nof a quantum associated bundle\nformulated in \\bmq\\ and point out the\npossibility of using the axiomatic definition\nof a frame bundle to try to define its\nnoncommutative analogue (cf.\\\nSection~5.1 in \\cite{bmq}).\n\n\\parindent0pt\n\\section{Preliminaries}\n\nThe notation used throughout this article is quite\nstandard and \nnot much different from that\nof \\bmq . Nevertheless, to eschew any possible misunderstanding or\nconfusion, we enclose a table of basic notations:\n\n\\bi\n\n\\item [{$[\\! ...]_{X}$}] an equivalence class\ndefined by $X$\n\n\\item [$\\delta_{m,n}$] equals 1 iff $m=n$, and 0\notherwise (Kronecker symbol)\n\n\\item [$\\frak g\\;\\;\\;\\:$] Lie algebra of a Lie group $G$\n\n\\item [$k\\;\\;\\;\\,$] field of characteristic zero \n(except for Proposition~\\ref{spera})\n\n\\item [$\\ot\\;\\;\\;$] tensor product over $k$\n\n\\item [$\\tau\\;\\;\\;\\,$] flip ($\\tau (u\\ot v):= v\\ot u$)\n\n\\item [$\\Omega\\sp{1}\\! A\\;$] first order universal\ndifferential calculus ($\\Omega\\sp{1}\\! A:=\\kr\\,\nm_{A}$ , $da:=1\\ot a-a\\ot 1 $)\n\n\\item [$\\Omega A\\;\\,$] differential envelope of $A$ \n\n\\item [$\\Omega (\\! A)$] differential algebra over $A$ (i.e.~a quotient of $\\hO A$\nby some differential ideal)\n\n\\item [$\\; m_{X}\\;$] multiplication on $X$, or in $X$\n(We will simply write $m$ for the multiplication in a Hopf algebra.)\n\n\\item [$\\Delta\\;\\;\\;$] comultiplication: $\\de (a)=a\\1\\ot a\\2$\n(Sweedler sigma notation with suppressed summation sign,\ncf.\\ Section~1.2 in~\\cite{swe})\n\n\\item [$\\de_{n}\\;$] comultiplication applied $n$\ntimes (Due to coassociativity we do not have\nto remember where \\de\\ is put in consecutive\ntensor products; $\\de\\sb na=a\\1\\ot\\cdots\\ot a\\sb{(n+1)}\\,$.)\n\n\\item [$\\eta_{Y}\\;\\;$] unit map of an algebra $Y$ (often suppressed)\n\n\\item [$\\epsilon\\;\\;\\;\\,$] counit (unless otherwise\nobvious from the context)\n\n\\item [$S\\;\\;\\,$ ] antipode, i.e.~$a\\1 S(a\\2)=\\epsilon (a)=S(a\\1)a\\2\\,$\n\n\\item [$A\\sp{op}\\;$] algebra identical with algebra $A$ as a vector space\nbut with the multiplication defined by $m\\sb{A\\sp{op}}=m\\sb A\\ci\\tau$\n\n\\item [$\\rho_{R}\\;\\;$] right coaction: $\\rho_{R}(x) = x\\0\\ot x\\1 $\n(Sweedler sigma notation for comodules with \nsuppressed summation sign, cf.\\ p.32--3 in~\\cite{swe})\n\\item [$\\dr\\;\\,$] right coaction on the\n`total space' of a quantum principal bundle\n\n\\item [$\\Delta_{\\cal R}\\;\\,$] right coaction on a right-covariant\ndifferential algebra of\nthe `total space' given by $\\mbox{$\\forall\\, m\\in{\\Bbb N}:\\;$}\n\\mbox{$\\dsr\n(p\\sb 0dp\\sb 1\\cdots dp\\sb m)$}=\\mbox{$(p\\sb 0)\\0d(p\\sb 1)\\0\\cdots\nd(p\\sb m)\\0\\ot (p\\sb 0)\\1\\cdots (p\\sb m)\\1$}$, where for\nany \\mbox{$n\\in\\{ 0,\\cdots ,m\\}$}, \\mbox{$(p\\sb n)\\0\\te (p\\sb n)\\1\n=\\dr p\\sb n$} (A differential algebra is right-covariant iff\n \\dsr\\ determined by\nthe above formula is well-defined; cf.\\ (21) and Section~4.2 in~\\bmq .)\n\n\\item [$ad_{R}\\;$] $:= (id\\ot m)\\ci (id\\ot S\\ot id)\\ci\n(\\tau\\ot id)\\ci\\de_{2}$ (right adjoint coaction;\n$ad\\sb Ra=a\\2\\te S(\\! a\\1\\! )a\\3\\,$)\n\n\\item [$P\\sp{co A}$] the space of right coinvariants ($P\\sp{co A}:=\\{p\\in P |\n\\dr p=p\\te 1\\}$)\n\n\\item [$*_{\\rho}\\;\\;\\,$] convolution: $\\forall\\, f\\!\\in\\!\n\\mbox{Hom}\\sb k(Q,X),\ng\\!\\in\\! \\mbox{Hom}\\sb k(A,Y): f*_{\\rho}g\n= \\m\\ci (f\\te g)\\ci\\rho_{R}$, where $(Q,\\rho\\sb R)$ is a right\n$A$-comodule and $\\m :X\\te Y\\ra Z$ is a multiplication map\n(If $\\rho_{R}$ equals $\\dsr$, \\dr\\ or \\de ,\nwe will use $*_{\\cal R}$, $*_{R}$ or $*$ respectively\nto denote the corresponding convolution.)\n\n\\item [$f\\sp{-1}\\;$] unless otherwise obvious from\nthe context, convolution inverse of $f$, i.e.~ \\\\\n $f\\sp{-1}(a\\1)f(a\\2)=(f\\sp{-1}\\! *\\! f)(a)\n=\\epsilon (a)=(f\\! *\\! f\\sp{-1})(a)=f(a\\1)f\\sp{-1}(a\\2)\\,$\\\\\n(In general, one has:\n $x\\sb{(0)}\\ot\\cdots\\ot x\\sb{(n)}\\ot\nf(x\\sb{(n+1)})g(x\\sb{(n+2)})\\ot\nx\\sb{(n+3)}\\ot\\cdots\\ot x\\sb{(m)}\\\\\n=x\\sb{(0)}\\ot\\cdots\\ot x\\sb{(n)}\\ot\n(f*g)(x\\sb{(n+1)})\\ot x\\sb{(n+2)}\\ot\\cdots\\ot\nx\\sb{(m-1)}\\;$ and\\\\\n${\\cal F}(x\\0)g(x\\1)\\ot x\\2\\ot\\cdots\\ot x\\sb{(m)}\n=({\\cal F}*\\sb\\rho g)(x\\0)\\ot x\\1\\ot\\cdots\\ot x\\sb{(m-1)}$.)\n\n\\ei\n\nAll algebras are assumed to be unital and associative.\nNow, let us recall the basic notions and\nconstructions of \\bmq\\ necessary to\nestablish the language used in this paper.\n\\bde [4.9 \\bmq ]\\label{qpbdef}\nLet $P$ be an algebra over a field $k$, $A$ a\nHopf algebra over the same field, $N_{P}\n\\inc\\Omega\\sp{1}\\! P$ a $P$-bimodule\ndefining the first order differential calculus\n\\op , $M_{A} \\inc\\ker\\,\\epsilon$ an $ad\\sb R$-invariant right\nideal defining the bicovariant differential calculus\n\\oa , and\n\\[\n\\dr : P\\lra P\\ot A\n\\]\nan algebra homomorphism making $P$ a right $A$-comodule\n algebra.\\\\ Then $(P,A,\\dr ,N_{P},M_{A})$ is called a quantum\nprincipal bundle iff:\n\n\\be\n\n\\item $\\can :P\\ot P\\ni t\\stackrel{\\mbox{\\em\\scriptsize def.}}{\\longmapsto}\n(m\\te id)\\cc (id\\te\\dr )\\, t\\in P\\ot A$ is a surjection (freeness condition),\n\n\\item $\\dsr (N_{P})\\inc N_{P}\\ot A$\n(right covariance of the differential\nstructure),\n\n\\item $\\can (N_{P})\\inc P\\ot M_{A}$\n (fundamental vector field compatibility condition),\n\n\\item $\\ker\\,\\mbox{\\em\\sf T} \\inc P\\hO\\sp1(P\\sp{co A}) P$ (exactness condition), \nwhere \n$\\hO\\sp1(P\\sp{co A}) := \\Omega\\sp 1 P\\sp{co A}\/(N\\sb\n P\\cap\\Omega\\sp 1 P\\sp{co A})$\n\\[\n\\mbox{and $\\;$ \\em\\sf T}: \\op\\ni{[\\alpha ]}\n_{N_{P}}\\stackrel{\\mbox{\\em\\scriptsize def.}}{\\longmapsto}\n\\llp(id\\te\\pi_{A})\\cc \\can\\lrp\\,\\alpha \\in P\\ot (\\ker\\,\\epsilon\\, \/\\, M_{A})\n\\]\n (the map $\\pi\\sb A :\n\\ker\\,\\epsilon\\ra\\ker\\,\\epsilon\\, \/\\, M_{A}$\nis the canonical projection, and\n$\\alpha\\!\\in\\!\\ker\\, m\\sb P$).\n \\ee\n\\ede\nFor simplicity, as well as to emphasize the\nanalogy with the classical situation, a quantum\nprincipal bundle is often denoted by\n$P(B,A)$, where $B:=P\\sp{co A}$ is the `base space' of the bundle. \nThe map {\\sf T} (denoted by\n$\\widetilde{~}\\sb{N\\sb P}$\nin \\bmq ) can\nbe more explicitly described by the formula\n\\[\n\\hsp{31mm}\\T (pdq)=pq\\0\\ot\n[q\\1 ]\\sb{M\\sb A} - pq\\ot 1 .\\hsp{44mm}\n\\mbox{(cf.\\ (24) in \\bmq )}\n\\]\n\n\\bre\\label{snake}{\\em\nLet $\\;\\T\\!\\sb U:\\mbox{Ker}\\, m\\sb P\\ra P\\ot\\mbox{Ker}\\,\\he\\;$ and\n$\\;\\T\\!\\sb{NM}:N\\sb P\\ra P\\ot M\\sb A\\;$ be the appropriate restrictions \nof \\cnl . It is straightforward to check that the following\ndiagrams are commutative diagrams (of left \\mbox{$P$-modules}) with exact rows\nand columns:\\\\\n\n\\beq\\label{d1}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\n\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb{U}&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb R\n&\\mapright{}&0&&&\\phantom{.}\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&\\mbox{Ker}\\, m\\sb P&\\mapright{}\n&P\\ot P&\\mapright{m\\sb P}&P&\\mapright{}&0\\cr\n&&\\mapdown{\\T\\!\\sb{U}}&&\\mapdown{\\T\\!\\sb R}&&\\mapdown{id}\\cr\n0&\\mapright{}&P\\ot\\mbox{Ker}\\,\\he\n&\\mapright{}&P\\ot A&\\mapright{m\\sb P\\ci (id\\ot\\he)}\n&P&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&\\mbox{Coker}\\,\\T\\!\\sb{U}&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb R\n&\\mapright{}&0&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\n\\beq\\label{d2}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb{NM}&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb U\n&\\mapright{}&\\mbox{Ker}\\,\\T\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&N\\sb P&\\mapright{}\n&\\Omega\\sp 1P&\\mapright{\\pi\\sb P}&\\op&\\mapright{}&0\\cr\n&&\\mapdown{\\T\\!\\sb{NM}}&&\\mapdown{\\T\\!\\sb U}&&\\mapdown{\\T}\\cr\n0&\\mapright{}&P\\ot M\\sb A&\\mapright{}&P\\ot\\mbox{Ker}\\,\\he&\\mapright{id\\ot\\pi\\sb A}\n&P\\ot(\\mbox{Ker}\\,\\he\\, \/\\, M\\sb A)&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&\\mbox{Coker}\\,\\T\\!\\sb{NM}&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb U\n&\\mapright{}&\\mbox{Coker}\\,\\T&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\nApplying the Snake Lemma (e.g., see Section~1.2 in \\cite{aki}) to both diagrams,\nwe obtain the following two exact sequences:\n\\beq\\label{s1}\n0\\lra\\mbox{Ker}\\,\\T\\!\\sb U\\lra\\mbox{Ker}\\,\\T\\!\\sb R\\lra 0\n\\lra\\mbox{Coker}\\,\\T\\!\\sb U\\lra\\mbox{Coker}\\,\\T\\!\\sb R\\lra 0\n\\phantom{vhvhvjhvhvsccsj......}\n\\eeq\n\\beq\\label{s2}\n0\\lra\\mbox{Ker}\\,\\T\\!\\sb{NM}\\lra\\mbox{Ker}\\,\\T\\!\\sb U\\lra\\mbox{Ker}\\,\\T \n\\lra\\mbox{Coker}\\,\\T\\!\\sb{NM}\\lra\\mbox{Coker}\\,\\T\\!\\sb U\\lra\\mbox{Coker}\\,\\T\\lra 0\n\\eeq\\ \\\\\nObserve that the freeness condition means exactly that $\\mbox{Coker}\\,\\T\\!\\sb R=0$,\nwhich, by (\\ref{s1}), is equivalent to $\\mbox{Coker}\\,\\T\\!\\sb U=0$ (see (33) in\n\\bmq ). Note also that $\\mbox{Ker}\\,\\T\\!\\sb{NM}=N\\sb P\\cap\\mbox{Ker}\\,\\T\\!\\sb U$.\n}\\hfill{$\\Diamond$}\\ere\n\n\\bco\\label{dcor}\n$\\!$Let $(P,A,\\dr ,N_{P},M_{A})$ be a quantum\nprincipal bundle. Then \\mbox{$\\can(\\! N\\sb P\\! )\\! =\\! P\\te M\\sb A$} \n(cf.~Example~4.11 in \\bmq\\ and the discussion below it).\n\\eco\n\n{\\it Proof.} From the exactness condition, we know that \n$\\mbox{Ker}\\,\\T=\\pi\\sb P(P\\Omega\\sp 1\\! B.P)$. On the other hand, since\n$P\\Omega\\sp 1\\! B.P\\inc\\mbox{Ker}\\,\\T\\!\\sb U$ and the map\n$\\pi\\sb U:\\mbox{Ker}\\,\\T\\!\\sb U\\ra\\mbox{Ker}\\,\\T$ in (\\ref{s2}) is a restriction\nof $\\pi\\sb P$ to $\\mbox{Ker}\\,\\T\\!\\sb U$, we can conclude that $\\pi\\sb U$ is\nsurjective. Consequently, by the freeness condition and the exactness of (\\ref{s2}),\n$\\mbox{Coker}\\,\\T\\!\\sb{NM}=0$, i.e.~\\cnl$(\\! N\\sb P\\! )=P\\te M\\sb A$.\n\\epf\\\\\n\nThe above corollary makes the following definition of a trivial quantum principal\nbundle equivalent to the definition proposed in Example~4.11 in \\bmq .\n\n\\bde\\label{trbu}\nA quantum principal bundle $(P,A,\\dr ,N\\sb P,M\\sb A)$ is called trivial iff there\nexists a convolution invertible map\n(trivialization) $\\Phi\\in \\mbox{\\em Hom}\\sb k(A,P)$ such that\n\\beq\\label{trcov}\n\\dr\\ci\\Phi = (\\Phi\\ot id)\\ci\\de\n\\eeq\n(i.e.~$\\Phi$ is right-covariant) and $\\Phi (1) = 1$. In such a case, $P$ is also\ncalled a crossed product or cleft extension (see p.273 in \\cite{sch2}).\n\\ede\n\\bde [1.1 \\cite{sch2}]\\label{galois}\nLet $P$ be a right $A$-comodule algebra and $B$ be the algebra of all right\ncoinvariants. The comodule $P$ is called an $A$-Galois extension iff the canonical\nleft \\mbox{$P$-algebra} and right $A$-coalgebra map\n\\[\n\\mbox{\\em\\sf T}\\!\\sb B:=(m\\sb P\\te id)\\ci(id\\te\\sb B\\dr):\nP\\ot\\sb B P\\ni p\\ot\\sb Bq\\longmapsto pq\\0\\ot q\\1\\in P\\ot A\n\\]\nis bijective. \n\\ede\n\\bpr\\label{diagram}{\\em\\footnote{\nThis proposition is implicitly proved in \\cite{tbt} (see Lemma~3.2 and the text\nabove it). The diagrammatic proof presented here was created during the author's\ndiscussion with Markus Pflaum.}}\nLet $P$, $A$ and $B$ be as above. A comodule algebra $P$ is an $A$-Galois extension\nif and only if $P(B,A)$ is a quantum principal bundle with the universal calculus.\n\\epr\n{\\it Proof.} Consider the following commutative diagram (of left $P$-modules)\nwith exact rows and columns:\\\\\n\n\\beq\\label{d3}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\n\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&P\\hO\\sp1\\! B.P&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb R\n&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb B&&&\\phantom{.}\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&P\\hO\\sp1\\! B.P&\\mapright{}\n&P\\ot P&\\mapright{}&P\\ot\\sb B P&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{\\T\\!\\sb R}&&\\mapdown{\\T\\!\\sb B}\\cr\n0&\\mapright{}&0\n&\\mapright{}&P\\ot A&\\mapright{id}\n&P\\ot A&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&0&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb R\n&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb B&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\nAgain, we can apply the Snake Lemma to obtain an exact sequence\n\\beq\\label{s3}\n0\\lra P\\hO\\sp1\\! B.P\\lra\\mbox{Ker}\\,\\T\\!\\sb R\\lra\\mbox{Ker}\\,\\T\\!\\sb B\\lra 0\\lra\n\\mbox{Coker}\\,\\T\\!\\sb R\\lra\\mbox{Coker}\\,\\T\\!\\sb B\\lra 0\\; .\n\\eeq\nAssume first that $P$ is an $A$-Galois extension. Then \n$\\mbox{Ker}\\,\\T\\!\\sb B=0=\\mbox{Coker}\\,\\T\\!\\sb B$, and, from the exactness of \n(\\ref{s3}), we can infer that $\\mbox{Coker}\\,\\T\\!\\sb R=0$ (freeness condition)\nand $\\mbox{Ker}\\,\\T\\!\\sb R=P\\hO\\sp1\\! B.P\\,$. \nOn the other hand, by the exactness of (\\ref{s1}),\nwe have $\\mbox{Ker}\\,\\T\\!\\sb U=\\mbox{Ker}\\,\\T\\!\\sb R\\,$. Hence\nthe exactness condition follows, and we can conclude that $P(B,A)$ is a quantum\nprincipal bundle with the universal calculus.\n\nConversely, assume that $P(B,A)$ is a quantum\nprincipal bundle with the universal calculus. Then \n$\\mbox{Ker}\\,\\T\\!\\sb R=\\mbox{Ker}\\,\\T\\!\\sb U=P\\hO\\sp1\\! B.P$\nand $\\mbox{Coker}\\,\\T\\!\\sb R=0\\,$. Consequently, again due to the exactness of\n(\\ref{s3}), we have that $\\mbox{Ker}\\,\\T\\!\\sb B=0=\\mbox{Coker}\\,\\T\\!\\sb B\\,$,\ni.e.~$P$ is an $A$-Galois extension.\n\\epf\n\n\\bde [\\bmq ]\\label{condef}\nA left $P$-module projection $\\Pi$ on \\op\\ is called a connection on\\linebreak\n $P(B,A)$~iff\n\\be\n\\item $\\ker\\,\\Pi = \\Omega\\sp{1}_{hor}(P)$\n($\\,\\mbox{\\em Im}\\,\\Pi$ is called the space of vertical\n forms),\n\\item $\\dsr\\ci\\Pi = (\\Pi\\ot id)\\ci\\dsr$ (right covariance).\n\\ee\\ede \n\nDue to Proposition 4.10 in \\bmq , a connection\nform can be defined in the following way:\n\n\\bde \\label{confordef}\nA $k$-homomorphism $\\omega : A\\ra\\op$ is called a\nconnection form on $P(B,A)$ iff it satisfies the\nfollowing properties:\n\\be\n\\item $\\omega (k\\oplus M_{A}) = 0$ (compatibility with\nthe differential structure),\n\\item $\\mbox{\\em\\sf T}\n\\ci\\omega = (id\\ot\\pi_{A})\\ci \\llp 1\\ot\n(id - \\epsilon)\\lrp$ (fundamental vector field condition),\n\\item $\\dsr\\ci\\omega = (\\omega\\ot id)\\ci\nad_{R}$ (right adjoint covariance).\n\\ee\\ede\n\nFor every $P(B,A)$, there is a one-to-one\ncorrespondence between connections and\nconnection forms. In particular, the connection\n$\\Pi\\sp{\\omega}$ associated to\na connection form $\\omega$ is given by the\nformula: $\\Pi\\sp{\\omega}=m\\sb{\\Omega\\sp 1(P)}\\ci\n(id\\ot\\omega )\\ci\\T$ ((47) in \\bmq ).\nSince $\\Pi\\sp{\\omega}$\nis a left $P$-module homomorphism, to calculate\n$\\Pi\\sp{\\omega}$ it suffices to\nknow its values on exact forms, and on exact forms\n(47)\\bmq\\ simplifies to\n\\beq\\label{confor}\n\\Pi\\sp\\omega\\ci d = id*_{R}\\omega .\n\\eeq\n\nThe following four definitions are based on Appendix~A\\bmq .\n\n\\bde\nLet $\\Omega (P)$ be any differential algebra\nhaving \\op\\ as in Definition~\\ref{qpbdef}. For all $n\\in\\Bbb{N}$, the space \n$\\llp P\\ob P\\lrp\\sp{n}$ is called the space of horizontal\n$n$-forms and is denoted by $\\Omega\\sp{n}_{hor}(P)$.\nThe space of horizontal \\mbox{0-forms} is identified with~$P$.\n\\ede\n\n\\bde\nLet $\\Omega (B)$ be the differential algebra obtained by the restriction of\n$\\Omega (P)$. For all $n\\in\\Bbb{N}$, the space $\\Omega\\sp{n}\\!\n(\\! B)P$ is called the\nspace of strongly horizontal $n$-forms and is denoted\nby $\\Omega\\sp{n}_{shor}(P)$. The space of strongly horizontal \\mbox{0-forms}\nis identified with $P$.\n\\ede\n\nNote that in the classical case $\\Omega\\sp{*}_{hor}(P)$\nand $\\Omega\\sp{*}_{shor}(P)$ coincide.\n\n\\bde\\label{tfdef}\n Let $\\, (V,\\rho_{R})\\,$ be a right $A\\sp{op}$-comodule algebra \n(see Remark~\\ref{bialgebra}). Then\\linebreak \n\\mbox{$\\phi\\in \\mbox{\\em Hom}_{k}\\llp V,\\Omega (P)\\lrp$}\n is called a pseudotensorial form on $P$ iff\n\\[\n\\dsr\\ci\\phi = (\\phi\\ot id)\\ci\\rho_{R}\\, .\n\\]\nA pseudotensorial form taking values in $\\Omega\\sp{*}_{hor}(P)$ \n(in $\\Omega\\sp{*}_{shor}(P)$) is called a tensorial (strongly tensorial)\n form on $P$. The space of all pseudotensorial,\ntensorial and strongly tensorial \\mbox{$n$-forms} ($n\\geq 0$) will be denoted by\n $PT_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$,\n$T_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$ and\n $ST_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$ respectively.\n\\ede\n\n\\bde [(68)\\bmq ]\\label{Ddef}\nLet $\\Pi$ be a connection on $P$. The $k$-homomorphism\n$D$ from $\\Omega\\sp{*}\\! P$ to $\\Omega_{hor}\\sp{*+1}\\! P$\ngiven by\n\\beq\\label{689}\nD : p\\sb{0}dp\\sb{1}\\cdots dp\\sb{n}\\longmapsto\n(id-\\Pi )(dp\\sb{0})\\cdots (id-\\Pi )(dp\\sb{n})\\, ,\n\\eeq\nwhere $n\\geq 0\\,$, is called the exterior covariant derivative associated to~$\\Pi$.\n\\ede \n\nTo complete this vocabulary review, we recall some basic definitions used\nin the Yang--Mills theory on projective modules. We choose here right rather\nthan left modules, but one should bear in mind that the formulation of this\nformalism for left modules is analogous.\n\n\\bpr[cf.~p.369 in \\cite{ms}]\\label{spera}\nLet $\\cal B$ be an associative unital algebra over a commutative ring $k$.\nLet $\\cal L\\sb k$ be a $k$-Lie subalgebra of the space of all $k$-derivations of\n$\\cal B$, and let $\\cal E$ be any right $\\cal B$-module admitting a connection. \nIf $\\hO(\\! \\cal B)$ is a differential graded subalgebra of \n\\mbox{$\\cal B\\oplus\\bigoplus\\sb{n\\geq 1}\\sp\\infty \n\\mbox{\\em Hom}\\sb k(\\mbox{\\large$\\wedge$}\\sp n\\cal L\\sb k,\\, \\cal B)$} \nwith the differential given by (see the first section in~\\cite{dkm}):\n\\bea\n(d\\ha)(X\\sb 0,X\\sb 1,\\cdots ,X\\sb n)\\!\\!\\!\n&=&\\!\\!\\!\\!\\sum\\sb{0\\leq i\\leq n}(-)\\sp iX\\sb i\\ha\n(X\\sb 0,\\cdots,X\\sb{i-1},X\\sb{i+1},\\cdots ,X\\sb n)\\\\\n&+&\\!\\!\\!\\!\\!\\!\\sum\\sb{0\\leq r\\,$}),\nlet $\\cal I\\sb0$ be a two-sided $*$-ideal of \\pf\\ generated by\n\\begin{eqnarray*}\n&&\\left\\{x\\sb m-x\\sb m\\sp*\\, ,\\;\\mbox{$\\sum\\sb j$}a\\sb jx\\sb j\\sp2-r\\sp2\\right\\},\\;\nm,j\\in\\{1,2,3\\}\\, ,\\; a\\sb j,r\\in\\Bbb R\\, ,\\; a\\sb j,r>0\\, ,\n\\end{eqnarray*}\nand let $\\cal I\\sb1$ be a two-sided $*$-ideal of \\pf\\ generated by\n\\begin{eqnarray*}\n&&\\mbox{\\large\\{}x\\sb m-x\\sb m\\sp*\\, ,\\; x\\sb1\\sp2+x\\sb2\\sp2+\n2\\mbox{$\\frac{1+q\\sp4}{(1+q\\sp2)\\sp2}$}x\\sb3\\sp2-1\\, ,\\; \nx\\sb1x\\sb2-x\\sb2x\\sb1-2i\\mbox{$\\frac{1-q\\sp4}{(1+q\\sp2)\\sp2}$}x\\sb3\\sp2\\, ,\\\\ \n&&\\ \\; x\\sb1x\\sb3-\\mbox{$\\frac{q\\sp{-2}+q\\sp2}{2}$}x\\sb3x\\sb1-\ni\\mbox{$\\frac{q\\sp{2}-q\\sp{-2}}{2}$}x\\sb3x\\sb2\\, ,\\;\nx\\sb2x\\sb3-\\mbox{$\\frac{q\\sp{-2}+q\\sp2}{2}$}x\\sb3x\\sb2+\ni\\mbox{$\\frac{q\\sp{2}-q\\sp{-2}}{2}$}x\\sb3x\\sb1\\mbox{\\large\\}}\\, ,\\\\\n&&\\ \\; m\\in\\{1,2,3\\}\\, ,\\; q\\in\\Bbb R\\, ,\\; 1\\geq |q|>0\\, .\n\\end{eqnarray*}\n(The second generator of $\\cal I\\sb1$ resembles the right-hand side of the formula\n(164) in \\cite{sm} that describes the metric on the $q$-Minkowski space discussed\nin Section~7.2 of~\\cite{sm}.)\nThe algebras $\\pf \/\\cal I\\sb0$ and $\\pf \/\\cal I\\sb1$ can be\nregarded as noncommutative two-spheres. Indeed, $\\pf \/\\cal I\\sb0$ is \n`the most noncommutative two-sphere', and $\\pf \/\\cal I\\sb1$ corresponds to\nthe equator sphere ($c=\\infty$) given by (7b) in \\cite{po} \n(see Remark~\\ref{equator}). Obviously, for $q=\\pm 1$, the algebra \n$\\pf \/\\cal I\\sb1$ corresponds to the usual $S\\sp2$. Since both $\\pf \/\\cal I\\sb0$\nand $\\pf \/\\cal I\\sb1$ can be used in the same way to construct connections\non a noncommutative Hopf fibration, we denote, for the sake of brevity,\n$\\pf \/\\cal I\\sb\\nu$ by \\pn , where $\\nu\\in\\{0,1\\}$. We also put\n$x\\sb{0j}=[x\\sb j]\\sb{\\cal I\\sb0}\\,$, $x\\sb{1j}=[x\\sb j]\\sb{\\cal I\\sb1}\\,$,\n\\mbox{$a\\sb{0j}=a\\sb j$}, $j\\in\\{1,2,3\\}\\, ,\\; \na\\sb{11}=1\\, ,\\, a\\sb{12}=1\\, ,\\, a\\sb{13}=2\\frac{1+q\\sp4}{(1+q\\sp2)\\sp2}\\, ,\\,\nr\\sb0=r\\, ,\\, r\\sb1=1\\,$, and thus define the coefficients \n$\\{a\\sb{\\nu j}\\, ,\\, r\\sb\\nu\\}\\sb{\\nu\\in\\{0,1\\},\\, j\\in\\{1,2,3\\}}\\,$.\nUnless stated otherwise, all the following statements of this example will be\nvalid for any of the two values of $\\nu$. The proposition below allows one\nto turn \\pn\\ into a right $A$-comodule algebra, where \n$A=\\mbox{Map}(\\Bbb Z\\sb2,\\Bbb C)$.\n\n\\bpr\\label{coaction}\nLet \\pf\\ and $A$ be as above. Also, let \\dr\\ be a coaction of $A$ on \\pf\\ \nmaking it a right $A$-comodule algebra. If \n$\\dr\\ci *=(*\\ot\\overline{\\phantom{a}})\\ci\\dr\\, $, where $\\overline{\\phantom{a}}$\ndenotes the complex conjugation, and\n\\beq\\label{co}\n\\dr :\\pf\\ni x\\sb j\\longmapsto x\\sb j\\ot (1-2\\hd)\\in\\pf\\te A\\, ,\n\\eeq\nwhere \\hd\\ is the map such that $\\hd (-1)=1$ and $\\hd (1)=0$, then \n$\\dr (\\cal I\\sb\\nu)\\inc\\cal I\\sb\\nu\\te A$.\n\\epr\n{\\it Proof.} Clearly, $\\dr (x\\sb j-x\\sb j\\sp*)\\in\\cal I\\sb\\nu\\te A\\,$ for any\n$j\\in\\{ 1,2,3\\}$. Also, for any $j,l\\in\\{1,2,3\\}$, we have \n\\[\n\\dr (x\\sb jx\\sb l)=\\dr (x\\sb j)\\dr (x\\sb l)=\nx\\sb jx\\sb l\\ot (1-2\\hd)\\sp2=x\\sb jx\\sb l\\te 1\\, .\n\\]\nHence $\\dr (\\cal I\\sb\\nu)\\inc\\cal I\\sb\\nu\\te A$, as claimed.\n\\epf\\\\\n\nIt follows now that a $*$-algebra homomorphism \n$\\hD\\sb\\nu :\\pn\\ra\\pn\\te A$ given by the formula \n$\\hD\\sb\\nu x\\sb{\\nu j}=x\\sb{\\nu j}\\ot (1-2\\hd)$ makes \\pn\\\na right $A$-comodule algebra.\n\n\\bpr\\label{bundle}\nLet \\pn\\ be a right $A$-comodule algebra as above. Then \n$(\\pn ,A,\\hD\\sb\\nu ,0,0)$ is a quantum principal bundle with the universal\ndifferential calculus.\n\\epr\n{\\it Proof.} By Proposition~\\ref{diagram}, it suffices to show that the\ncanonical map $\\mbox{\\sf T}\\!\\sb B:\\pn\\te\\sb{B\\sb\\nu}\\pn\\ra\\pn\\te A$, where\n$B\\sb\\nu:=\\pn\\sp{co A}$, is bijective. First, however, let us prove the\nfollowing\n\\ble\\label{bundlel}\nLet $B\\sb\\nu$ be as above. Then $B\\sb\\nu$ is the space spanned by monomials\nfrom \\pn\\ whose total degree is even, i.e.\n\\beq\\label{bn}\nB\\sb\\nu =\\left\\{\\sum\\sb{k\\geq 1}\\;\\sum\\sb{i\\sb1,...,i\\sb{2k}}\na\\sb{i\\sb1...i\\sb{2k}}x\\sb{\\nu i\\sb1}\\cdots x\\sb{\\nu i\\sb{2k}}\\in\\pn\\, \n\\mbox{\\Large$|$}\\; \na\\sb{i\\sb1...i\\sb{2k}}\\in\\Bbb C,\\, i\\sb1,...,i\\sb{2k}\\in\\{1,2,3\\}\\right\\}\\, .\n\\eeq\n\\ele\n{\\it Proof.} To simplify notation, let us denote the right hand side of \n(\\ref{bn}) by $\\widetilde{B\\sb\\nu}$. Thanks to (\\ref{co}), it is clear that\nevery element of $\\widetilde{B\\sb\\nu}$ is right coinvariant. It is also clear\nthat every element of \\pn\\ can be written as \n$b\\sb0+\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc\\widetilde{B\\sb\\nu}$. (Observe that any number\n$c\\in\\Bbb C$ can be expressed as \n$c=cr\\sb\\nu\\sp{-2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\\sp2\\in\n\\widetilde{B\\sb\\nu}$.) Furthermore, since\n\n\\[\n\\hD\\sb\\nu p-p\\te 1=b\\sb0\\te 1+\\sum\\sb{j=1}\\sp3\n(b\\sb jx\\sb{\\nu j}\\te 1-2b\\sb jx\\sb{\\nu j}\\te\\hd )-b\\sb0\\te 1-\\sum\\sb{j=1}\\sp3\nb\\sb jx\\sb{\\nu j}\\te 1=-2\\left(\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}\\right)\\te\\hd\\, ,\n\\]\n\nwe can conclude that $\\hD\\sb\\nu p=p\\te 1\\imp p=b\\sb0\\in\\widetilde{B\\sb\\nu}\\,$.\nHence $B\\sb\\nu=\\widetilde{B\\sb\\nu}$, as claimed.\n\\epf\\\\\n\nNow, consider a left \\pn -module map \n$\\widetilde{\\T }\\!\\sb B :\\pn\\te A\\ra\\pn\\te\\sb{B\\sb\\nu}\\pn $ given by the formula:\n\n\\[\n\\widetilde{\\T }\\!\\sb B (1\\te a)\\longmapsto\\left\\{\n\\begin{array}{ll}\n1\\te\\sb{B\\sb\\nu}1 & \\mbox{for $a=1$}\\\\\n\\mbox{$\\frac{1}{2}(1\\te\\sb{B\\sb\\nu}1-\nr\\sb\\nu\\sp{-2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\\ot\\sb{B\\sb\\nu}x\\sb{\\nu i})$} \n& \\mbox{for $a=\\hd $}\\, .\n\\end{array}\n\\right.\n\\]\n\nRecall that every element of \\pn\\ can be written as \n$b\\sb0+\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu$. Therefore, since\n$\\widetilde{\\T }\\!\\sb B\\ci\\T\\!\\sb B$ is a left \\pn -module map, it suffices to\ncheck that \n\\[\n(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)\\llp 1\\ot\\sb{B\\sb\\nu} \n(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp=\n1\\ot\\sb{B\\sb\\nu} (b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\n\\]\nfor arbitrary $\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu$. \nWith the help of Lemma~\\ref{bundlel}, we have\n\\bea\n&&\n(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)\\llp 1\\ot\\sb{B\\sb\\nu} \n(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp\n\\\\ &&\n=b\\sb0(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)(1\\ot\\sb{B\\sb\\nu}1)\n\\mbox{$+\\sum\\sb{j=1}\\sp3$}\nb\\sb j(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)(1\\ot\\sb{B\\sb\\nu}x\\sb{\\nu j})\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb j\n\\widetilde{\\T }\\!\\sb B(x\\sb{\\nu j}\\te 1-2x\\sb{\\nu j}\\te\\hd )\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}\nb\\sb j\\llp x\\sb{\\nu j}\\ot\\sb{B\\sb\\nu} 1\n-2x\\sb{\\nu j}\\widetilde{\\T }\\!\\sb B(1\\te\\hd )\\lrp\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}\n\\llp b\\sb jx\\sb{\\nu j}\\ot\\sb{B\\sb\\nu}1-b\\sb jx\\sb{\\nu j}\\ot\\sb{B\\sb\\nu}1\n+r\\sb\\nu\\sp{-2}\\mbox{$\\sum\\sb{i=1}\\sp3$}(a\\sb{\\nu i}b\\sb jx\\sb{\\nu j}x\\sb{\\nu i}\n\\ot\\sb{B\\sb\\nu}x\\sb{\\nu i})\\lrp\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}\\mbox{$\\sum\\sb{i=1}\\sp3$}\nr\\sb\\nu\\sp{-2}a\\sb{\\nu i}\\ot\\sb{B\\sb\\nu}b\\sb jx\\sb{\\nu j}x\\sb{\\nu i}\\sp2\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j}\n\\mbox{$\\sum\\sb{i=1}\\sp3$}r\\sb\\nu\\sp{-2}a\\sb{\\nu i}x\\sb{\\nu i}\\sp2)\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\, .\n\\eea\nThus we have shown that $\\widetilde{\\T }\\!\\sb B\\ci\\T\\!\\sb B=id$.\nFurthermore, it is straightforward to verify that\n$\\T\\!\\sb B\\ci\\widetilde{\\T }\\!\\sb B=id$. Hence $\\widetilde{\\T }\\!\\sb B$ is \nthe inverse of $\\T\\!\\sb B$ and consequently $\\T\\!\\sb B$ is bijective,\nas needed.\n\\hfill{$\\rule{7pt}{7pt}$}\n\n\\bre\\label{finite}\\em\nNote that, since $\\mbox{Map}(\\Bbb Z\\sb2,\\Bbb C)$ is finite dimensional,\nthe injectivity of $\\T\\!\\sb B$ follows immediately from its surjectivity\nand Theorem~1.3 in \\cite{sch2}.\n\\hfill{$\\Diamond$}\\ere\n\n\\bre\\label{equator}\\em\nRecall that a classical point of an algebra $B$ over a field $k$ is defined\nas an algebra homomorphism from $B$ to $k$. For $q\\neq\\pm 1$, the space of all\nclassical points of $P\\!\\sb1$ is parameterized by all pairs $(x,y)\\in\\Bbb R\\sp2$\nsubject to the relation $x\\sp2+y\\sp2=r\\sb1=1$. Any such pair yields an algebra\nhomomorphism $f:P\\!\\sb1\\ra\\Bbb C$ via the formulas \n$f(x\\sb{11})=x$, \\mbox{$f(x\\sb{12})=y$}, $f(x\\sb{13})=0\\,$. It is clear that the \nclassical `subspace' of $P\\!\\sb1$ is precisely its equator. Hence the name \n`equator sphere'. (To see the correspondence between $P\\!\\sb1$ and the \n$C\\sp*$-algebra defined by (7b)\\cite{po}, put $\\mu=q$ and \n\\beq\\label{corr}\nx\\sb{11}=\\frac{i}{2}(B\\sp*-B)\\, ,\\; x\\sb{12}=-\\frac{1}{2}(B\\sp*+B)\\, ,\\;\nx\\sb{13}=\\frac{-1-q\\sp2}{2}A\\, ;\n\\eeq\n cf.~the beginning of Section~7 in \\cite{po}.)\nNow, note that the quantum sphere employed in Section~5.2 of~\\bmq\\ \n($c=0$ in \\cite{po})\ncan be, in the same manner, regarded as a north pole sphere. On the other hand,\nthe quantum principal bundles considered in Proposition~\\ref{bundle} were \nconstructed to generalize the usual Hopf fibration $S\\sp2\\ra RP\\sp2$ \n(set $q=\\pm 1$ in the bundle $(P\\!\\sb1,A,\\hD\\sb1,0,0)$) where\n$\\Bbb Z\\sb2$ moves the points on the sphere to their antipodal counterparts\n(see p.69 in \\cite{tr}). \nOn the north pole sphere used in \\bmq , there is no other classical point \nto which the north pole could be moved under the free action of $\\Bbb Z\\sb2\\,$.\nThis is why, in order to deform the \nHopf fibration $S\\sp2\\ra RP\\sp2$, we used here the equator sphere instead. \n\\footnote{\nI am grateful to Stanis\\l aw Zakrzewski for explaining these things to me.}\n\\hfill{$\\Diamond$}\\ere\n\n\\bpr\\label{sconnection}\nLet $\\pn (B\\sb\\nu ,A)$ be a quantum principal bundle as in \nProposition~\\ref{bundle} and Lemma~\\ref{bundlel}. Also, let\n$\\ho\\in\\mbox{\\em Hom}\\sb{\\Bbb C}(A,\\hO\\sp1\\pn )$ be a homomorphism defined by\nthe formula\n\\[\n\\ho (a)=\\left\\{\n\\begin{array}{ll}\n0 & \\mbox{for $a=1$}\\\\\n\\mbox{$-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3\na\\sb{\\nu i}x\\sb{\\nu i}dx\\sb{\\nu i}$} \n& \\mbox{for $a=\\hd $}\\, .\n\\end{array}\n\\right.\n\\]\nThen \\ho\\ is a strong connection form on $\\pn (B\\sb\\nu ,A)$.\n\\epr\n{\\it Proof.} To prove that \\ho\\ is a connection form it suffices to verify that\n$(\\T\\cc\\ho )\\hd =1\\te\\hd\\ $. (Other conditions of Definition~\\ref{confordef} are\nimmediately satisfied.) We have\n\n\\bea\n(\\T\\cc\\ho )\\,\\hd\n&=&\n-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3\na\\sb{\\nu i}x\\sb{\\nu i}\\T\\!\\sb R(1\\te x\\sb{\\nu i}-x\\sb{\\nu i}\\te 1)\n\\\\ &=&\n-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\n(x\\sb{\\nu i}\\te 1-2x\\sb{\\nu i}\\te\\hd\\ -x\\sb{\\nu i}\\te 1)\n\\\\ &=&\n1\\ot\\hd\\, . \\phantom{+\\frac{1}{2}\\ot 1}\n\\eea\nHence \\ho\\ is indeed a connection form. Our next step is to show that\n\\ho\\ is strong (see Definition~\\ref{scdef}). With the help of formula\n(\\ref{confor}) and the Leibniz rule, for any \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu\\,$, we have\n\\bea\n(\\hP\\sp\\omega\\cc d)(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\n&=&\n-2\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}\\ho (\\hd )\n\\\\ &=&\nr\\sb\\nu\\sp{-2}\\sum\\sb{j=1}\\sp3\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}b\\sb jx\\sb{\\nu j}\nx\\sb{\\nu i}dx\\sb{\\nu i}\n\\\\ &=&\nr\\sb\\nu\\sp{-2}\\sum\\sb{j=1}\\sp3\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}b\\sb j\n\\llp d(x\\sb{\\nu j}x\\sb{\\nu i}\\sp2)-d(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\lrp\n\\\\ &=&\n\\sum\\sb{j=1}\\sp3b\\sb jdx\\sb{\\nu j}-r\\sb\\nu\\sp{-2}\\!\\!\\!\\sum\\sb{i,j\\in\\{1,2,3\\}}\n\\!\\!\\! a\\sb{\\nu i}b\\sb jd(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\, .\n\\eea\nApplying the Leibniz rule again, we obtain\n\\[\n(id-\\hP\\sp\\omega)\\llp d(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp\n=\ndb\\sb0+\\sum\\sb{j=1}\\sp3db\\sb j.x\\sb{\\nu j}\n+r\\sb\\nu\\sp{-2}\\!\\!\\!\\sum\\sb{i,j\\in\\{1,2,3\\}}\\!\\!\\!\na\\sb{\\nu i}b\\sb jd(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\in\\hO\\sb{shor}\\sp1\\pn\\, .\n\\]\nTaking advantage of the fact that any $p\\in\\pn\\ $ can be expressed as\n$b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu\\,$, we can conclude that \\ho\\\nis strong.\n\\epf\\\\\n\n\\bpr\\label{wconnection}\nLet $\\pn (B\\sb\\nu ,A)$ and \\ho\\ be as in the proposition above. A homomorphism\n\\linebreak\\mbox{$\\widetilde{\\ho }\\in\\mbox{\\em Hom}\\sb{\\Bbb C}(A,\\hO\\sp1\\pn )$} \n defined by\nthe formula\n\\[\n\\widetilde{\\ho }(a)=\\left\\{\n\\begin{array}{ll}\n0 & \\mbox{for $a=1$}\\\\\n\\ho (\\hd )+dx\\sb{\\nu l}\\sp2 \n& \\mbox{for $a=\\hd $}\\, ,\n\\end{array}\n\\right.\n\\]\nwhere $l\\in\\{1,2,3\\}$, is a connection 1-form of a connection that is not strong.\n\\epr\n{\\it Proof.} Let $l$ be any fixed element of $\\{1,2,3\\}$. Since \n$\\dsr(dx\\sb{\\nu l}\\sp2)=dx\\sb{\\nu l}\\sp2\\ot 1$ and $\\T(dx\\sb{\\nu l}\\sp2)=0$,\nit is clear that $\\widetilde{\\ho }$ is a connection 1-form. To prove that\n$\\hP\\sp{\\tilde{\\omega}}$ is not a strong connection, we will demonstrate that\n$(id-\\hP\\sp{\\tilde{\\omega}})(dx\\sb{\\nu l})\n\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\, $.\nWith the help of formula (\\ref{confor}), we have:\n\\[\n(id-\\hP\\sp{\\tilde{\\omega}})(dx\\sb{\\nu l})=dx\\sb{\\nu l}+2x\\sb{\\nu l}\n\\widetilde{\\ho }(\\hd )=dx\\sb{\\nu l}+2x\\sb{\\nu l}\\ho (\\hd )\n+2x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2=(id-\\hP\\sp\\omega)(dx\\sb{\\nu l})\n+2x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\n\\]\nTherefore, as $\\hP\\sp\\omega$ is a strong connection, it is enough to show that\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\, $.\nTo this end, let us put \n$B\\sb\\nu\\sp c:=\\left\\{\\sum\\sb{j=1}\\sp3b\\sb j.x\\sb{\\nu j}\\in\\pn\\ |\\; \nb\\sb j\\in B\\sb\\nu\\, ,\\; j\\in\\{1,2,3\\}\\right\\}$. Clearly, \n$\\pn =B\\sb\\nu +B\\sb\\nu\\sp c\\,$. An argument similar to that used in the proof of\nLemma~\\ref{bundlel} shows that $B\\sb\\nu\\cap B\\sb\\nu\\sp c=0$. \nIt follows now that $\\pn =B\\sb\\nu\\oplus B\\sb\\nu\\sp c\\,$. As we consider here\nthe universal calculus on \\pn , we have an isomorphism\n$\n\\psi :\\hO\\sp1\\pn\\ni dp.u\\to [p]\\sb{\\Bbb C}\\ot u\\in\\pn \/\\Bbb C\\ot\\pn\\, .\n$\nFurthermore, we have\n\\begin{eqnarray}\\label{split}\n\\hO\\sp1\\pn \n&=&\n\\psi\\sp{-1}\\llp\\psi (\\hO\\sp1\\pn )\\lrp \n\\nonumber \\\\ &=&\\psi\\sp{-1}\n(B\\sb\\nu \/\\Bbb C\\ot\\pn )\\oplus\\psi\\sp{-1}\n\\llp (B\\sb\\nu\\sp c\\oplus\\Bbb C)\/\\Bbb C\\ot\\pn\\lrp\n\\nonumber \\\\ &=&\ndB\\sb\\nu .\\pn\\oplus d(B\\sb\\nu\\sp c\\oplus\\Bbb C).\\pn \n\\nonumber \\\\ &=&\n\\hO\\sp1B\\sb\\nu .\\pn\\oplus d(B\\sb\\nu\\sp c).\\pn\\, .\n\\end{eqnarray}\nOn the other hand, taking into account the isomorphism\n$\\hO\\sp1\\pn\\cong\\pn\\ot (\\pn \/\\Bbb C)$, one can show \n(with some help of the representation presented in Point~III.(a) of Proposition~4\nin \\cite{po} and formulas (\\ref{corr})) that\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\neq 0$. Therefore, since\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2=dx\\sb{\\nu l}\\sp3-dx\\sb{\\nu l}.x\\sb{\\nu l}\\sp2\n\\in d(B\\sb\\nu\\sp c).\\pn\\,$, we can conclude, by virtue of (\\ref{split}),\nthat $x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\,$,\n as desired.\n\\epf\\\\\n\n\\bre\\label{vertical}\\em\nThe strong connection form \\ho\\ defined in Proposition~\\ref{sconnection} is\nnon-trivial. Indeed, taking (\\ref{confor}) into account, one can see that\n\\[\n\\hP\\sp{\\omega}(dx\\sb{\\nu l})=-2x\\sb{\\nu l}\\ho (\\hd )=r\\sb\\nu\\sp{-2}\n\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu l}x\\sb{\\nu i}dx\\sb{\\nu i}\\, ,\n\\]\nwhere, as before, $l$ is any fixed element of $\\{1,2,3\\}$.\nOn the other hand, it can be checked that \n$\\left\\{[x\\sb{\\nu i}]\\sb{\\Bbb C}\\right\\}\\sb{i\\in\\{1,2,3\\}}$\nare linearly independent and that $x\\sb{\\nu l}\\sp2\\neq 0$. \n(Again, one can take advantage of the representation presented in Point~III.(a) \nof Proposition~4 in \\cite{po} and formulas (\\ref{corr}).) Hence, with the\nhelp of an isomorphism $\\hO\\sp1\\pn\\cong\\pn\\ot (\\pn \/\\Bbb C)$, it follows that\n$\\hP\\sp\\omega(dx\\sb{\\nu l})\\neq 0$. Consequently, the space of vertical forms\n(i.e.~Im $\\hP\\sp\\omega$) is non-zero.\n\\hfill{$\\Diamond\\;\\blacklozenge$}\\ere \n\\eex\n\nWe end our display of examples with a strict monoidal category\n(see Section~6.1 in~\\cite{shst}) example of a weak connection.\n\n\\bex\\label{formal}\\em\nSimilarly to Example~\\ref{z4}, \nthis example is concerned with a translation of the concept\nof the canonical connection on a homogeneous space to a different set-up.\nOnly this time, the groups employed are neither\nAbelian, nor finite. The former makes our bundle look more interesting,\nthe latter forces us to replace the algebraic tensor product by the\nappropriate dual of the Cartesian product. More precisely, let\n$\\frak M$ be the image of the category of sets under the contravariant\nfunctor $\\mbox{Map}(\\,\\cdot\\, ,k)$. The tensor product $\\ot\\sb{\\frak M}$\ndefined by\n\\[\n\\mbox{Map}(X,k)\\ot\\sb{\\frak M}\n\\mbox{Map}(Y,k)=\\mbox{Map}(X\\!\\times\\! Y,k)\n\\]\nmakes $\\frak M$ a strict monoidal category. Moreover, if $X$ is a group,\nthen $\\mbox{Map}(X,k)$ is an \\mbox{$\\frak M$-Hopf} algebra, where the\ndefinition of an \\mbox{$\\frak M$-Hopf} algebra is the same as\nthat of a Hopf algebra but with the tensor product taken to be\n$\\ot\\sb{\\frak M}\\,$. In what follows we define several gauge theoretic\nnotions in the setting of the category $\\frak M\\,$:\n\\be\n\\item $\\Omega\\sp 1\\sb{\\frak M}X:=\\{\nF\\!\\in\\!\\mbox{Map}(X\\!\\times\\! X,k)\\, |\\,\n\\forall\\, x\\!\\in\\! X:F(x,x)=0\\}$ (\\mbox{$\\mbox{Map}(X,k)$-bimodule} of\n\\mbox{$\\frak M$-differential} \\mbox{\\mbox{1-forms}})\n\n\\item $\\Delta\\sb{\\frak M-R}:=R\\sp *$ (right coaction), where\n$R:X\\times G\\ra X$ is a right free action of the group $G$ on $X$\n(e.g., see p.55 in~\\cite{tr}; $G$ will denote\na group and $R$ its right free action throughout the rest of this example)\n\n\\item $\\Delta\\sb{\\frak M-{\\cal R}}:{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}}\\ra\\{\n K\\!\\in\\!\\mbox{Map}(X\\!\\times\\! X\n\\!\\times\\! G,k)\\; |\\;\\forall\\; x\\!\\in\\! X,\\, g\\!\\in\\!\n G:K(x,x,g)=0\\}$ (right coaction\non \\mbox{$\\frak M$-differential} \\mbox{\\mbox{1-forms}}), where\n$\\mbox{\\large$\\forall$}\\; F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}}\\lc\\,\nx,y\\!\\in\\! X\\lc\\, g\\!\\in\\! G:\n(\\Delta\\sb{\\frak M-{\\cal R}}F)(x,y,g):=F\\llp R(x,g)\\bc R(y,g)\\lrp$\n\n\\item A triple $\\llp\\mbox{Map}(X,k)\\lc\\mbox{Map}(G,k)\\lc\n\\Delta\\sb{\\frak M-R}\\lrp$ is called an \\mbox{$\\frak M$-principal}\nbundle and, for simplicity, denoted by $(X,G,R)$.\n\n\\item ${\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}} :=\\mbox{\\large$\\{$} F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}} \\;\n|\\;\\mbox{\\large$\\forall$}\\; x\\!\\in\\!\n X\\lc\\, g\\!\\in\\! G: F\\llp R(x,g)\\lc\\, x\\lrp\n=0\\mbox{\\large$\\}$}$ (horizontal \\mbox{$\\frak M$-differential}\n\\mbox{\\mbox{1-forms}})\n\\item $B:=\\mbox{Map}(X\/G,k)$ (base space of $(X,G,R)$)\n\\item ${\\mbox{$\\Omega\\sp 1\\sb{\\frak M-shor}X$}} :=\\mbox{\\large$\\{$} F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}}\\;\n|\\;\\mbox{\\large$\\forall$}\\; x,y\\!\\in\\!\nX\\lc\\, g\\!\\in\\! G:\nF\\llp R(x,g)\\lc\\, y\\lrp =F(x,y)\\mbox{\\large$\\}$}$ (strongly\nhorizontal \\mbox{$\\frak M$-differential}\n\\mbox{\\mbox{1-forms}})\n\\item Let $(X,G,R)$ be an \\mbox{$\\frak M$-principal} bundle and let\n\\[\n\\widetilde\\Pi :=\\widetilde\\Pi\\sb\n1\\times\\widetilde\\Pi\\sb 2\n\\in\\mbox{$\\mbox{Map}(X\\!\\times\\! X,X\\!\\times\\!\n X)$}\n\\]\n be an idempotent satisfying\nthe following conditions:\n\\bi\n\\item [a)] $\\mbox{\\large$\\forall$}\\; x,y\\in\n X:\\widetilde\\Pi\\sb 1(x,y)=x$,\n\\item [b)] $\\mbox{\\large$\\forall$}\\; F\\in{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}} :\\;\nF\\ci\\widetilde\\Pi =0\\iff F\\in{\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}}$,\n\\item [c)] $\\mbox{\\large$\\forall$}\\; x,y\\!\\in\\! X\\lc\\, g\\!\\in\\!\nG: \\widetilde\\Pi\\sb 2\\llp\nR(x,g)\\,\\lc\\, R(y,g)\\lrp =R\\llp\\widetilde\\Pi\n\\sb 2(x,y)\\,\\lc\\, g\\lrp$.\n\\ei\nThen $\\widetilde\\Pi\\sp *$ is called an\n\\mbox{$\\frak M$-connection}\nand is denoted by $\\Pi\\sb{\\frak M}$.\n\\item An \\mbox{$\\frak M$-connection} $\\Pi\n\\sb{\\frak M}$ is called a strong\n \\mbox{$\\frak M$-connection} iff\n\\[\n\\mbox{\\large\\boldmath$\\forall$}\\,\n x,y\\!\\in\\! X\\lc\\; g\\!\\in\\! G:\\;\n\\widetilde\\Pi\\sb 2\\llp R(x,g)\\,\\lc\\, y\\lrp\\\\\n =\n\\widetilde\\Pi\\sb 2(x,y)\\, .\n\\]\n\\ee\nThe above definitions were constructed so that the \\mbox{$\\frak M$-objects}\nthus defined become the corresponding\n`quantum objects' (the universal differential calculus assumed)\nwhen both $X$ and $G$ are finite and the \\mbox{$\\frak M$-tensor} product\nis the same as the algebraic tensor product. In particular,\none can rethink and equivalently\ndescribe Example~\\ref{z4} in \\mbox{$\\frak M$-terms}.\n Doing so blurs the view of general principles making that example work,\nbut it allows one to have a better insight into its concrete mathematical fabric.\n\nClearly, if $H$ is a subgroup of $G$ acting\non $G$ on the right by the group\nmultiplication (let us denote this action by\n$R\\sb G$), then\n$(G,H,R\\sb G)$ is an \\mbox{$\\frak M$-principal}\n bundle. Furthermore,\nany surjection \\mbox{$j:G\\ra H$} satisfying\n $j\\ci\\imath = id$, where\n\\mbox{$\\imath :H\\hookrightarrow G$} is the\n inclusion, and\n\\beq\\label{jad}\n\\forall\\, g\\in G,h\\in H:\\, j(h\\sp {-1}gh)=h\\sp\n {-1}j(g)h\n\\eeq\nyields an \\mbox{$\\frak M$-connection} on\n$(G,H,R\\sb G)$. Indeed, let\n$\\widetilde\\Pi\\sp j(g,r):=\\llp g,\\,\ngj(g\\sp{-1}r)\\lrp$, for any\n$g,r\\in G$. Then, for all $g,r\\in G\\bc\\,\n h\\in H$, we have:\n\\be\n\\item $(\\widetilde\\Pi\\sp j)\\sp 2(g,r)=\\lblp g,\\,\n gj\\llp g\\sp{-1}gj(g\\sp{-1}r)\\lrp\\lbrp\n=\\llp g,\\, gj(g\\sp{-1}r)\\lrp =\\widetilde\\Pi\\sp j(g,r)$,\nwhere the middle equality is implied by the formula $j\\ci\\imath =id$.\n\n\\item $\\widetilde\\Pi\\sp j\\sb 1(g,r)=g$ (obvious).\n\n\\item $F\\in\\Omega\\sp 1\\sb{\\frak M-hor} G\n\\Rightarrow F\\ci\\widetilde\\Pi\\sb j=0$ (obvious).\n\n\\item For any $F\\in\\mbox{Map}( G\\!\\times\\! G,k)$, the implication\n\\[\nF\\ci\\widetilde\\Pi\\sp j=0\\Rightarrow\nF\\in\\Omega\\sp 1\\sb{\\frak M-hor} G\n\\]\nis a consequence of the fact that\n$\\widetilde\\Pi\\sb j$ restricted to\n\\mbox{$\\bigcup\\sb{g\\in G}(g,gH )$}\ncoincides with the inclusion of\n\\mbox{$\\bigcup\\sb{g\\in G}(g,gH )$} in $ G\\times G$.\n\n\\item $\\widetilde\\Pi\\sp j\\sb 2(gh,rh)=\nghj(h\\sp{-1}g\\sp{-1}rh)=gj(g\\sp{-1}r)h\n=\\widetilde\\Pi\\sp j\\sb 2(g,r)h$, where\nthe middle step follows from\n(\\ref{jad}).\n\\ee\nHence, $(\\widetilde\\Pi\\sp j)\\sp *$ is an \\mbox{$\\frak M$-connection}.\nMoreover, much as in Example~\\ref{z4}, if\n$(\\widetilde\\Pi\\sp j)\\sp *$ is a strong\n\\mbox{$\\frak M$-connection}, then, for all $g\\in G , h\\in H$, $hj(g)=j(hg)$.\n\nNext, we proceed to consider a special case of $G$ and $H$.\nTo do so, first we need a definition of an algebraic formal group:\n\n\\bde [cf.\\ \\cite{ha} and Appendix~A in \\cite{mgtqg}]\\label{afg}\nLet $\\frak g$ be a finite dimensional Lie algebra ($\\dim \\frak g=n$),\nand let $\\{ E\\sp\\nu\\}\\sb{\\nu\\in\\{ 1,\\cdots ,n\\}}$\nbe a basis of $\\frak g$.\nAlso, let $F$ be the formal group law\n(see, e.g., Section~9.1 and Section~1.1 in~\\cite{ha}) given by the\nBaker--Campbell--Hausdorff formula determined by $\\frak g$\nand the basis $\\{ E\\sp\\nu\\}\\sb{\\nu\\in\\{ 1,\\cdots ,n\\}}$\n(see Appendix~A in~\\cite{mgtqg}), so that to every\npair of $n$-tuples of formal power series in a finite number of\nvariables with no free term we can assign another such $n$-tuple, i.e.\n\\bea\n&&\nF\\lblp\n\\llp p\\sb 1(t\\sb 1,\\cdots ,t\\sb l)\\lc\\cdots\\lc p\\sb n(t\\sb 1,\\cdots\n,t\\sb l)\\lrp\\lbc\n\\llp q\\sb 1(s\\sb 1,\\cdots ,s\\sb m)\\lc\\cdots\\lc q\\sb n(s\\sb 1,\\cdots\n,s\\sb m)\\lrp\\lbrp \\\\ &&\n=\\llp r\\sb 1(t\\sb 1,\\cdots ,t\\sb l,s\\sb 1,\\cdots ,s\\sb m)\\lc\\cdots\\lc\nr\\sb n(t\\sb 1,\\cdots ,t\\sb l,s\\sb 1,\\cdots ,s\\sb m)\\lrp\\, .\n\\eea\nSymbolically, we will write \\mbox{\n$\\llp\\, p\\sb 1(t\\sb 1\\, ,\\,\\cdots\\, ,\\, t\\sb l)\\,\\lc\\,\\cdots\\,\\lc\\,\n p\\sb n(t\\sb 1\\, ,\\,\\cdots\\, ,\\, t\\sb l)\\,\\lrp$}\nas $\\exp (p\\sb\\nu E\\sp\\nu )$ and \\linebreak\\mbox{$F\\llp\\exp\n(p\\sb\\nu E\\sp\\nu )\\lc\\exp (q\\sb\\mu E\\sp\\mu )\\lrp$}~as~\\mbox{\n$\\exp (p\\sb\\nu E\\sp\\nu )\\exp (p\\sb\\nu E\\sp\\nu )$}.\n (The Einstein convention of summation over repeating indices is\nassumed here and throughout the rest of this example.)\nThe group generated with the use of $F$ by the $n$-tuples\n$\\{\\exp (t\\sb mX)\\}\\sb{m\\in{\\Bbb N},X\\in\\frak g}\\,$, where\n$\\{ t\\sb m\\}\\sb{m\\in{\\Bbb N}}$\nare formal power series in one variable with all\nbut the linear coefficients vanishing, is called the algebraic formal group\nassociated to $\\frak g$ and is denoted by $E\\sb{\\frak g}$.\n\\ede\n\nNow, let $G={\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$, $H={\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$\nand let \\mbox{$j:{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}\\ra{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$} be the surjection defined by\n$j\\llp\\exp (p\\sb l E\\sp l +p\\sb\\mu E\\sp\\mu)\\lrp = \\exp (p\\sb l E\\sp l)$,\nwhere $\\{E\\sp l\\}\\sb{l\\in\\{ 1,2,3\\}}$ is a fixed basis of ${\\frak s\\frak u}(2)$, \nand $\\{ E\\sp\\mu\\}\\sb{\\mu\\in\\{ 1,2,3\\}}$ is a fixed basis of\n$i{\\frak s\\frak u}(2)$. It is clear that \n$({\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} ,{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}} ,R\\sb{E\\sb{\\frak s\\frak l(2,\\Bbb C)}})$ is an\n\\mbox{$\\frak M$-principal} bundle, and $j\\ci\\imath = id$.\nThus, to see that $j$ induces an\n\\mbox{$\\frak M$-connection}, it suffices to prove the following:\n\n\\ble\\label{lemex}\n$\\;\\mbox{\\large$\\forall$}\\, g\\in {\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} ,\n h\\in {\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\, : \\; j(h\\sp{-1}gh)=h\\sp{-1}j(g)h$\n\\ele\n{\\it Proof.} Since the formal power series\ndetermining elements of ${\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$ are generated by the\nBaker--Campbell--Hausdorff formula, we know that\n\\bea\n\\mbox{\\large$\\forall$}\\, g\\!\\in\\!{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}\\;\\,\\mbox{\\large$\\exists$}\n\\,\\epsilon\\sb g>0,n\\!\\in\\!{\\Bbb N}\\;\\,\\mbox{\\large$\\forall$}\\,\n\\overline{x}\\in\\{ (z\\sb 1,\\cdots ,z\\sb n)\\in\\Bbb R\\sp n\\, |\n\\; z\\sb 1\\sp 2+\\cdots +z\\sb n\\sp 2<\\epsilon\\sb g\\sp 2\\}:&&\n\\\\ \n V\\sb{\\overline{x}}(g):=\\exp\\llp p\\sb l\n(\\overline{x})E\\sp l + p\\sb\\mu (\\overline{x})E\\sp\n\\mu\\lrp\\in SL(2,{\\Bbb C})\\, .&&\n\\eea\nThis means that the formal power series\n$\\{ p_{l},p_{\\mu}\\}\\sb{l,\\mu\\in\\{ 1,2,3\\}}$\ndefining $g$ are convergent when evaluated at\n any $\\overline{x}$ in an $\\epsilon$-neighborhood\nof $0\\in {\\Bbb R}\\sp n$.\nNow, let $g$ be an arbitrary element of ${\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$\nand $h$ be any element of\n${\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$ of the form $h=\\exp (tX)$, where \n\\mbox{$X\\!\\in\\!{\\frak s\\frak u}(2)$} and $t$\nis understood as a formal power series in one variable. Also, let \n$\\epsilon :=\\min\\{\\epsilon\\sb{h\\sp{-1}j(g) h}\\; ,\\;\n\\epsilon\\sb{j(h\\sp{-1} gh)}\\} $,\n $ \\overline{y} :=(\\overline{t} ,\\overline{x})\\in\n{\\Bbb R}\\sp{n+1}$ and $\\widetilde{j}$ be\nthe map defined on all elements of $SL(2,{\\Bbb C})$ of the form\n $\\exp(t\\sb l E\\sp l + t\\sb\\mu E\\sp\\mu)$, \n$\\forall\\, l,\\mu :\\, t\\sb l,t\\sb\\mu\\in {\\Bbb R}, $ \nby the formula\n \\[\n\\widetilde{j} \\llp\\exp (t\\sb l\n E\\sp l+t\\sb\\mu E\\sp \\mu )\\lrp =\\exp\n(t\\sb l E\\sp l)\\, .\n\\]\nThen, using the fact that the splitting ${\\frak s\\frak l}(2,{\\Bbb C}) = \n{\\frak s\\frak u}(2)\\oplus i{\\frak s\\frak u}(2)$ is $ad$-invariant,\nfor every $\\overline{y}$ of length smaller than $\\epsilon$, we have\n\\bea\n&& (V\\sb{\\overline{y}} \\cc j)\\,\\llp\\exp( -tX)\\exp( p\\sb l\n E\\sp l+p\\sb\\mu E\\sp\\mu)\\exp(tX)\\lrp \\\\\n&& = (\\widetilde{j}\\cc V\\sb{\\overline{y}})\\,\n\\llp\\exp(-tX)\\exp(\np\\sb lE\\sp l +p\\sb\\mu E\\sp\\mu)\\exp(tX)\n\\lrp \\\\\n&& =\\widetilde{j}\\lblp\\exp( -\\overline{t}X)\\exp\n\\llp p\\sb l\n (\\overline{x})E\\sp l + p\\sb\\mu(\n\\overline{x})E\\sp\\mu\\lrp\\exp(\\overline{t}X)\\lbrp \\\\\n&& =\\widetilde{j}\\lblp\\exp\\llp p\\sb l\n(\\overline{x})\n\\exp(-\\overline{t}X) E\\sp l\\exp(\\overline{t}X) +\np\\sb\\mu(\\overline{x}) \\exp( -\\overline{t}X)\n E\\sp\\mu\n\\exp(\\overline{t}X)\\lrp\\lbrp \\\\\n&& = \\exp\\llp p\\sb l(\\overline{x})\\exp(-\n\\overline{t}X)\n E\\sp l\\exp(\\overline{t}X)\\lrp \\\\\n&& = \\exp(-\\overline{t}X)\\exp\\llp p\\sb\nl(\\overline{x})E\\sp l\\lrp\\exp(\\overline{t} X) \\\\\n&& = V\\sb{\\overline{y}} \\llp\\exp( -tX)\\exp( p\\sb\nl E\\sp l)\\exp(tX)\\lrp \\\\\n&& = V\\sb{\\overline{y}} \\lblp\\exp( -tX)\n j\\llp\\exp( p\\sb l\nE\\sp l+p\\sb\\mu E\\sp\\mu)\\lrp\\exp(tX)\n\\lbrp .\n\\eea\nThus, the formal power series defining\n$j(h\\sp{-1} gh)$ and $h\\sp{-1}j(g)h$\ncoincide when evaluated on an open\n neighborhood of $0\\in{\\Bbb R}\\sp {n+1}$,\n and hence are identical. To end the proof,\nwe need to note that $ad:\n{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\ra\\mbox{Aut}({\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} )$ is a\nhomomorphism and, since every element of\n${\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$ is generated by elements of the form\n$\\exp(tX)$, the formula\n$j(h\\sp{-1}gh) = h\\sp{-1}j(g)h$ is \nvalid for all $g\\in{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$ and $h\\in{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\,$, as claimed.\n\\hfill $\\Box$\n\\bigskip\n\nFinally, since $h=\\exp (tY)$, $g=\\exp(sZ)$,\n where $Y:=\\mbox{\\scriptsize $\\pmatrix{0&1\\cr -1&0}$}$, \n$Z:=\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&-1}$}$, do not satisfy\n$j(hg) = hj(g)$, the \\mbox{$j$-induced}\n\\mbox{$\\frak M$-connection} is non-strong, as\ndesired.\\footnote{\nI am very grateful to Philip Ryan \nfor pointing out that $Y,Z$\nprovide the desired counterexample to the formula $j(hg) = hj(g)$.}\n\nIt seems proper to mention at this point\n that it would be interesting to see to what extent gauge theory on\nquantum principal bundles can\nwork in some more interesting categories and whether \nthe monoidal reconstruction (see\n Section~8.2 in \\cite{shst}) can be\nextended to reconstruct bundles and\nconnections.\n\\hfill{$\\blacklozenge$}\\eex\n\n\\section{Gauge Transformations}\n\nThe next natural step is to determine how\nstrong connections behave under quantum gauge transformations. To do so,\nwe must first define gauge transformations of a quantum principal bundle.\nOne can define the group of quantum\ngauge transformations as the group of convolution-invertible elements\nof $\\mbox{Hom}\\sb k(A,P)$ which intertwine\n$\\Delta\\sb R$ with $ad\\sb R$, and satisfy $f(1)=1$. (The same definition is\nconsidered in Proposition~5.2 of~\\cite{tbt}.)\nThen one can define their action on connection forms in a way analogous\n to the action of their classical counterparts on the classical connection\n forms (see \\cite{booss}). Quantum gauge\ntransformations defined in this manner\ngeneralize locally defined quantum gauge transformations\ndiscussed in Section~3 of~\\bmq\\ (cf.\\ Section~3 in~\\cite{md4}).\n\n\\bde\\label{gtrans}\nLet $P(B,A)$ be a quantum principal bundle.\nA $k$-homomorphism \\mbox{$f:A\\ra P$} is called a gauge\ntransformation iff\\vspace*{3mm}\n\n\\hsp{5mm} 1. $f$ is convolution invertible,\n\n\\hsp{5mm} 2. $\\dr\\ci f = (f\\ot id)\\ci ad\\sb R,$\n\n\\hsp{5mm} 3. $f(1)=1$.\n\\ede\n\n\\bpr\\label{gtprop}\nThe set of all gauge transformations of a\nquantum principal bundle\nis a group with respect to convolution.\nWe denote this group by $GT(P)$.\n\\epr\n{\\it Proof.} A routine sigma notation calculation\nverifies the following lemma:\n\n\\ble\\label{adlem}\n$\\,$ Let $\\,\\{ (Q\\sb i,\\rho\\sb i)\\}\\sb{i\\in\\{ 1,2\\} }\\,$,\n$\\,V\\sb{12}\\,$ and $\\,m\\sb{12}\\,$ be as in Lemma~\\ref{r-inv}.\nThen, for all\\linebreak\n\\mbox{$\\mbox{\\em f}\\sb 1\\in\\mbox{\\em Hom}\\sb k(A,Q\\sb 1)$},\n\\mbox{$\\mbox{\\em f}\\sb 2\n\\in\\mbox{\\em Hom}\\sb k(A,Q\\sb 2)$},\n\\[\n\\llp (\\mbox{\\em f}\\sb 1\\te id)\\ci ad\\sb R\\lrp *\n\\llp (\\mbox{\\em f}\\sb 2\\te id)\\ci ad\\sb R\\lrp =\n\\llp (\\mbox{\\em f}\\sb 1\\! *\\!\\mbox{\\em f}\\sb 2)\\ot\nid\\lrp\\ci ad\\sb R\\, .\n\\]\n\\ele\n\nHence, since the map\n\\[\n(\\dr\\ci ):\\mbox{Hom}\\sb k(A,P)\\ni\\mbox{f}\\longmapsto\\dr\\ci\\mbox{f}\\in\n\\mbox{Hom}\\sb k(A,P\\te A)\n\\]\n is an algebra homomorphism (cf.\\ Lemma~4.0.2 in \\cite{swe}),\nthe set of all gauge\n transformations is closed\nunder the convolution. Furthermore,\nit follows from the same reason that\n\\mbox{$\\dr\\ci\\mbox{f}\\,\\sp{-1}=(\\dr\\ci\\mbox{f}\\, )\\sp{-1}$}. Therefore,\nas $f(1)=1$ implies $f\\sp{-1}(1)=1$,\nby putting $\\mbox{f}\\sb 1=f$ and $\\mbox{f}\\sb 2=f\\sp{-1}$\nin Lemma~\\ref{adlem}, we can also conclude the existence of the inverse.\n \\hfill $\\Box$\n\\bigskip\n\nWhen defined in this way,\nquantum gauge transformations are unwilling to preserve the\nproperty $\\omega(M\\sb A) = 0$ (see Definition~\\ref{confordef} and\nProposition~\\ref{gact}) defining a connection\nform $\\omega$ on a bundle with a general differential calculus. This is the case\neven if one assumes that the gauge transformations\nsatisfy an additional condition\n$\\llp(f\\ot f\\sp{-1})\\ci\\de\\lrp (M_{A})\\inc N_{P}\\,$.\n(A related discussion can be found around formula (48)\nin \\cite{bmq}; note that this condition\nis satisfied in the classical situation,\nin which $M_{A}=(\\mbox{Ker}\\,\\epsilon )\\sp{2}$ ---\nsee Example 1 on p.132 in \\cite{wor}.) Therefore,\n when dealing with quantum gauge transformations, we will assume the universal\ndifferential calculus.\n\n\\bpr\\label{gact}\nLet $f\\in GT(P)$ and $\\omega\\in\\cal C(P)$, where $\\cal C(P)$ denotes the space of\nall connection forms on a quantum principal bundle $(P,A,\\dr ,0,0)$. \nThen the formula\n\\[\nG\\!\\sb f\\omega =f*\\omega * f\\sp{-1} + f*(d\\ci f\\sp{-1})\n\\]\n(cf.~(20) in \\cite{wz}) defines an action $G:GT(P)\\times\\cal C(P)\\lra\\cal C(P)\\,$.\n\\epr\n{\\it Proof.} Let us verify first that $G\\!\\sb f\\omega$ is indeed a connection form.\n\\be\n\\item $G\\!\\sb f\\omega (k) = 0$. Obvious.\n\\item Taking into account that \\T\\ is a\nleft $P$-module morphism (see Point~4 in\nDefinition~\\ref{qpbdef}) and remembering that for the universal differential\ncalculus $N\\sb P\\! =\\! 0\\! =\\! M\\sb A$, one can see that\n\n\\bea\n&& (\\T\\ci G\\sb{\\! f}\\omega)(a) \\\\\n&& = f(a\\sb{(1)})(\\T\\cc\\omega)(a\\2)\n(\\Delta\\sb R\\cc f\\sp{-1})(a\\3 ) \\\\\n&& + \\llp(m\\ot id)\\ci (id\\ot\\dr)\\lrp\\llp\nf(a\\sb{(1)})\\ot\n f\\sp{-1}(a\\2) - \\epsilon(a)\\ot 1\\lrp \\\\\n&& = \\lblp f(a\\sb{(1)})\\ot\n\\llp a\\2-\\epsilon(a\\2)\n\\lrp\\lbrp\\llp f\\sp{-1}(a\\4 )\\ot\nS(a\\3 )a\\sb{(5)}\\lrp \\\\\n&& + f(a\\sb{(1)})f\\sp{-1}(a\\3 )\\ot\nS(a\\2)a\\4 - 1\\ot\\epsilon(a) \\\\\n&& = f(a\\sb{(1)}) f\\sp{-1}(a\\3 )\n \\ot\\llp\\epsilon(a\\sb {(2)})\na\\4 - S(a\\2)a\\4 \\lrp \\\\\n&& + f(a\\sb{(1)})f\\sp{-1}(a\\3 )\\ot\nS(a\\2)a\\sb\n{(4)} - 1\\ot\\epsilon(a) \\\\\n&& = 1\\ot\\llp a-\\epsilon(a)\\lrp .\n\\eea\n\\item $\\dsr\\ci G\\sb{\\! f}\\omega = (G\\sb{\\! f}\\omega\\ot id)\\ci ad\\sb R$\ncan be proved much as Proposition~\\ref{gtprop}.\n\\ee\nNow, to complete the proof, it suffices to note that\n$G\\sb{\\! f*g}\\omega = G\\sb{\\! f}G\\sb{\\! g}\\omega$.\n\\hsp{\\fill}$\\Box$\n\\bigskip\n\nThe action of the gauge group $GT(P)$ on the space of connections can be derived\nfrom its action on connection forms. It is explicitly described by\n\n\\bpr\\label{gtpi}\nLet $GT(P)$ be as in Proposition~\\ref{gact}.\nDenote by ${\\cal P}(P)$ the space of\nall connections on $P$, and by $\\Upsilon$ the bijection providing the\ncorrespondence between connections and connection forms, i.e., let\n\\[\n\\Upsilon : {\\cal P}(P)\\ni\\Pi\\longmapsto\\sigma\\sb\\Pi\\ci\\llp 1\\ot\n (id -\\epsilon )\\lrp\\in\\cal C(P)\\, ,\n\\]\nwhere $\\sigma\\sb\\Pi :\nP\\ot\\mbox{\\em Ker}\\,\\epsilon\\ra\\Omega\\sp 1\\! P$ is\nthe unique left \\mbox{$P$-module} homomorphism satisfying\n \\mbox{$\\mbox{\\em\\sf T}\\ci\\sigma\\sb\\Pi =id$} and\n\\mbox{$\\sigma\\sb\\Pi\\ci\\mbox{\\em\\sf T}=\\Pi$}.\n{\\em\\footnote{ \nSee Proposition~4.4 and the paragraph above it in \\bmq .}}\nThen the map\n${\\cal G} : GT(P)\\times{\\cal P}(P)\\lra{\\cal P}(P)$ given by\n\\beq\\label{gpi}\n{\\cal G}\\sb f\\Pi\n= m\\sb{\\Omega\\sp 1P}\\ci\\lblp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\lbrp +\n m\\sb{\\Omega\\sp 1P}\\ci\\llp id\\,\\ot\\, f\\! *\\! (d\\!\n\\ci\\! f\\sp{-1})\\lrp\\ci\\mbox{\\em\\sf T}\n\\eeq\nis the action of the gauge group $GT(P)$ on the space of connections\n${\\cal P}(P)$, and the following diagram commutes:\n\\bcd\nGT(P)\\times{\\cal P}(P) @>(id,\\Upsilon)>> GT(P)\\times\\cal C(P)\\\\\n@V{\\cal G}VV @VGVV\\\\\n{\\cal P}(P) @>\\Upsilon >>\\cal C(P)\\\\\n\\ecd\n\\epr\n{\\it Proof.} Both assertions of the proposition follow from the formula:\n\\beq\\label{gtpifor}\n\\mbox{\\large$\\forall$}\\,\nf\\in GT(P),\\,\\Pi\\in{\\cal P}(P):\\; {\\cal G}\\sb f\\Pi =\n(\\Upsilon\\sp{-1}\\ci G\\!\\sb f\\ci\\Upsilon) (\\Pi )\n\\eeq\nSince both sides of the above equation are left \\mbox{$P$-module}\nhomomorphisms, in order to prove~(\\ref{gtpifor}),\nit suffices to show that,\nfor any $f\\in GT(P)$ and $\\Pi\\in\\cal P(P)$, we have\n\\[\n(\\cal G\\sb f\\Pi )\\ci d =\n\\llp(\\Upsilon\\sp{-1}\\cc G\\!\\sb f\\cc\\Upsilon) (\\Pi )\\lrp\\ci d\\, ,\n\\]\nthat is, \n\\bea\\label{gtpifor2}\n\\hsp{2.25mm} &&\n m\\sb{\\Omega\\sp 1P}\\ci\\Lblp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\Lbrp\\ci\\mbox{\\sf T}\n\\ci d\\hsp{57mm}\n\\mbox{(\\theequation )}\\addtocounter{equation}{1} \\\\\n&&\n = m\\sb{\\Omega\\sp 1P}\\ci\\lblp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\lbrp\\ci d\n\\hsp{29.5mm} \\mbox{(pure gauge terms cancel).}\n\\eea\nTo do so, let us calculate the value of the left hand side of\n (\\ref{gtpifor2})\non arbitrary $p\\in P$:\n\\bea\n&&\n\\LAblp m\\sb{\\Omega\\sp 1P}\\ci\\LAlp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\LArp\\ci\\mbox{\\sf T}\n\\ci d\\LAbrp (p)\n\\\\ &&\n=\\LAblp m\\sb{\\Omega\\sp 1P}\\ci\\LAlp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\LArp\\LAbrp\n(p\\0\\ot p\\sb{(1)} - p\\ot 1)\n\\hsp{11mm} \\mbox{(see (24)\\bmq )}\n\\\\ &&\n=p\\0f(p\\sb{(1)})\\sigma\\sb\\Pi\\llp1\\ot p\\2-\\epsilon (p\\2)\n\\ot 1\\lrp f\\sp{-1}(p\\3 )\n\\\\ &&\n=\\sigma\\sb\\Pi\\llp p\\0f(p\\sb{(1)})\\ot p\\2-\np\\0f(p\\sb{(1)})\\epsilon (p\\2)\n\\ot 1\\lrp f\\sp{-1}(p\\3 )\n\\\\ &&\n=\\llp\\sigma\\sb\\Pi\\ci\\mbox{\\sf T}\\ci d\\ci\n (id\\! *\\sb R\\! f)\\lrp (p\\0)\\,\nf\\sp{-1}(p\\sb{(1)})\n\\hsp{41 mm} (\\mbox{$\\, id\\! *\\sb{\\! R}\\! f$ is right-covariant})\n\\\\ &&\n=\\lblp\\llp\\Pi\\cc d\\cc\n (id\\! *\\sb R\\! f)\\lrp*\\sb Rf\\sp{-1}\\lbrp\\, (p)\n\\\\ &&\n = \\Lblp m\\sb{\\Omega\\sp 1P}\\ci\\Llp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\Lrp\\Lbrp (1\\ot p - p\\ot 1)\n\\\\ &&\n = \\Lblp m\\sb{\\Omega\\sp 1P}\\ci\\Llp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\Lrp\\ci d\\Lbrp (p)\n\\eea\n\nHence, we can conclude that (\\ref{gtpifor2}) is true,\nand the proposition\nfollows. \\hfill{$\\Box$}\n\n\\bre{\\em\nThe left \\mbox{$B$-module} isomorphism $(id*\\sb Rf):P\\lra P$\ncan be regarded as\na quantum version of a gauge transformation \n understood as an appropriate\ndiffeomorphism of the total space of a principal bundle\n(see p.339 in \\cite{booss} and Definition~5.1 in~\\cite{tbt}).\nIn fact, the map $f\\to id*\\sb Rf$ is a group isomorphism\n(see Corollary~5.3 in \\cite{tbt}).\n(Observe that $(id*\\sb Rf\\sp{-1})\\ci\n(id*\\sb Rf)=id=(id*\\sb Rf)\\ci (id*\\sb Rf\\sp{-1}).\\,$)\n\\hfill{$\\Diamond$}}\\ere\n\nHaving defined and described quantum gauge\ntransformations and their\naction on the space of connections and connection forms,\n we can now show that these\ntransformations preserve the strongness of\n a connection, i.e.\n(provided $M\\sb A= 0 =N\\sb P$) their action\nis well-defined on the space\n\\spp\\ of all strong connections on a\nquantum bundle $P$. (Obviously, their action\n is then also well-defined on the space $\\cal S\\cal C(P)$\nof all strong connection forms.)\n\n\\bpr\nLet $P(B,A)$ be a quantum principal\n bundle with the universal differential calculus. Then\n$\\mbox{\\large$\\forall$}\\, f\\in GT(P)$:\n $\\Pi\\in\\spp \\,\\Leftrightarrow\\, \\cal G\\sb f\\Pi \\in\\spp$.\n\\epr\n{\\it Proof.}\nNote first that, for any right $A$-comodule coaction\n\\mbox{$\\rho\\sb{\\cal C}:{\\cal C}\\lra{\\cal C}\\ot A$}, any \\linebreak\n\\mbox{$\\alpha\\sb 1\\in\\mbox{Hom}\\sb k\\llp {\\cal C},\\Omega\n\\sp m(X)\\lrp$}\nand \\mbox{$\\alpha\\sb 2\\in\\mbox{Hom}\\sb k\\llp{\\cal C},\n\\Omega\\sp n(X)\\lrp$} (where $m\\geq 0,\\, n\\geq 0$\nand $\\Omega (X)$ is any differential algebra), there is the familiar\ngraded Leibniz rule:\n\\[\nd\\ci (\\alpha\\sb 1\\! *\\sb{\\rho\\sb{\\cal C}}\\!\\alpha\\sb 2)=(d\\ci\\alpha\\sb\n 1)*\\sb{\\rho\\sb{\\cal C}}\\alpha\\sb 2\n+ (-)\\sp m\\alpha\\sb 1*\\sb{\\rho\\sb{\\cal C}}(d\\ci\\alpha\\sb 2)\n\\]\nNow, using similar calculations as in the proof of Proposition~\\ref{gtpi} \nand the Leibniz rule for the convolution, one can see that,\nfor any $f\\!\\in\\!\n GT(P),\\,\\Pi\\!\\in\\! {\\cal P}(P)$,\n\\bea\n\\hsp{20mm} && (id-{\\cal G}\\sb f\\Pi)\\ci d\n\\hsp{90mm} \\mbox{(see (\\ref{gpi}))}\n\\\\ &&\n= d - \\llp\\Pi\\ci d\\ci (id\\! *\\sb R\\! f)\\lrp *\\sb Rf\\sp{-1}\n- id*\\sb Rf*(d\\cc f\\sp{-1}) \\\\\n\\hsp{14mm} && = \\llp d\\ci (id\\! *\\sb R\\! f)\\lrp\\, *\\sb R\\, f\\sp{-1}\n- \\llp\\Pi\\ci d\\ci (id\\! *\\sb R\\! f)\\lrp *\\sb Rf\\sp{-1}\\\\\n\\hsp{14mm} && = \\llp (id-\\Pi )\\ci d\\ci\n(id\\!*\\sb R\\! f)\\lrp\\, *\\sb R\\, f\\sp{-1} \\, .\n\\eea\nHence, $\\mbox{\\large$\\forall$}\\,\nf\\!\\in\\! GT(P):\\,\\Pi\\!\\in\\!\\spp \\Rightarrow {\\cal G}\\sb\nf\\Pi\\!\\in\\!\\spp $, and since ${\\cal G}$ is a\ngroup action on ${\\cal P}(P)$, $\\mbox{\\large$\\forall$}\nf\\!\\in\\! GT(P):\\, {\\cal G}\\sb f\\Pi\\!\\in\\!\\spp\n\\Rightarrow \\Pi\\!\n\\in\\!\\spp $. \\hfill $\\Box$\n\\bigskip\n\n\\section{Curvature}\n\nWe are ready now to examine the properties of the exterior\n covariant derivative (see Definition~\\ref{Ddef})\nassociated with a strong connection.\n This will lead to the definition of a (global)\n curvature form on a quantum principal bundle. \nThe following proposition and corollaries describe\nthe composition of the covariant exterior derivative with strongly\ntensorial differential forms\n(see Definition~\\ref{tfdef}). They and Proposition~\\ref{curadj} are analogous\nto the corresponding local (i.e.~valid only for trivial quantum bundles)\nstatements made in \\bmq . \nAs we do not know how to characterize all\n differential algebras for which $D$ can be well-defined,\nwe will simply assume here, whenever needed, that $\\Omega(P)$ \nis a right-covariant differential algebra such that $D$ is well-defined. \n\n\\bpr [cf.\\ (17) and (76) in \\bmq ]\\label{Dformula}\nLet $(P,A,\\dr ,N\\sb P,M\\sb A)$ be a quantum principal bundle with a\nconnection form $\\omega$, and let $\\hO (P)$ be a differential algebra\nsuch that\\linebreak \n$\\op = \\hO\\sp1\\! P\/N\\sb P$ and the exterior covariant derivative\n$D\\sp\\omega$ associated to \\ho\\ is well-defined by formula~(\\ref{689}). \nThen, for all \n\\mbox{$\\phi\\in ST\\sb\\rho\\llp V,\\Omega\\sp n(\\! P)\\lrp ,\\, n\\in \\{0\\}\\cup{\\Bbb N}$},\n\\[\n D\\sp\\omega\\ci\\phi =d\\ci\\phi - (-)\\sp n \\phi *\\sb\\rho\\omega\\, .\n\\]\n\\epr\n{\\it Proof.}\nNote that, for any $d\\alpha\\in\\Omega\\sp n(B)$,\n$n\\in{\\Bbb N}$, $p\\in P$, we have\n\\bea\n\\hsp{6mm}&& (d-D\\sp\\omega)(d\\alpha .p)\n\\\\ &&\n=(-)\\sp nd\\alpha dp - (-)\\sp nd\\alpha .(id - \\Pi\\sp\\omega )(dp)\n\\hsp{70mm}\\mbox{(see (\\ref{confor}))}\n\\\\ &&\n=(-)\\sp nd\\alpha .p\\0\\omega(p\\sb{(1)})\n\\\\ &&\n=(-)\\sp n(id*\\sb{\\cal R}\\omega )(d\\alpha .p)\\, .\n\\eea\nOn the other hand, for all $v\\in V$, $\\,\\phi (v)$\ncan be written as a finite sum\n$\\sum\\sb id\\alpha\\sb i.p\\sb i$ for some closed differential forms\n$d\\alpha\\sb i\\in\\Omega\\sp n(B)$ and 0-forms\n$p\\sb i\\in P$. Hence, with the help of Lemma~\\ref{rcovlem}, we obtain\n\\[\n \\llp(d-D\\sp\\omega)\\cc\\phi\\lrp\\, (v)\n=\\llp(-)\\sp n(id*\\sb{\\cal R}\\omega )\\cc\\phi\\lrp\\, (v)\n=(-)\\sp n(\\phi *\\sb\\rho\\omega )(v)\\, ,\n\\]\nand the assertion follows.\n\\hfill{$\\Box$}\n\n\\bco [cf.\\ (7)\\cite{bmq}]\\label{Dcor}\nLet $\\hO (P)$ be as in the proposition above. \nIf $D$ is the exterior covariant derivative\nassociated to a strong connection, then, for every \\nin ,\nwe have $D\\ci\\lblp ST\\sb\\rho\\llp V, \\Omega\\sp n(\\! P)\\lrp\\lbrp\\inc ST\\sb\\rho\\llp\nV,\\Omega\\sp{n+1}(\\! P)\\lrp .$\n\\eco\n{\\it Proof.}\nIn the same way as in the case of the differential envelope, it can be directly\ncalculated that $D$ is always right-covariant (see Appendix~A in \\bmq ), and thus,\nfor all connections, $D$ composed with a pseudotensorial differential\n form is a tensorial differential form, i.e. \n\\[\n\\mbox{\\large\\boldmath$\\forall$}\\, n\\in\\{ 0\\}\\!\\cup\\!{\\Bbb N} :\\;\nD\\ci\\lblp PT\\sb\\rho\\llp V, \\Omega\\sp n(\\! P)\\lrp\\lbrp\\inc T\\sb\\rho\\llp\nV,\\Omega\\sp{n+1}(\\! P)\\lrp\\, .\n\\]\n Hence, to prove the assertion of this corollary,\nit suffices to note that, if $D$ is associated to a strong connection, then,\nas can be seen from the second line of the first calculation in\nProposition~\\ref{Dformula}, $D$ evaluated on a strongly horizontal differential\nform yields a strongly horizontal differential form.\n\\hfill{\\mbox{$\\Box$}}\n\n\\bre\\label{scdefrem}\\em\nWith the help of $D$ one can equivalently define a strong\nconnection as a connection\nwhose exterior covariant derivative maps $P$ into strongly horizontal forms,\ni.e.\\ $D(P)\\inc\\Omega\\sp 1\\sb{shor}(P)$. Obviously, since $D$ can be defined on $P$\nfor any differential calculus, such a definition works on any quantum\nprincipal bundle.\n\\hfill{$\\Diamond$}\\ere\n\n\\bco [cf.\\ Section~3 in \\cite{bmq}]\\label{D2cor}\nLet \\ho\\ be a strong connection form, and let $D\\sp\\omega$ and $\\hO (P)$ be as in \nProposition~\\ref{Dformula}. Then\n\\beq\\label{D2gen}\n\\mbox{\\large$\\forall$}\\,\\phi\\in ST\\sb\\rho\\llp\nV,\\Omega(P)\\lrp\\; :\\; (D\\sp\\omega)\\sp 2\\ci\\phi\n= -\\phi *\\sb\\rho(d\\ci\\omega +\\omega*\\omega).\n\\eeq\n\\eco\n\nFormula (\\ref{D2gen}) is, in particular, true for a classical principal bundle\n$\\widetilde{P}(M,G)$. (A classical principal bundle is a special case of a\nquantum principal bundle when we replace the algebraic tensor product by the\nappropriately completed tensor product.)\nIt is a generalization of the classical formula\n\\beq\\label{D2clas}\nD\\sp 2\\alpha =\n\\varrho\\sp{'}(F)\\wedge\\alpha\\, ,\n\\eeq\n where $\\alpha$ is a differential form on $\\widetilde{P}$\n with values in a finite dimensional vector space $W$, the homomorphism\n\\mbox{$\\varrho\\sp{'} : \\frak g\\ra\\frak g\\frak l(W)$} \nis the Lie algebra representation induced\nby a homomorphism \\mbox{$\\varrho :G \\ra GL(W)$},\nand $F$ is the curvature form of a connection defining $D$ (e.g., \nsee (19) in~\\cite{tr}). More precisely,\n taking $V$ to be the tensor algebra of the dual vector space\n$W\\sp*$, defining $\\rho\\sb R$~by\n\\[\n\\mbox{\\large$\\forall\\,$} v\\in W\\sp*\\, :\\,\n\\rho\\sb R(v)(w,g)=v\\llp\\varrho(g\\sp{-1})w\\lrp\\, ,\n\\]\nputting \n\n\\[\n\\phi\\sb\\alpha (v) : \n\\left\\{\\begin{array}{ll}\n{\\displaystyle\\bigwedge\\sp{\\deg\\alpha}T\\sb\n{\\tilde{p}}\\widetilde{P}\\ni X\\sb{{\\tilde{p}}}\n\\longmapsto\\left\\{\n\\begin{array}{ll}\n(v\\cc\\alpha) (X\\sb{{\\tilde{p}}})\n& \\mbox{for $v\\in W\\sp*$}\\\\\n0 & \\mbox{otherwise}\n\\end{array}\n\\right.} & \\mbox{for $\\deg\\alpha > 0$}\\\\ \n\\phantom{.} & \\phantom{.}\\\\\n{\\displaystyle\\widetilde{P}\\ni\\widetilde{p}\n\\longmapsto(\\check{v}\\cc\\alpha)\n(\\widetilde{p})\\in\\Bbb R}\n & \\mbox{for $\\deg\\alpha = 0\\,$,}\\\\\n\\end{array}\\right.\n\\]\\ \\\\\n\nwhere $\\check{v}$ is the polynomial function on $W$ corresponding to the tensor $v$,\nand remembering that every $\\frak g$-valued\ndifferential form $\\vartheta$ on $\\widetilde{P}$ can be viewed as an\n\\mbox{$\\epsilon_{C\\sp{\\!\\infty\\!}(\\! G\\!)}$-d}erivation\n\\mbox{$\\psi_{\\vartheta} : C\\sp{\\infty}(G)\\ra\\bigwedge\\sp*(\\widetilde{P})$} \ngiven by $\\psi_{\\vartheta}(a)X_{\\tilde{p}} = \\vartheta(X_{\\tilde{p}}) a$\n(or $\\psi_{\\vartheta}(a)\\widetilde{p} =\\vartheta (\\widetilde{p}) a$\n if $\\deg\\alpha = 0$), we can rewrite (\\ref{D2clas}) as\n\\[\nD\\sp 2\\cc\\phi\\sb\\alpha =\n-\\phi\\sb\\alpha\\! *\\sb\\rho\\! F\\, .\n\\]\nNote that, since every vector space\n$W$ is a Lie group and there is a canonical\nisomorphism between $W$ and the Lie algebra of $W$,\nwe could define $\\phi\\sb\\alpha$ in the\nsame way as we define~$\\psi\\sb\\vartheta$. But then we would not have\n$\\phi\\sb\\alpha (1) = \\delta\\sb{0,\\deg\\alpha}$, which we need in Appendix~A\n(see Proposition~\\ref{asssecprop}).\nIn the classical case, we can replace $F$ by $\\; d\\ci\\omega +\n\\omega*\\omega\\;$ or\n$\\; D\\ci\\omega\\;$ in the last formula, but we need to put\n$\\; F=d\\ci\\omega + \\omega *\\omega\\;$ to obtain (\\ref{D2gen}).\nAlso, formula (\\ref{D2clas}) can be considered as a motivating factor in defining\n the curvature of a connection on a projective module \nas the square of a covariant derivative\n(see p.554 in \\cite{conbook}; \ncovariant derivative $\\equiv$ connection on a projective\nmodule). Therefore, it is natural to define the (global)\ncurvature form of a connection $\\Pi\\sp\\omega$\nin the following way:\n\n\\bde \\label{curdef}\nLet $\\omega$ be a connection \\mbox{1-form} on a\nquantum principal bundle. The\ndifferential form \\mbox{$d\\cc\\omega +\n\\omega\\! *\\!\\omega$} is called the\ncurvature form of $\\omega$ and is denoted by\n$F\\sb\\omega\\,$.\n\\ede\n Clearly, if $\\omega$ is a strong connection\n form, then, at least in the case of a trivial\n bundle, for any differential algebra\n$\\Omega(P)$, even if $D$ is not well-defined,\n $F\\sb\\omega$ is horizontal\n(\\mbox{$F\\sb\\omega =\\Phi\\sp{-1}\\!\\! *\\! F\\sb\n \\beta\\! *\\!\\Phi$},\nwhere $\\Phi$ and $\\beta$ are as in\nProposition~\\ref{scprop} and $F\\sb\\beta :=\nd\\cc\\beta +\\beta\\! *\\!\\beta$\nis a local curvature\nform, cf.~(11)\\cite{bmq}). Moreover, unlike the\nexpression $D\\cc\\omega$,\nthe so-defined\ncurvature form has the desired (at least from\nthe point\n of view of Yang--Mills theory) transformation\nproperties, i.e.~we have:\n\\nopagebreak\n\\bpr [cf.\\ (13)\\cite{bmq}, (20)\\cite{wz}]\\label{curadj}\nLet $P(B,A)$ be a quantum principal bundle with\nthe universal differential calculus,\n$\\omega\\in\\cal C(P)$ and $f\\in GT(P)$. Then\n\\[\nF\\sb{G\\!\\sb f\\omega} =\nf*F\\sb\\omega*f\\sp{-1}\\, .\n\\]\n\\epr\n{\\it Proof.} Straightforward.\\hfill{$\\Box$}\n\n\\section{\\mbox{\\boldmath$U\\sb q(2)$}--Yang--Mills Theory on a Free Module}\n\nTo begin with, let us show that it is possible to define an action functional\non quantum bundle connections in such a way that, at least in the case of\na trivial quantum bundle, \nit agrees with the Yang--Mills\n action functional constructed in Section~1 of both \\cite{conri} and \\cite{ri}.\n(Clearly, we assume that the `base space' of a quantum bundle is an algebra over\n which modules are considered in~\\cite{conri,ri}.)\nConsiderations in \\cite{bmc} and Proposition~\\ref{curadj}\n suggest that, if $A$ is a matrix quantum group and $T$ its matrix of generators\n(fundamental representation, cf.\\ p.628 in~\\cite{cmp}),\nit is reasonable to define \nan action on a quantum principal bundle $P(B,A)$ to be\n(compare with the Lagrangian given by (6.65) in \\cite{md2} or (4.1) in \\cite{md4})\n\\beq\\label{ym}\nYM(\\omega) = - \\llp{\\cal T}\\ci Tr\\ci (F\\sb\\omega*F\\sb\\omega)\\lrp(T),\n\\eeq\nwhere \n$Tr$ is the usual matrix trace, and\n\\mbox{${\\cal T}:\\Omega P\\ra k$} is a linear map vanishing on\n$[P,\\Omega P]$. (To ensure that we have an ample supply\nof connections, throughout this section we will\nuse the universal differential calculus.)\nClearly, remembering the property of $T$ that,\nfor any $\\mbox{f}\\sb 1$ and $\\mbox{f}\\sb 2\\,$,\n$\n(\\mbox{f}\\sb 1\\! *\\! \\mbox{f}\\sb 2)(T\\sb{ab})=\\sum\\sb c\n\\mbox{f}\\sb 1(T\\sb{ac})\\, \\mbox{f}\\sb 2(T\\sb{cb})\\, ,\n$\none can see that\n\\[\n\\mbox{\\large$\\forall$}\\,\nf\\!\\in\\! GT(P) , \\omega\\!\\in\\!\\cal C(P) :\\,\nYM(G\\!\\sb f\\omega) = YM (\\omega)\\, .\n\\]\nSimilarly, if $P(B,A)$\nis a trivial bundle with a trivialization\n$\\Phi$ (which we will also assume for\nthe rest of this section), then\nfor any $\\omega\\,$,\n\\beq\\label{ymbeta}\nYM(\\omega) = -\\llp {\\cal T}\\ci\nTr\\ci(\\Phi*F\\sb\\omega*F\\sb\\omega\n*\\Phi\\sp{-1})\\lrp(T) =\n-\\llp {\\cal T}\\ci Tr\\ci\n(F\\sb\\beta*F\\sb\\beta)\\lrp(T) =:YM(\\beta )\\, ,\n\\eeq\n where $\\beta$ is given by formula (\\ref{betadef})\n(see Remark~\\ref{secrem}) and $F\\sb\\beta $\nis the curvature form associated to it (see the end of the previous section).\nAs to \\ct , observe that one should expect to have a lot of information\nvested in it: projection from the universal to a non-universal calculus and\nmetric (Hodge star). In what follows, we specify \\ct\\ in such a way as to\nincorporate the Yang--Mills functional presented in \\cite{conri,ri} into the quantum\nbundle framework. One ought to bear in mind that to obtain a $q$-deformed\nYang--Mills theory in the spirit of quantum groups, one should invent another\n\\ct\\ or change the formula (\\ref{ym}) altogether. Here, however, we investigate\nwhat effects can entail from the deformation of the structure group alone.\n\n\\bde\\label{tld}\nLet \\ctb\\ be a faithful invariant trace on $B$ (as in \\cite{conri,ri}),\n$\\cal L$ be a finite dimensional Lie subalgebra of $\\,\\mbox{\\em Der}(B)$, \n$\\{ X\\sb l\\}\\sb{l\\in\\{1,...,\\mbox{\\em\\scriptsize dim}\\cal L\\}}$ be its basis, and\n$\\widetilde{X}\\sb l=X\\sb l\\ot id$,\\linebreak\n $l\\in\\{1,...\\, ,\\mbox{\\em dim}\\,\\cal L\\}$. \nWe put\n\\[\n\\ct\\ (\\alpha )=\n\\left\\{\n\\begin{array}{ll}\n\\hsp{3mm}(\\ctb\\te\\epsilon)\\; (\\alpha) & \\mbox{if $\\deg\\alpha =0$}\\\\\n\\phantom{1} & \\phantom{1}\\\\\n\\hsp{3mm}\\displaystyle 0 & \\mbox{if $\\deg\\alpha =2m-1$ }\\\\\n\\phantom{1} & \\phantom{1}\\\\\n{\\displaystyle\\sum\\sb{l\\sb 1<\\cdots 0$.}}\n\\[\\Delta T =T\\buildrel .\\over\\ot T,\\;\\;\n\\epsilon (T) =I,\\;\\; S(T)=T\\sp\\dagger ,\\]\nwhere\n\\[\nT=\\pmatrix{a&b\\cr -q\\sp{-1}b\\sp*t\\sp* &\n a\\sp*t\\sp*}\\,\n \\]\nand $\\stackrel{.}{\\ot}$ is the matrix\nmultiplication with the product of its\nentries replaced by the tensor product $\\ot\\,$;\nand\n\\[\n\\Delta t =t\\ot t , \\;\\; \\epsilon(t) =1\n,\\;\\; S(t)=t\\sp*,\\;\\; S(t\\sp*) =t.\n\\]\n\\ede\n\nThe next step is to establish an equivalence between a certain\nclass of strong connections on a trivial $U\\sb q(2)$-bundle\nand the space of hermitian connections on the associated free module.\nSince any $\\beta\\in\n\\mbox{Hom}\\sb{{\\Bbb C}}(U\\sb q(2),\\Omega B)$ satisfying\n$\\beta({\\Bbb C})=0$ is\na connection form on the base space $B$ of a\n trivial $U\\sb q(2)$ bundle (see Remark~\\ref{secrem}\n and the paragraph below) and the\n$\\beta$-induced connection $\\nabla$ on $B\\sp 2$\n depends only on $\\beta(T)$, we will\nrestrict our attention to connection forms\nthat are, in some way, uniquely\ndetermined by $\\beta(T)$. In the classical situation, $\\beta$ is an\n\\mbox{$\\epsilon$-derivation} and thus is automatically\ndetermined by $\\beta (T)$.\nMimicking this classical differential geometry formula for $\\beta$,\nwe define an auxiliary map \n\\mbox{$\n\\widetilde\\hb :\\Bbb C\\langle a,b,t,a\\sp*, b\\sp*,t\\sp*,1\\rangle\n\\ra\\hO\\sp1\\! B\n$}\nby\\\\\n\n\\begin{eqnarray}\\label{betafor}\n&& \n\\hsp{-2mm}\\widetilde{\\beta}(\na\\sp{k\\sb1}a\\sp{*\\, l\\sb1}b\\sp{m\\sb1}b\\sp{*\\, n\\sb1}t\\sp{p\\sb1}t\\sp{*\\, r\\sb1}\n\\cdots \na\\sp{k\\sb s}a\\sp{*\\, l\\sb s}b\\sp{m\\sb s}b\\sp{*\\, n\\sb s}t\\sp{p\\sb s}t\\sp{*\\, r\\sb s})\n\\nonumber \\\\ && \\ \\nonumber \\\\ &&= \n\\left\\{\\begin{array}{ll}\n0 &\\mbox{for $\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i\\geq 2$ }\n\\\\ \\ & \\ \\\\\nq\\sp{\\sum\\sb{i=j+1}\\sp s(l\\sb i-k\\sb i)}\\;\\widetilde{b} \n& \\mbox{for $m\\sb j=1$, $j\\in\\{ 1,...,s\\}$, \n$\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0$\\phantom{........}}\n\\\\ \\ & \\ \\\\\nq\\sp{\\sum\\sb{i=j+1}\\sp s(l\\sb i-k\\sb i)}\\;\\widetilde{b}\\,\\sp* \n&\\mbox{for $n\\sb j=1$, $j\\in\\{ 1,...,s\\}$, \n$\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0$ }\n\\\\ \\ & \\ \\\\\n{\\displaystyle\\sum\\sb{i=1}\\sp sk\\sb i\\widetilde{a}\n+\\sum\\sb{i=1}\\sp sl\\sb i\\widetilde{a}\\,\\sp*\n+\\sum\\sb{i=1}\\sp sp\\sb i\\widetilde{t} \n+\\sum\\sb{i=1}\\sp sr\\sb i\\widetilde{t}\\,\\sp*}\n& \\mbox{for $\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0\\, ,$}\n\\end{array} \\right .\n\\end{eqnarray}\\ \\\\\n\nwhere, a priori, $\\widetilde{a}\\, ,\\,\\widetilde{b}$ and $\\widetilde{t}$\nare any differential forms in $\\Omega\\sp 1\\! B$. \nOne can verify that\\linebreak\n \\mbox{$\\widetilde\\hb (\\cal J+\\cal J\\sp*)=0$} if\n$\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\n\\widetilde{t}+\\widetilde{t}\\,\\sp*$. Hence, since\n\\[\n\\widetilde\\beta \n(aa\\sp*+bb\\sp*-1)=0=\\widetilde\\beta (tt\\sp*-1)\\;\\;\\mbox{\\boldmath$\\Rightarrow$}\n\\;\\;\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\\widetilde{t}+\\widetilde{t}\\,\\sp*\\, ,\n\\]\nwe can conclude that (\\ref{betafor})\ndefines $\\hb\\in\\mbox{Hom}\\sb{\\Bbb C}(U\\sb q(2),\\hO\\sp1\\! B)$ such that\n$\\hb (1)=0$\n(i.e., a \\mbox{$U\\sb q(2)$-connection} form) if and only if\n$\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\\widetilde{t}+\\widetilde{t}\\,\\sp*$.\nMoreover, it is straightforward to check that\n\\beq\\label{dag}\n\\beta(T)\\sp\\dagger\\pmatrix{1&0\\cr 0&q}+\\pmatrix{1&0\\cr 0&q}\n\\beta(T) =0\\, ,\n\\eeq\n and conversely, that for every $M\\in M\\sb 2\\llp\\Omega\\sp1\\! B\\lrp$ satisfying \n(\\ref{dag}) we can find unique\n$\\widetilde{a}$, $\\widetilde{b}$, $\\widetilde{t}$ such that $\\beta (T) = M$\n(cf.~Proposition~1 in~\\cite{av}).\nThus we have proved the following:\n\n\\bpr\\label{hermitian}\nLet \n\\mbox{$\\cal C\\sb B=\\{\\hb\\!\\in\\!\\mbox{\\em Hom}\\sb{\\Bbb C}(U\\sb q(2),\\hO\\sp1\\! B)\n\\, |\\;\n\\hb\\ \\mbox{satisfies (\\ref{beta2}) for some}\\;\\widetilde{a},\n\\widetilde{b},\\widetilde{t}\\in\\hO\\sp1\\! B\\}$} and \n$\\cal C\\cal C(B\\sp2)$ denote the space of Hermitian connections on $B\\sp2$\n(see Point~1 and Point~3 of Definition~\\ref{pmdef}), where\n\\beq\\label{beta2}\n\\beta(a\\sp ka\\sp{*l}b\\sp m b\\sp{*n}t\\sp\npt\\sp{*r}):= \\left\\{\\begin{array}{ll}\n0 &\\mbox{for $m+n\\geq 2$ }\\\\\n\\widetilde{b} & \\mbox{for $m=1$, $n=0$}\\\\\n\\widetilde{b}\\,\\sp* &\\mbox{for $m=0$, $n=1$\n }\\\\\n(k-l)\\widetilde{a}+(p-r)\\widetilde{t}\n & \\mbox{for $m=n=0\\, ,$}\n\\end{array} \\right .\n\\eeq\nand $\\langle\\; ,\\;\\rangle$ is\ngiven by the formula $\\langle\\xi , \\zeta\\rangle =\\xi\\sp\\dagger \n\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}\\zeta$.\nThe map \n\\[\n\\psi :\\cal C\\sb B\\ni\\hb\\longmapsto d+\\hb (T)\\in\\cal C\\cal C(B\\sp2)\n\\]\nis a bijection.\n\\epr\n\n\\bco\\label{hercor}\nLet $P(B,U\\sb q(2))$ be a trivial quantum principal bundle with the\nuniversal differential calculus. There exists a one-to-one correspondence\nbetween the elements of $\\cal C\\cal C(B\\sp2)$, i.e.~the universal\ncalculus connections on $B\\sp2$ that are compatible with the Hermitian\nmetric given by the matrix $\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}$, \nand the strong connections\non $P(B,U\\sb q(2))$ whose `pullback' on the base space $B$ (given\nby (\\ref{betadef})) satisfies (\\ref{beta2}).\n\\eco\n\nThis way, we obtain the Yang--Mills theory of connections compatible with the\n\\mbox{$q$-d}ependent Hermitian structure on $B\\sp 2$. Now, in order to handle\nthe critical points of $YM$ the same way they were dealt with in \\cite{ri},\nlet us assume (as in \\cite{ri}, p.535--6) that $B$ is a smooth dense \n$*$-subalgebra of a $C\\sp*$-algebra, and that it is equipped with a faithful\ninvariant trace~\\ctb . As was argued in Section~1 of\n\\cite{conri}, the Yang--Mills functional\ndoes not depend on the choice of\na Hermitian metric --- we can gauge $q$ out of the picture. \nHence, the critical points of the \\mbox{$U\\sb q(2)$--Yang--Mills} action\nfunctional are simply the critical points\nof the Yang--Mills action (see p.536 in~\\cite{ri}), if there are any,\n`rotated' by the appropriate $q$-dependent gauge\ntransformation (see Section~1 in~\\cite{conri}). \n(More explicitly, as the Hermitian metric is given by\n$\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}$, \nthe corresponding gauge transformation is\n$\\nabla\\to\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&\\sqrt{q}}$}\\sp{-1}\n\\nabla\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&\\sqrt{q}}$}$.)\nConsequently, we have:\n\n\\bco\\label{moduli}\nIn the above described setting, the $U\\sb q(2)$\nand $U(2)$--Yang--Mills theories have\n the same moduli spaces of critical points (cf.\\ Section~4 in\n\\cite{ri} and p.582 in~\\cite{conbook}). \n\\eco\n\nAnother way to remove $q$\nfrom the picture is to alter (\\ref{beta2}) by replacing $\\widetilde{b}\\,\\sp*$ by\n$q\\widetilde{b}\\,\\sp*$. Then, however, we would lose the\ngeometrical interpretation of the\n action of the $\\mbox{$q$-deformation}$ of $U(2)$\non the space of\ncompatible connections as the action of\n the gauge transformation. Also, the formula\n$\\beta(T\\sp*) = \\beta(T)\\sp* $ \nwould no longer be\ntrue. (Caution: at least in the general case, we cannot\nclaim that $\\beta$ is a \\mbox{$*$-morphism} even when it commutes\nwith the $*$ on the generators.) In any case, we can see\n that the Yang--Mills theory remains unchanged\nfor any $q\\in{\\Bbb R}\\sb +$. A~similar\n situation was discussed in the context of\nquantum group gauge theories on classical\n spaces in the last two paragraphs of\nSection~2 in \\cite{bmc}.\n\n\\bre\\label{sl}\\em\nThe reason for employing in the considerations\nabove $U\\sb q(2)$ rather than \n$SU\\sb q(2)$ is that, when using $SU\\sb q(2)$, formula\n(\\ref{beta2}) entails\n$Tr\\,\\beta (T) = 0$ for all $\\beta$, and although the tracelessness of\n $\\beta (T)$ is automatically\npreserved by the entire $GL(B\\sp2)$ in the\nclassical case, we cannot claim the\nsame in general (cf.~Introduction and Section~3 in~\\cite{arar}). \nNor can we claim that the\ntracelessness of $\\beta (T)$ is preserved by the \ngauge group $U(B\\sp2)$ of unitary automorphisms of $B\\sp 2$\n(see Remark~\\ref{ua}). Besides, $U\\sb q(2)$-connections\ngiven by (\\ref{beta2}) can be regarded as $\\langle \\; ,\\;\\rangle$-compatible\n connections, and vice versa, which makes a clear analogy with\nthe classical situation, where Hermitian metrics are $U(2)$-structures\n(cf.\\ p.13 in \\cite{tr}), and $U(2)$-connections are automatically compatible\n with the corresponding Hermitian structures (cf.~p.94--5 in~\\cite{tr}).\n\\hfill{$\\Diamond$}\\ere\n\n Finally, let us mention that\nan alternative approach would be to work with a non-universal differential\ncalculus instead of assuming (\\ref{betafor}), which is put in\nthe theory by hand. But that is yet another story.\n\\bigskip\n\\bigskip\n\n{\\parindent=0pt\n{\\bf\\Large Appendix}\\vspace*{-6pt}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n Physics of QCD jets is one of the most important testing grounds\nfor the theory of strong interactions. The perturbative evolution of a\nquark-gluon jet is at present well understood (see, e.g., \\cite{BPQ}). The known\ndetails include in particular the famous colour coherence phenomenon which\nleads to a probabilistic picture of timelike jet evolution where the daughter\npartons are emitted into a gradually shrinking cones. The nonperturbative\naspects of a jet evolution are much less understood. From the experience\nof a sum rule approach applied to the analysis of heavy resonance properties\n \\cite{SVZ} it is clear, that when the virtuality of a daughter parton is\n of order of\n1 {\\mbox {GeV}} one has to take into account the nonperturbative corrections\ndue to the nonperturbative vacuum fields giving rise to the QCD vacuum\ncondensates.\nAnother situation in which one deals with dissipative effects is a jet\npropagation in, e.g., nuclear medium. Although physically these situations\nare very different, we expect the corresponding theoretical formalism\nto be the same.\n\n In both cases the new interaction vertices lead to the appearance of new\ndimensionful parameters. This means in turn that a scaling description\nof jet evolution where the interaction vertices are determined only by\ndimensionless quantities (energy or virtuality ratios) is no longer valid.\nPhysically what happens is a beginning of a string formation taking energy\nfrom the perturbative component and converting it to nonperturbative degrees\nof freedom. The first model description of a jet evolution taking into account\nthe nonperturbative energy loss was proposed long ago by Dremin \\cite{Dr}.\n It was\nbased on the analogy with the physics oh high energy electromagnetic showers\nin the medium, where apart from a scale-invariant evolution due to a photon\nbremmstrahlung and pair creation there appears a scale-noninvariant\nenergy dissipation due to the ionization of the atoms in the medium \\cite{Be}.\nLater the corresponding modified evolution equations were analytically solved\nin \\cite{DL1}, where the expressions for the parton multiplicities in quark and\n gluon\njets and the energy loss by the perturbative component were computed.\nA discussion of this approach and also of the latest Monte-Carlo calculations,\n\\cite{GE} where the nonperturbative component was described by QCD effective\nlagrangeans can be found in the recent review \\cite{DL2}. A serious drawback\n of Dremin equations \\cite{Dr} is, however, a disregard of the kinematical\nand color interference effects in the jet evolution which are known to be\nof crucial importance for describing the jet characteristics. The main goal of\n the present paper is to introduce a formalism\nproviding a possibility of an analytical treatment of the jet evolution\ntaking into account both color interference and nonperturbative energy loss.\n\nBefore writing down the modified evolution equations let us recall, that color\ncoherence in QCD jets can be described using various approximations. The simplest\nis a leading logarithmic one (DLA, see \\cite{BPQ}). The calculations in this\napproximation are relatively simple, but the energy of the jet is not conserved.\nThis is due to the eikonal description used in this approximation, when the\nenergy loss of the projectile is considered to be negligible. A more refined\nanalysis is made within a modified leading logarithmic approximation (MLLA),\nwhere the total energy is conserved. The Dremin equations \\cite{Dr}\ncorrespond to a case, where the energy is perturbatively conserved. Thus to\ncalculate an absolute value of the dissipative energy loss one has to modify\nan MLLA formalism. The corresponding equations can be written (see Appendix A),\nbut unfortunately we were not able to find their solutions. Thus in what\nfollows we shall concentrate on a simpler DLA case, where the energy\nnonconservation is built in already at the level of perturbation theory.\nIn this case one can not calculate the absolute energy loss, but the relative\none is fairly well-defined.\n\n\n\\section{DLA Equation for the Generating Functional}\n\nLet us turn to a description of a formalism which generalizes the known\neffective methods of describing the QCD jet evolution including color\ncoherence effect \\cite{BPQ} by providing a possibility of considering\nthe new possible nonperturbative sources of energy dissipation.\n\nTechnically it is most convenient to use a generating functional.\n$G(\\{u\\})$ from which one can calculate both an exclusive cross section\n\\begin{equation}\nd\\sigma^{excl}_N=\\left(\\prod^N_{i=1}\\,d^3k_i\\,\\frac{\\delta}{\\delta u(k_i)}\n\\right)\\,G(\\{u\\})\\Big|_{u=0}d\\sigma_0.\n\\end{equation}\nand an inclusive one\n\\begin{equation}\nd\\sigma^{incl}_N=\\left(\\prod^N_{i=1}\\,d^3k_i\\,\\frac{\\delta}{\\delta u(k_i)}\n\\right)\\,G(\\{u\\})\\Big|_{u=1}d\\sigma_0.\n\\end{equation}\nIn the following we shall use the notations\n$dK=\\gamma_0^2\\frac{dk}{k}\\,\\frac{d^2k_\\perp}{2\\pi k^2_\\perp}$ for the\nusual DLA integration measure and $\\Gamma(p,\\theta)$ for the possible gluon\nemission phase space domain, where $p$ is an initial gluon energy and\n$\\theta$ is an opening angle of a jet. Let us remind, that in DLA\napproximation the generating functional $G(p,\\theta;\\{u\\})$ satisfies\nthe following master equation\n\n\\begin{flushleft}\n$G(p,\\theta;\\{u\\}) = u(p)e^{-w(p,\\theta)}+$\n\\end{flushleft}\n\\begin{equation}\\label{Master}\n\\int_{\\Gamma(p,\\theta)} {dk \\over k}\n{d^2 k_{\\perp} \\over {{k}_{\\perp}}^2} {2 C_V \\alpha_s \\over \\pi}\ne^{-w(p,\\theta)+w(p,\\theta_k)} G(p,\\theta_k;\\{u\\}) G(k,\\theta_k;\\{u\\}) ,\n\\end{equation}\nwhere\n$$\n w(p,\\theta)= \\int_{\\Gamma(p,\\theta)} dK\n$$\nis a total probability of an emission of a gluon having an energy $p$\ninto the cone having an opening angle $\\theta$, and $\\exp (-w(p,\\theta))$\nis a formfactor giving a probability of a non-emission of the same gluon.\nThen a sandard procedure (\\cite{BPQ}) leads to the following equation\non the generating functional:\n\\begin{equation}\\label{Gold}\nG(p,\\theta;\\{u\\})=u(p)\\,exp\\left(\\fspace{p}\\,[G(k,\\theta_k;\\{u\\})-1]\\right).\n\\end{equation}\n\nLet us now discuss the possible nonperturbative modifications of the equations\nfor the generating functional. The aim is to describe a partial convertion\nof the perturbative degrees of freedom (gluons) and their energy\ninto the nonperturbative\nones by the process presumably analogous to the ionization in the case of\nelectromagnetic showers discussed in the Introduction. Physically the\npicture for the jet is that the dissipation pumps some energy from the\ncone corresponding to the perturbative jet evolution reducing the gluon\nmultiplicity and energy inside it. Thus we have some additional nonperturbative\nvertex coupling the perturbative gluons with the nonperturbative vacuum or\nnuclear medium.\nThe presence of such a coupling can be related to reducing a probability of an\nemission of a perturbative gluon by taking away a certain portion of its energy\nbefore the perturbative emission. Techically it is convenient to introduce a\nfollowing modification of the gluon emission probability:\n\\begin{equation}\n w(p,\\theta;\\beta)= \\int_{\\Gamma(p-\\beta,\\theta)} dK,\n\\end{equation}\nwhere the constant $\\beta$ describes the additional damping of a perturbative\n gluon emission due to a dissipative interaction with the nonperturbative\nvacuum fluctuations or nuclear medium introduced by compressing a possible\n phase space domain\nfor the gluon emission from $\\Gamma(p,\\theta)$ to $\\Gamma(p-\\beta,\\theta)$.\n The corresponding modification of the master equation\n(\\ref{Master}) reads\n\\begin{flushleft}\n$G(p,\\theta;\\{u\\}) = u(p)e^{-w(p,\\theta;\\beta)}+$\n\\end{flushleft}\n\\begin{equation}\n\\int_{\\Gamma(p-\\beta,\\theta)} {dk \\over k}\n{d^2 k_{\\perp} \\over {{k}_{\\perp}}^2} {2 C_V \\alpha_s \\over \\pi}\ne^{-w(p,\\theta;\\beta)+w(p,\\theta_k;\\beta)}\nG(p,\\theta_k;\\{u\\}) G(k,\\theta_k;\\{u\\}) ,\n\\end{equation}\nThen instead of the Eq.~(\\ref{Gold}) we get\n\\begin{equation}\\label{Gnew}\nG(p,\\theta;\\{u\\})=u(p)\\,exp\\left(\\fspace{p-\\beta}\\,[G(k,\\theta_k;\\{u\\})-1]\n\\right),\n\\end{equation}\nwhere the initial conditions for this equation read $G(p,\\theta;\\{u\\})|_{u=1}=1$.\n\nWe suggest that $\\beta \\ll p$ for all physically meaningful values of energy\nand in the following we shall consider the terms of the first order in the\n small parameter $\\varepsilon=\\beta\/p$. In Appendix B we illustrate\nthe thus arising perturbation theory by deriving the differential equation on\n the\nleading correction to the generating functional.\n\n\\section{Mean multiplicity}\n\nLet us begin this section by looking at the simplest jet characteristic, a\nmean multiplicity of partons in a jet (in the following we are considering only\ngluons). Following the standard steps we obtain a following integral equation\non the mean multiplicity ${\\bar{n}}(p,\\theta)$:\n\\begin{equation}\n{\\bar{n}}(p,\\theta)=1+\\fspace{p-\\beta}\\,{\\bar{n}}(k,\\theta_k).\n\\end{equation}\\label{phas}\nIn the following it will be convenient to rewrite the integration over the phase\nspace as\n\\begin{equation}\n\\fspace{p-\\beta}=\\int_{\\Gamma(p-\\beta,\\theta)}\\frac{dk}{k}\\,\n\\frac{d^2k_\\perp}{2\\pi k^2_\\perp}\\gamma_0^2=\n\\int\\limits^{y-\\varepsilon}_0 d\\xi\\,\\int\\limits_0^\\xi dy'\\gamma_0^2,\n\\end{equation}\nwhere\n$$\ny=\\ln\\frac{p\\theta}{Q_0},\ny'=\\ln\\frac{k\\theta_k}{Q_0},\n\\xi=\\ln\\frac{k\\theta}{Q_0}.\n$$\nWorking in the first order in $\\varepsilon$ we can write\n$$\n{\\bar{n}} (p,\\theta)={\\bar{n}}_0 (y)+\\varepsilon {\\bar{n}}_1 (y).\n$$\nThe equations for the functions ${\\bar{n}}_0(y)$ and ${\\bar{n}}_1(y)$\n take the form\n\\begin{equation}\n{\\bar{n}}_0 (y)=1+\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_0 (y')\n\\end{equation}\nand\n\\begin{equation}\n {\\bar{n}}_1 (y)=\n\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_1 (y')\\,(e^{y-y'}-1)\\,-\\,\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_0 (y')\n\\end{equation}\nThe first one has a well known solution\n$$\n{\\bar{n}}_0 (y)= \\mbox{ch}(\\gamma_0 y),\n$$\nand the second one can be rewritten in the differential form\n\\begin{equation}\\label{dif}\n{\\bar{n}}_1 ''(y)- {\\bar{n}}_1'(y)\n=\\gamma_0^2 {\\bar{n}}_1 (y)+\\gamma_0^2(\\mbox{ch}(\\gamma_0y)\n-\\gamma_0\\mbox{sh}(\\gamma_0 y))\n\\end{equation}\nwith the initial conditions\n$ {\\bar{n}}(0)=0 , {\\bar{n}}'(0)=-\\gamma_0^2.$\nThe solution reads\n\\begin{equation}\n\\bar n_1=\\gamma_0^2\\mbox{ch}(\\gamma_0y)-\\gamma_0\\mbox{sh}(\\gamma_0y)+\n+\\gamma_0\\frac{\\lambda_2e^{\\lambda_1y}-\\lambda_1e^{\\lambda_2y}}\n{\\sqrt{1+4\\gamma_0^2}}.\n\\end{equation}\nIn the limit of $(y\\gg1,\\gamma_0\\ll1)$ we have\n\\begin{equation}\\label{n1}\n{\\bar{n}}\\approx\\frac{1}{2}e^{\\gamma_0y}(1-\\varepsilon\\gamma_0-\n2\\varepsilon\\gamma_0^3e^y)=\n\\frac{1}{2}\\left({\\frac{Q}{Q_0}}\\right)^{\\gamma_0}\\left(1-\\gamma_0\n\\frac{\\beta}{p}-2\\gamma_0^3\\frac{\\beta\\theta}{Q_0}\\right)\n\\end{equation}\nThe above result is in accord with the expectation based on physical reasoning.\nNamely, the dissipation should lead to a decrease in perturbative multiplicity.\nLet us note, that from Eq. (\\ref{n1}) it is clear, that in order for\nperturbation theory to be applicable one should have\n$\\beta \\ll p\/\\gamma_0$ and $\\beta \\ll Q_0\/\\gamma_0^3$. This means that apart\nfrom the expected expansion parameter $\\varepsilon$ there appears a new one\n$\\varepsilon'= 2\\gamma_0^2\\frac{\\beta\\theta}{Q_0}$. An interesting feature\nof the above answer is a multiplicative combination of the dissipation scale\n$\\beta$ and a perturbative coupling constant hidden in $\\gamma_0$.\n\n\\section {Energy distrubution of particles in a jet}\n\nLet us now turn to a computation of a one-particle energy distribution function\n$\\bar D(k,\\theta)$. It can be obtained from the generating functional in the\nfollowing way:\n\\begin{equation}\n\\bar D(k,\\theta)=k\\frac{\\delta}{\\delta u(k)}G(p,\\theta;\\{u\\})\\Big|_{u=1},\n\\end{equation}\nMaking use of (\\ref{Gnew}) and (\\ref{phas}) we get\n\\begin{equation}\\label{en}\n\\bar D(l,y)=\\delta(l)+\\int\\limits^{l-\\varepsilon}_0 dl' \\int\\limits^y_0 dy'\n\\gamma_0^2 \\bar D(l',y'),\n\\end{equation}\nwhere $l=\\ln(p\/k)=\\ln(1\/x)$ and $l'=\\ln(k'\/k)$. The integration is performed\nover $k'$ and $\\theta_{k'}$. Let us stress that in the above equation (\\ref{en})\nthe distrubution function $\\bar D$ is also $\\beta$ - dependent. Expanding\nit in the small parameter $\\varepsilon$\n$$\n\\bar D(l,y)=\\bar D_0(l,y)+\\varepsilon \\bar D_1(l,y)\n$$\nwe rewrite (\\ref{en}) in the form\n\\begin{equation}\n\\bar D_0(l,y)+\\varepsilon\\bar D_1(l,y)=\\delta(l)+\n\\int\\limits^{l-\\varepsilon}_0 dl' \\int\\limits^y_0 dy'\\gamma_0^2\n(\\bar D_0(l,y)+\\varepsilon e^{l-l'}\\bar D_1(l,y)).\n\\end{equation}\nConsidering the zero and first order in $\\varepsilon$ we have\n\\begin{equation}\\label{D0eq}\n\\bar D_0(l,y)=\\delta(l)+\\int\\limits^{l}_0 dl' \\int\\limits^y_0 dy'\n\\gamma_0^2 \\bar D_0(l',y').\n\\end{equation}\n\\begin{equation}\\label{D1eq}\n\\bar D_1(l,y)=\\int\\limits^{l}_0 dl' \\int\\limits^y_0 dy'\\gamma_0^2 e^{l-l'}\n\\bar D_1(l',y')-\\int^y_0 dy'\\gamma_0^2 \\bar D_0(l,y').\n\\end{equation}\nThe solution of equation (\\ref{D0eq}) reads\n\\begin{equation}\n\\bar D_0(l,y)=\\delta(l)+\\gamma_0\\sqrt{\\frac{y}{l}}I_1(2\\gamma_0\\sqrt{yl})\n\\end{equation}\nand that for $\\bar D_1$ is (for details see Appendix C):\n\\begin{equation}\\label{ensol}\n\\bar D_1(l,y)=-y\\gamma_0^2\\delta(l)-\\gamma_0^2(e^l+1)\\frac{y}{l}\nI_2(2\\gamma_0\\sqrt{yl})+\\gamma_0^3\\left(\\frac{y}{l}\\right)^{3\/2}(e^l-1)\nI_3(2\\gamma_0\\sqrt{yl}).\n\\end{equation}\nThe plot of the resulting distribution for the jet energy $20 {\\mbox { GeV}}$\nis given at Fig.~1. We see that, as anticipated, the effect of dissipation\nshows itself through the noticeable reduction of distribution function.\nLet us also mention that here we also have a multiplicative dependence on both\nperturbative and nonperturbative factors.\n\n\n\\section{Conclusions}\n\nIn this paper we have proposed a phenomenological nonperturbative\nmodification of the equations describing the evolution of QCD jets\nand taking into account the crucial feature of the colour coherence\nof the QCD cascades\nby accounting for nonperturbative energy dissipation. The corresponding\nmodification of the MLLA and DLA formalism exploiting the generating\n functional was proposed. The calculation of simplest jet characteristics\nsuch as mean multiplicity and energy spectrum of particles in a jet in a\nDLA approximation has demonstrated an expected decrease in multiplicity\n and corresponding changes in the energy distribution.\n\n An interesting feature of the result is an unusual\nperturbative damping of the introduced nonperturbative energy dissipation\nappearing in the multiplicative dependance on some power of the QCD coupling\nconstant times the dissipative scale.\nIt is tempting to relate this feature with the successes of the soft\nblanshing hypothesis, where it is assumed that the nonperturbative\neffects are not crucially essential for the jet characteristics. It is\ninteresting to see, whether such damping is present within a more realistic\nMLLA description. Work in this direction is in progress.\n\n\n\\section{Acknowledgements}\nWe are grateful to I.M.~Dremin and I.V.~Andreev for the useful discussions.\nA.L. is grateful to P.V.~Ruuskanen for kind hospitality in the University\nof Helsinki, where part of this work was done. The research was supported by\nRussian Fund for Fundamental Research, Grant 93-02-3815.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nPolymer materials, while possessing some unique and attractive qualities, such as low weight, high strength, resistance to chemicals, ease of processing, are for the most part\ninsulators. If methods could be devised to turn common insulating polymers into conductors, that would open great prospects for using such materials in many more areas than they are\ncurrently used. These areas may include organic solar cells, printing electronic circuits, light-emitting diodes, actuators, supercapacitors, chemical sensors, and biosensors \\cite{Long_2011}.\n\n\nSince the reliable methods for carbon nanotubes (CNT) fabrication had been developed in the 1990s, growing attention has been paid to\nthe possibility of dispersing CNTs in polymers, where CNTs junctions may form a percolation network and turn insulating polymer into a good conductor when a percolation threshold is overcome.\nAn additional benefit of using such polymer\/CNTs nanocomposites instead of intrinsically conducting polymers, such as polyaniline \\cite{Polyaniline}\nfor example, is that dispersed CNTs, besides providing electrical conductivity, enhance\npolymer mechanical properties as well.\n\nCNTs enhanced polymer nanocomposites have been intensively investigated experimentally, including composites conductivity \\cite{Eletskii}. As for the theoretical research in this area, the results are more modest. \nIf one is concerned with nanocomposite conductivity, its value \ndepends on many factors, among which are the polymer type, CNTs density, nanocomposite preparation technique, CNTs and their junctions geometry, a possible presence of defects\nin CNTs and others. Taking all these factors into\naccount and obtaining quantitatively correct results in modeling is a very challenging task since the resulting conductivity is formed at different length scales: at the microscopic level it is influenced by the\nCNTs junctions contact resistance and at the mesoscopic level it is determined by percolation through a network of CNTs junctions. Thus a consistent multi-scale method for the modeling of conductivity,\nstarting from atomistic first-principles calculations of electron transport through CNTs junctions is necessary.\n\nDue to the complexity of this multi-scale task, the majority of investigations in the area are carried out in some simplified forms, this is especially true for the underlying part of the modeling: determination\nof CNTs junction contact resistance. For the contact resistance either experimental values as in \\cite{Soto_2015} or the results of phenomenological Simmons model as in \\cite{Xu_2013, \nYu_2010, Jang_2015, Pal_2016} are usually taken, or even an arbitrary value of contact resistance reasonable by an order of magnitude may be set \\cite{Wescott_2007}. In \\cite{Bao_2011, Grabowski_2017} the tunneling probability through a CNT junction is modeled using a rectangular potential barrier and the quasi-classical approximation.\n\nThe authors of \\cite{Castellino_2016} employed an oversimplified two-parameter expression for contact resistance, with these parameters fitted to the experimental data.\nThe best microscopic attempt, that we are aware of, is using the semi-phenomenological tight-binding approximation for the calculations of contact resistance \\cite{Penazzi_2013}. But in \\cite{Penazzi_2013} just\nthe microscopic part of the nanocomposites conductivity problem is addressed, and the conductivity of nanocomposite is not calculated. Moreover, in \\cite{Penazzi_2013} the coaxial CNTs configuration is only considered,\nwhich is hardly realistic for real polymers.\n\nThus, the majority of investigations are concentrated on the mesoscopic part of the task:\nrefining a percolation model or phenomenologically taking into account different geometry peculiarities of CNTs junctions. Moreover, comparison with experiments is missing in some publications on this topic.\nThus, a truly multi-scale research, capable of providing quantitative results comparable with experiments, combining fully first-principles calculations of contact resistance on the microscopic level with a percolation model on \nthe mesoscopic level seems to be missing. \n\nIn our previous research \\cite{comp_no_pol}, we proposed an efficient and precise method for fully first-principles calculations of CNTs contact resistance and combined it with a Monte-Carlo statistical percolation model to\ncalculate the conductivity of a simplified example network of CNTs junctions without polymer filling. \nIn the current paper, we are applying the developed approach to the modeling of conductivity of the CNTs enhanced polymer polyimide R-BAPB. \n\nR-BAPB (Fig. \\ref{fig_RBAPB-struct}) is a novel polyetherimide synthesized using 1,3-bis-(3$'$,4-dicarboxyphenoxy)-benzene (dianhidride R) and 4,4$'$-bis-(4$''$-aminophenoxy)diphenyl (diamine BAPB). It is thermostable polymer with extremely high thermomechanical properties (glass transition temperature $T_g= 453-463$~K, melting temperature $T_m= 588$~K, Young's modulus $E= 3.2$~GPa) \\cite{Yudin_JAPS}. This polyetherimide could be used as a binder to produce composite and nanocomposite materials demanded in shipbuilding, aerospace, and other fields of industry. The two main advantages of the R-BAPB among other thermostable polymers are thermoplasticity and crystallinity. R-BAPB-based composites could be produced and processed using convenient melt technologies.\n\nCrystallinity of R-BAPB in composites leads to improved mechanical properties of the materials, including bulk composites and nanocomposite fibers. It is well known that carbon nanofillers could act as nucleating agents for R-BAPB, increasing the degree of crystallinity of the polymer matrix in composites. As it was shown in experimental and theoretical studies\n\\cite{Yudin_MRM05, Larin_RSCADV14, Falkovich_RSCADV14,Yudin_CST07}, the degree of crystallinity of \ncarbon nanofiller enhanced R-BAPB may be comparable to that of bulk polymers.\n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_Larin1\/RBAPB.png}\n\\caption{The chemical structure of R-BAPB polyimide.} \n\\label{fig_RBAPB-struct}\n\\end{figure}\n\nOrdering of polymer chains relative to nanotube axes could certainly influence a conductance of the polymer filled nanoparticle junctions. However, it is expected that such influence will depend on many parameters, including the structure of a junction, position, and orientation of chain fragments on the nanotube surface close to a junction, and others. Taking into account all of these parameters is a rather complex task that requires high computational resources for atomistic modeling and ab-initio calculations, as well as complex analysis procedures. Thus, on the current stage of the study, we consider only systems where the polymer matrix was in an amorphous state, i.e. no sufficient polymer chains ordering relative to nanotubes were observed.\n\n\n\n\n\n\\section*{Description of the multiscale procedure} \n\nThe modeling of polymer nanocomposite electrical conductivity is based on a multi-scale approach, in which different simulation models are used at different scales. For the electron transport in polymer composites with a conducting filler, the lowest scale corresponds to the contact resistance between tubes. The contact resistance is determined at the atomistic scale by tunneling of electrons between the filler particles via a polymer matrix, and hence, analysis of contact resistance requires knowledge of the atomistic structure of a contact. Therefore, at the first step, we develop an atomistic model of the contact between carbon nanotubes in a polyimide matrix using the molecular dynamics (MD) method. This method gives us the structure of the intercalated polymer molecules between carbon nanotubes for different intersection angles between the nanotubes. One should mention, that since a polymer matrix is soft, the contact structure varies with time and, therefore, we use molecular dynamics to sample these structures. \n\nBased on the determined atomistic structures of the contacts between nanotubes in the polymer matrix we calculate electron transport through the junction using electronic structure calculations and the formalism of the Green's matrix. Since this analysis requires first-principles methods, one has to reduce the size of the atomistic structure of a contact to acceptable values for the first-principles methods, and we developed a special procedure for cutting the contact structure from MD results. First-principles calculations of contact resistance should be performed for all snapshots of an atomistic contact structure of MD simulations, and an average value and a standard deviation should be extracted. In this way, one can get the dependence of a contact resistance on the intersection angle and contact distance.\n\nUsing information about contact resistances we estimate the macroscopic conductivity of a composite with nanotube fillers. For this, we used a percolation model based on the Monte Carlo method to construct a nanotube network in a polymer matrix. In this model, we used distributions of contact resistances, obtained from the first-principles calculations for the given angle between nanotubes. Using this Monte Carlo percolation model one can investigate the influence of non-uniformities of a nanotube distribution on macroscopic electrical conductivity. \n\nIn the A section, we will describe the details of molecular dynamics modeling of the atomistic structure of contacts between nanotubes. In the B section, we present the details of first-principles calculations of electron transport for estimates of contact resistance. Finally, in the C section, we present the details of the Monte Carlo percolation model.\n\n\\subsection*{A. Preparation of the composite atomic configurations}\n\nInitially, two metallic CNTs with chirality (5,5) were constructed and separated by 6~\\AA. The CNTs consisted of 20 periods along the axis, and each one had the total length of 4.92~nm.\nThe broken bonds at the ends of the CNTs were saturated with Hydrogen atoms.\nThe distance 6 \\AA\\ was chosen, because starting with this distance polymer molecules are able to penetrate the space between CNTs. The three configurations of CNTs junctions were prepared: the first one with parallel CNTs axes (angle between nanotube axes $\\varphi=0^\\circ$),\nthe second one with the axes crossing at 45 degrees ($\\varphi=45^\\circ$), and the third one with perpendicular axes ($\\varphi=90^\\circ$).\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_em1.png}&\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_compress1.png}\\\\\n\\end{tabular}\n\\caption{The snapshots of the the nanocomposite system with the parallel orientation of carbon nanotubes at the initial state (left picture) and after the compression procedure (right picture). The black lines represent the periodic simulation cell.} \\label{fig_Larin-conf}\n\\end{figure}\n\nTo produce the polymer filled samples, we used \nthe procedure similar to that employed for the simulations of the thermoplastic polyimides and polyimide-based nanocomposites in the previous works\n\\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14, Nazarychev_MA16, Larin_RSCADV15}. First, partially coiled R-BAPB chains with the polymerization degree $N_p=8$, which corresponds to the polymer regime onset \\cite{Lyulin_MA13, Lyulin_SM14}, were added to the simulation box at random positions avoiding overlapping of polymer chains. This results in the initial configuration of samples with a rather low overall density ($\\rho\\sim 100~$ kg\/m$^3$) (Fig. \\ref{fig_Larin-conf}). Then the molecular dynamics simulations were performed to compress the systems generated, equilibrate them and perform production runs.\n\nThe molecular dynamics simulations were carried out using Gromacs simulation package \\cite{gromacs1, gromacs2}. The atomistic models used to represent both the R-BAPB polyimide and CNTs were parameterized using the Gromos53a6 forcefield \\cite{gromos}. Partial charges were calculated using the Hartree-Fock quantum-mechanical method with the 6-31G* basis set, and the Mulliken method was applied to estimate the values of the particle charges from an electron density distribution. As it was shown recently, this combination of the force field and particle charges parameterization method allows one to reproduce qualitatively and quantitatively the thermophysical properties of thermoplastic polyimides \\cite{Nazarychev_EI}. The model used in the present work was successfully utilized to study structural, thermophysical and mechanical properties of the R-BAPB polyimide and R-BAPB-based nanocomposites \\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14}.\n\nAll simulations were performed using the NpT ensemble at temperature $T=600$~K, which is higher than the glass transition temperature of R-BAPB. The temperature and pressure values were maintained using Berendsen thermostat and barostat \\cite{Berendses_1, Berendsen_2} with relaxation times $\\tau_T= 0.1$~ps and $\\tau_p=0.5$~ps respectively. The electrostatic interactions were taken into account using the particle-mesh Ewald summation (PME) method \\cite{PME1, PME2}. \n\nThe step-wise compression procedure allows one to obtain dense samples with an overall density close to the experimental polyimide density value ($\\rho \\approx 1250-1300$~kg\/m$^3$), as shown in Fig. \\ref{fig_Larin-conf}. The system pressure $p$ during compression was increased in a step-wise manner up to $p=1000$~bar and decreased then to $p=1$~bar. After compression and equilibration, the production runs were performed to obtain the set of polymer filled CNT junction configurations.\n\nAs the conductance of polymer filled CNT junctions is influenced by the density and structure of a polymer matrix in the nearest vicinity of a contact between CNTs, the relaxation of the overall system density was used as the system equilibration criterion. To estimate the equilibration time, the time dependence of the system density was calculated as well as the density autocorrelation function $C_\\rho(t)$:\n\\begin{equation}\nC_\\rho(t)=\\frac{\\langle \\rho(0) \\rho(t)\\rangle}{\\langle \\rho^2 \\rangle},\n\\end{equation}\nwhere $\\rho(t)$ is the density of the system at time $t$ and $\\langle \\rho^2 \\rangle$ is the average density of the system during the simulation.\n\n\nAs shown in Fig. \\ref{fig_Larin-density}a, the system density does not change sufficiently during simulation after the compression procedure. At the same time, the analysis of the density auto-correlation functions shows some difference in the relaxation processes in the systems studied (see Fig. \\ref{fig_Larin-density}b). In the case of the system where CNTs were placed parallel to each other ($\\varphi=0^\\circ$), $C_\\rho(t)$ could be approximated by the exponential decay function $C_\\rho(t)=\\exp(-t\/\\tau)$ with relaxation time $\\tau = 4$~ps. The density relaxation in the systems with crossed CNTs ($\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) was found to be slower. For these two systems density the auto-correlation functions could be approximated by a double exponential function $C_\\rho(t)=A\\exp(-t\/\\tau_1) + (1-A)\\exp(-t\/\\tau_2)$, and the relaxation times determined using this fitting were $\\tau_1=2.7$~ps and $\\tau_2=12.2$~ns (for $\\varphi=90^\\circ$), and $\\tau_1=9.5$~ps and $\\tau_2=24.6$~ns (in case of $\\varphi=45^\\circ$).\n\nNevertheless, the results obtained after the analysis of the system density relaxation allow us to choose the system equilibration time to be 100 ns, which is higher than the longest system density relaxation times determined by the density autocorrelation function analysis. \nThe same simulation time was used in our previous works to equilibrate the nanocomposite structure after switching on electrostatic interactions \\cite{Larin_RSCADV14, Nazarychev_EI, Larin_RSCADV15}. The equilibration was followed by the 150 ns long production run. To analyze the polymer filled CNT junction conductance, 31 configurations of each simulated system, separated by 5 ns intervals, were taken from the production run trajectory.\n\n\\begin{figure}[h]\n\\begin{tabular}{ccc}\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_left.pdf}\\\\\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_right.pdf}\\\\\n\\end{tabular}\n\\caption{The time dependence of the system density $\\rho$ (a) and the density auto-correlation functions $C_\\rho(t)$ (b) for the systems with various angles between nanotube axes $\\varphi$. The dots correspond to the calculated data. The solid lines correspond to the fitting of $C_\\rho(t)$ with the exponential (in case of $\\varphi=0^\\circ$) or double exponential (in case of of $\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) functions.} \n\\label{fig_Larin-density}\n\\end{figure}\n\nAfter the configurational relaxation is finished, we have to prepare polymer filled CNT junctions configuration for the first-principles calculations of contact resistance. \nThe method we used for the calculations of a contact resistance is based on the solution of the ballistic electronic transport problem, finding the Volt-Ampere characteristic $I(V)$ of a device and\nderiving the contact resistance from the linear part of $I(V)$ corresponding to the low voltages. For this purpose, we employed the Green's function method for \nsolving the scattering problem and the Landauer-Buttiker approach to find the current through a scattering region coupled to two semi-infinite leads, as described in \\cite{Datta}.\nSpecific details of how these techniques are applied in the case of crossed CNTs can be found in \\cite{comp_no_pol}.\n\n\\subsection*{B. The first-principles calculations of the contact resistance of CNTs junctions filled with polymer} \\label{sec_meth_fp}\n\nFor the preparation of a device for the electronic transport calculations, we first form that part of the device which consists of the atoms belonging to the CNTs used in\nthe CNTs+polymer relaxation. Regions with the same geometry as in \\cite{comp_no_pol}\nare cut from the initial 20-period long CNTs, and the rest of the atoms belonging to the CNTs are discarded. This is done to make possible a direct comparison of the results obtained\nfor the polymer filled CNTs junctions with the results for CNTs junctions without polymer reported in \\cite{comp_no_pol} for the same separation of CNTs equal to 6 \\AA. \n\nNote that\nthe CNTs parts of the scattering device contain atoms shifted from their positions in ideal CNTs due to the influence of the adjacent polymer molecules, and these shifts are time-dependent\nas a result of thermal fluctuations. \n\nThe cut regions contain two fragments of CNTs each 9 periods long, and in the case of the CNTs parallel configuration, the CNTs overlap by 7 periods.\nIn the nonparallel configurations, one of the CNTs is rotated around the axis perpendicular to the CNTs axes in the parallel configuration and passing through the\ngeometrical center of a device in the parallel configuration. \n\nAfter the construction of the CNTs part of the scattering region, we attach to it leads that consist of 5 period long fragments of an ideal CNT. The CNTs parts of the scattering regions with the attached leads for the three considered configurations are shown in Fig. \\ref{fig_CNT_conf}.\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig1\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The CNTs parts of the junctions. Left: the parallel configuration, center: CNTs axes are crossing at 45 degrees,\nright: the perpendicular configuration. The leads atoms are colored by green. } \\label{fig_CNT_conf}\n\\end{figure}\n\nAfter the preparation of the CNT parts of the junctions, we still have 17766 atoms in a device. A system with such a large number of atoms cannot be treated by fully first-principles\natomistic methods. On the other hand, keeping all those atoms for a precision calculation of the contact resistance of polymer filled CNTs junctions is not necessary,\nas only those polymer atoms which are close enough to a CNT will serve as tunneling bridges and give a contribution to the junctions conductivity. Thus,\nfor the calculations of the contact resistance, only those atoms were kept which are closer to the CNTs than a certain distance $d$. It has been established by numerical experiments\nthat if the value of $d$ is taken equal to the CNTs separation $d=6$ \\AA\\ this is quite sufficient, and taking into account more distant atoms does not change the contact resistance significantly.\n\nThe procedure of sorting the polymer atoms is as follows. In our molecular dynamics simulations we used 27 separate polymer molecules, consisting of 8 monomers each.\nIf at least one of the atoms of a polymer molecule was closer to the CNTs part of a junction than $d=6$ \\AA, the whole molecule was kept for a while, and discarded \notherwise. Having applied this first part of the procedure, we kept 4 polymer molecules for the parallel configuration, 8 molecules for the perpendicular configuration and\n11 molecules for the 45 degrees configuration. \n\nThen we looked at the monomers of each molecule that survived the first round of selection. The same procedure was applied\nto monomers as the one used earlier for molecules: if at least one of the monomer atoms was closer to the CNTs part of a junction than $d=6$ \\AA, \nthe whole monomer was kept for a while, and \ndiscarded otherwise. \n\nAfter the second round of selection with monomers was over, we dealt in the same manner with the individual residues comprising a monomer. The broken\nbonds that appeared in the second and third stages were saturated with hydrogen atoms. The described procedure resulted in the following numbers of atoms in the whole\ndevice, including the central scattering region and the leads: 881 for the parallel configuration, 1150 for the perpendicular configuration, and 1074 for the 45 degrees configuration.\nThe atomic configurations obtained using the described procedure for the first time steps in the corresponding series are presented in Fig. \\ref{fig_junction_conf}.\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig2\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The atomic configurations for the first-principles calculations of the polymer filled CNTs junctions contact resistance. The configurations are for the first time steps in the corresponding series.\nLeft: the parallel configuration, center: CNTs axes are crossed at 45 degrees,\nright: the perpendicular configuration. The Carbon atoms are gray, the Nitrogen atoms are blue, the Oxygen atoms are red, and the Hydrogen atoms are light gray. The leads atoms are colored by green. } \n\\label{fig_junction_conf}\n\\end{figure}\n\nA fully ab-initio method for electronic structure investigations utilizing a localized pseudo-atomic basis set, as described in \\cite{Ozaki_DFT} and implemented in \\cite{OpenMX}, \nwas used for the calculations of the electronic structures of the whole device and the leads.\nWe used basis set s2p2d1, the Pseudo Atomic Orbitals (PAO) cutoff radius equal to 6.0 a. u., and the cutoff energy of 150 Ry.\nThe pseudo-potentials generated according to Morrison, Bylander, and Kleinman scheme \\cite{pseudopot} were used. For the \ndensity functional calculations, the exchange-correlation functional was used in PBE96 form \\cite{PBE96}.\n\nUsing the electronic structures of the whole device and the leads we calculated energy the dependent transmission function\n through the device. Then the dependence $I(V)$ of the current $I$ on the voltage $V$ between the leads was determined\nwith the Green's function approach as described in detail in \\cite{Datta}. Finally, the Landauer-Buttiker approach was used to find the current through a polymer filled CNTs junction.\n\nSolving a scattering problem for a nano-device at arbitrary voltages is a computationally very complex task since it requires achieving self-consistency for both\nelectron density and induced electrostatic potential simultaneously. Fortunately, for contact resistance calculations one can take\nadvantage of the fact that the required voltages are very low. \n\nAccording to the experimental evidence, the size of a nanocomposite specimen used in conductivity experiments is about 10 mm \\cite{exp_CNT_length_1}, and the typical voltages applied across such specimen do not exceed 100 V \\cite{CNT_book}. Then for the size of a central scattering region about 1 nm, the voltage drop is about 10$^{-5}$ V, which is well within the range where the simplified approach is applicable.\n\nThe question of the modeling of quantum transport in the limit of low voltages was discussed in detail in \\cite{comp_no_pol}, where it was demonstrated that in the case of moderate voltages between leads,\nthe scattering probability $T(E)$ is not sensitive to the details of the electrostatic potential distribution $V({\\mathbf r})$ in the central scattering region, and\nsome physically reasonable approximation may be chosen for $V({\\mathbf r})$. \n\nThis is due to the fact that the difference of the Fermi functions\n$f(\\varepsilon - \\mu_L)$ and $f(\\varepsilon - \\mu_R)$ \nfor the left and right leads with corresponding the chemical potentials $\\mu_L$ and $\\mu_R$, present in the original Landauer-Buttiker formula:\n\\begin{equation}\nI=\\frac{2e}{h} \\int T(\\varepsilon) \\left (f(\\varepsilon - \\mu_L) - f(\\varepsilon - \\mu_R) \\right )d\\varepsilon ,\n\\label{LB_formula}\n\\end{equation}\nwhere $e$ is the elementary charge, $h$ is the Planck constant, $\\varepsilon$ is the electron energy and $T(\\varepsilon)$ \nis the energy dependent transmission probability,\n is reduced in this case to a very narrow and sharp peak centered at the Fermi level\nof the device. \n\nIn addition to the analysis performed in \\cite{comp_no_pol}, in this article, to verify the accuracy of our approach, we made contact resistance calculations for a simple test CNT junction in a coaxial configuration, using both the simplified method we suggest and the full NEGF method, where not only electron charge density but the electric potential was converged as well. The interlead voltage used in those test calculations was\nset to $10^{-4}$ V, and the gap between the CNTs tips was 0.94 \\AA. The atomic configurations for the test calculations are presented in Fig. \\ref{fig_NEGF_compare}. The consistent NEGF calculations produced 1.71~$\\cdot 10^{-5}$ S for the conductance of the junctions shown in Fig. \\ref{fig_NEGF_compare}, while modeling without searching for convergence of potential yielded 1.72 $\\cdot 10^{-5}$ S. \n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_conf_compare\/fig_compare.pdf}\n\\caption{The atomic configuration used in this work to validate the method of calculating quantum transport by comparing it againts the consistent NEGF approach.} \\label{fig_NEGF_compare}\n\\end{figure}\n\nThus, in our case, a very complex task of finding the $I(V)$ characteristic of a nano-device can be significantly simplified without the loss\nof precision. For the $I(V)$ calculations in the current paper we employed the abrupt potential model introduced in \\cite{comp_no_pol}: the potentials $V_L$ for the\nleft lead and $V_R$ for the right lead were set and were used for all atoms of the corresponding CNT to which that lead belonged. As for the polymer atoms, both $V_L$ and\n$V_R$ can be safely used for them, and at the considered voltages, adopting these two options, as we have checked by direct calculations, leads to the differences in\ncurrent not exceeding 0.1\\%.\n\n\\subsection*{C. The percolation model} \\label{sec_percolation}\nDetermination of the conductivity of a polymer-CNT system can be implemented in 2 stages. First, a percolation cluster is formed, and the second stage implies solving the matrix problem for a random resistor circuit (network). \n\nAt the first stage, the modeling area -- a cube of the linear size $L$ -- is filled with CNTs. For this task, permeable capsules (cylinders with hemispheres at the ends) with a fixed length and diameter were chosen as filling objects corresponding to CNTs. The cube is filled by the successive addition of CNTs until a fixed bulk density of CNTs \n\\begin{equation}\n\\eta = \\frac{((4\/3)\\pi R^3 + \\pi R^2 h)N}{L^3} \\label{cnt_vol_frac}\n\\end{equation}\nin the cube is reached, where $R$ is the radius of the cylinder and hemisphere, $h$ is the height of the cylinder, and $N$ is the number of CNTs in the cube. The percolation problem for permeable capsules was previously solved in \\cite{Xu_16}, and in \\cite{Schilling_15} capsules with a semipermeable shell were considered.\n\nIn the percolation problem, we use periodic boundary conditions as, for example, in \\cite{Bao_2011}. \nWe use the method of finding a percolation threshold based on the Newman and Ziff algorithm \\cite{Newman_2001}, where the identification of a percolation cluster is made at the stage of its formation.\n\n\nWhen a percolation cluster is formed, the obtained CNT configuration is transformed into a resistor circuit (2nd stage). \n\nThe contributions to a conductance matrix resulting from inner resistance of CNTs and the junctions tunneling resistance are usually discussed in connection with constructing conductivity percolation algorithms. Direct measurements of CNTs resistance per unit length are available. In \\cite{Chiodarelli_11}, the inner resistance of CNTs is estimated as $15 \\cdot 10^3$ $\\Omega\/\\mu$m. The results of \\cite{Ebbesen_96} give specific CNTs resistance in the range $(12-86) \\cdot 10^3$ $\\Omega\/\\mu$m. Taking into account that the characteristic CNTs lengths in nanocomposites are about several $\\mu$m \\cite{exp_CNT_length_1,CNT_book}, this results in the inner CNTs resistance approximately $10^4-10^5$ $\\Omega$, which is at least one order of magnitude less than the tunneling resistance obtained in this work. The specific results on tunneling resistance will be discussed below in section \\ref{RND}. Thus, in our percolation model, the inner resistance of CNTs is neglected, and only tunneling resistance of CNTs junctions is taken into account. This can significantly reduce the requirements for computational time. \n\nWhen contact resistance is determined only by tunneling, the principle of compiling the matrix for the percolation problem will be as follows. \nFirst, the matrix (N, N) is compiled from the bonds of percolation elements, where N is the number of CNTs participating in percolation. \nThen this matrix is transformed into the conductivity matrix according to the second Kirchhoff law \n\\begin{equation}\n\\sum_j G_{ij} (V_i-V_j) = 0 \\label {second_Kirchoff}\n\\end{equation}\n- the sum of currents for all internal elements of a percolation network is zero -- where $G_{ij}$ are the elements of the conductance matrix ${\\mathbf G}$ and $V_i$ is the component of the voltage vector ${\\mathbf V}$ corresponding to the $i$-th contact point in a percolation network \\cite{Kirkpatrick_73}. \nThe voltages on the left and right borders of a simulation volume are set to $V_L=1$ V and $V_R=0$ V, respectively. \n\nNow finding the conductivity of the system is reduced to the problem $\\mathbf{GV}=\\mathbf{I}$, where\n$\\mathbf{I}$ is the vector of the currents between the contact points. \n\nThe dimension of the matrix problem can be further reduced to \n(N-2, N-2) by excluding boundary elements. \n\nAfter solving the equation (\\ref{second_Kirchoff}), with the elements of the $\\mathbf{G}$ matrix obtained by the first-principles calculations, we obtain the voltage vector for all internal elements. \nThen, knowing this vector, we sum up all the currents on each of the boundaries. The currents on the left $I_L$ and right $I_R$ \nboundaries of a simulation volume are equal in magnitude and opposite in sign $I_L= - I_R$. Knowing these currents, we determine the conductance of the simulation system as \n$G=|I_L|\/(V_L-V_R) = |I_R|\/(V_L-V_R)$. \nAnd then the conductivity of the composite is calculated as $\\sigma=GL\/S$, where $L$ is the distance between the faces of a simulation volume where voltage is applied, and $S$ is the area of that kind of face. In our case, for the simulation volume of a cubic shape, $S=L^2$, and $\\sigma=G\/L$.\n\nTo calculate the conductivity, the following system parameters were selected: the length of a nanotube is $l=3$ $\\mu$m, the diameter of a CNT is $D=30$ nm, the aspect ratio $l\/D=100$, and the size of the system is 4 $\\mu$m. The same values were used in \\cite{Yu_2010}. We adopted those values to test our realization of the percolation algorithm against the previously obtained results \\cite{Yu_2010}.\nThen for the given parameters for each fixed tube density, the Monte Carlo method (100 implementations of various configurations of CNT networks) was used to calculate the system conductivity. \n\nThe quality of CNTs dispersion is one of the key factors that affect the properties of nanocomposites, and a lot of efforts is taken\nto achieve a homogeneous distribution of fillers \\cite{Mital_14}. In this work, we take into consideration the effect of inhomogeneity of a CNTs distribution on composite conductivity.\nThe spatial density of nanotubes $\\rho_{CNT}$, in this case, has one peak with a Gaussian distribution:\n\\begin{equation} \n\\rho_{CNT}=\\rho_0 \\cdot exp(-(\\mathbf {r-r_0})^2\/\\rho_\\sigma^2), \\label {agglo}\n\\end{equation}\nwhere ${\\mathbf r_0}$ coincides with the geometrical center of a simulation volume, and $\\rho_\\sigma = L\/12$. The value of the $\\rho_0$ parameter is chosen so that the CNTs volume fraction in the inhomogeneous case is the same as in the homogeneous distribution.\n\n\n\\section*{Results and discussion}\\label{RND}\nTo find the contact resistance of polymer filled CNTs junctions one first needs to find their Volt-Ampere characteristics $I(V)$ and to determine the voltage range where\n$I(V)$ is linear and is not sensitive to the specific distribution of an electrostatic potential in the scattering region. In Fig. \\ref{fig_I_V} the $I(V)$ plot\nfor the first time step in the atomic geometry series for the parallel configuration is shown.\n\\begin{figure}[b]\n\\begin{tabular}{lr}\n\\includegraphics[width=3.5cm]{figs\/I_V_test\/I_V.pdf} &\n\\includegraphics[width=3.5cm]{figs\/I_V_test_low_V\/I_V_low_V.pdf}\n\\end{tabular}\n\\caption{The Volt-Ampere characteristic for the polymer filled CNTs junction corresponding to the first time step in the series for the parallel configuration.\nLeft frame: maximum inter-lead voltage is $10^{-3}$ V, right frame: $10^{-4}$ V. The circles correspond to the results of calculations,\nthe lines are guides for an eye.} \n\\label{fig_I_V}\n\\end{figure}\n\nIt is clearly seen from Fig. \\ref{fig_I_V} that up to about $10^{-4}$ V the $I(V)$ characteristic is linear, and after that value, it starts to deviate from a\nsimple linear dependence. Thus, for the calculations of a contact resistance $R$ and its inverse, a junction conductivity $G$,\nwe used the electrical current values obtained for the inter-lead voltage equal to $10^{-4}$ V. Note, that according to our estimates in section \\ref{sec_meth_fp}, a characteristic voltage drop on the length of a CNTs tunneling junction is about $10^{-5}$ V which is well within the region where the linear $I(V)$ is observed.\n\nThe time dependences of the junctions conductances for the three considered configurations are presented in Fig. \\ref{fig_G_t}. One might expect that the \nshifts of both CNTs atoms and polymer atoms in the central scattering region due to thermal fluctuations would lead to fluctuations of junctions conductances $G$, but \nquantitative characteristics of this phenomenon such as minimum $G_{min}$, maximum $G_{max}$, mean values $\\langle G \\rangle$ and a standard deviation $G_\\sigma$ can only be captured by highly precise fully atomistic \nfirst-principles methods, like those employed in the current paper. The resulting fluctuations of conductance are very high. For the parallel CNTs configuration\nthe minimum value, $G_{min} = 2.4 \\cdot 10^{-8}$ S, and the maximum value, $G_{max} = 6.8 \\cdot 10^{-6}$ S, differ by more than two orders of magnitude, for the 45 degrees and perpendicular\nconfigurations the corresponding ratios are about 30. The same strong variations of conductance over time were reported in \\cite{Penazzi_2013} for the coaxial\nCNTs configuration, where the results were obtained using a semi-empirical tight-binding approximation. Thus, it is obvious that for the precise determination\nof a conductance of polymer filled CNTs junctions one needs to use fully atomistic approaches, and phenomenological methods taking atomic configurations into account \non the average are not reliable.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/G_t\/g_t.pdf}\n\\caption{The time dependence of the polymer filled junctions conductance $G$ in S. Red color correspond to the parallel configuration, the green lines -- to the perpendicular configuration\nand the blue lines -- the 45 degrees configuration. The results of the calculations are shown by circles, the saw-tooth lines serve as a guide for an eye.\nThe straight solid lines designate the mean values of conductance $\\langle G \\rangle$, and the dashed ones -- $\\langle G \\rangle \\pm G_\\sigma$.} \n\\label{fig_G_t}\n\\end{figure}\n\n\n\n\\begin{table*}[h]\n \\centering\n\\caption{The results of statistical analysis of the CNTs junctions conductances in S, for CNTs separation equal to 6 \\AA \\ and different values of the CNTs crossing angles $\\varphi$, \nwithout polymer from \\cite{comp_no_pol}, and with polyimide R-BAPB filling obtained in the current paper.\n}\n\\begin{tabular}{cccccc}\n\\textrm{$\\varphi$}&\n\\textrm{no polymer}&\n\\textrm{}&\n\\textrm{polymer present}&\n\\textrm{}&\n\\textrm{}\\\\\n\\textrm{}&\n\\textrm{the results of \\cite{comp_no_pol}}&\n\\textrm{$G_{min}$}&\n\\textrm{$G_{max}$}&\n\\textrm{$\\langle G \\rangle$}&\n\\textrm{$G_\\sigma$}\\\\\n\\hline\n& & & & &\\\\ \n0 & 3.6$\\cdot10^{-13}$ &2.4$\\cdot10^{-8}$ &6.8$\\cdot10^{-6}$ & 1.8$\\cdot10^{-6}$ & 1.6$\\cdot10^{-6}$\\\\\n& & & & & \\\\\n\\hline\n& & & & &\\\\ \n0.2$\\pi$ & 1.4$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.25$\\pi$ & --- &4.8$\\cdot10^{-9}$ &1.4$\\cdot10^{-7}$ &3.4$\\cdot10^{-8}$ &2.7$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.3$\\pi$ & 1.2$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.5$\\pi$ & 4.2$\\cdot10^{-14}$ &2.2$\\cdot10^{-9}$ &4.3$\\cdot10^{-8}$ & 1.4$\\cdot10^{-8}$ & 1.1$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\end{tabular}\n\\label{tab_cond}\n\\end{table*}\n\nTo assign the tunneling resistance to a polymer filled CNT junction the following algorithm was used. First, for each junction that had to be used in the percolation algorithm, a uniformly distributed random number $\\varphi$ in the range [0, $\\pi\/2$] was generated. The value of the intersection angle for that junction was assigned to the obtained random number. The mean values and standard deviations for CNTs tunneling resistances and conductances calculated for the different atomic configurations corresponding to the different time steps are known for $\\varphi = 0, \\pi\/4$, and $\\pi\/2$. Analyzing figure 4 of \\cite{comp_no_pol}, one can see that though an angle dependence of current and hence conductivity is a rather complex function, in the first approximation one can adopt a roughly piece-wise linear character for this function with the minimum located at $\\varphi=0.25\\pi$. Thus the logarithm of the mean value of conductance $\\mu_\\varphi$ for the generated $\\varphi$ was set by linear interpolation between the logarithms of the mean values of conductances for $\\varphi=0$ and $\\varphi=\\pi\/4$ or $\\varphi=\\pi\/4$ and $\\varphi=\\pi\/2$ presented in Table \\ref{tab_cond}. The same algorithm was applied to finding the standard deviation values $\\sigma_\\varphi$ for the generated $\\varphi$. After the statistical parameters for the generated $\\varphi$ are estimated, the conductivity of the junction is set to a random number generated using the normal distribution with the parameters $\\mu_\\varphi$ and $\\sigma_\\varphi$.\n\n\nIn \\cite{comp_no_pol}, the conductances were reported for the CNTs junctions with almost the same geometry as the CNTs parts of the devices considered in the current paper. The only difference \nbetween the configurations is that in this work the carbon atoms belonging to the CNTs part of the central scattering region are shifted somewhat from their equilibrium positions\ndue to the interaction with polymer. The maximum values of those shifts along the $x$, $y$, and $z$ coordinates lie in the range $0.2-0.5$ \\AA. This gives us the possibility to directly compare the current results to the data \nfrom \\cite{comp_no_pol} and,\nthus, elucidate the influence of polymer filling on the junctions conductance. The corresponding data and the results of a basic statistical\nanalysis for the case of the polymer filled junctions are provided in Tab. \\ref{tab_cond}.\n\nFirst, as was expected, filling CNTs junctions with polymer creates carrier tunneling paths and increases junctions conductance by 6-7 orders of magnitude.\nSecond, it is evident that the CNTs axes crossing angle is crucial for the junctions conductivity when polymer is present as it was the case without\npolymer \\cite{comp_no_pol}. At the same time, the sharp dependence of polymer filled junctions conductance on the CNTs crossing angle is somewhat different\nfrom the analogous dependence for junctions without polymers. While in the latter case this dependence is sharply non-monotonous, with a pronounced minimum\nat the angles around $0.25\\pi$, in the former case there is a significant difference between the conductance values for the parallel and nonparallel configurations, but\nthe configurations with the angle $\\varphi$ between CNTs angles equal to $0.25\\pi$ and $0.5\\pi$ have very close conductances, and their mean values averaged over\ntime $\\langle G \\rangle_{45}$ and $\\langle G \\rangle_{per}$ lie within the\nranges $\\langle G \\rangle \\pm G_\\sigma$ of each other. Moreover, in contrast to the geometries without polymer, for the polymer filled CNTs junctions $\\langle G \\rangle_{per}$ is lower than $\\langle G \\rangle_{45}$ by a factor of 2.4.\n\nNote also that for the parallel configuration, the polymer influence on the junction conductance is more pronounced than for the nonparallel ones.\nFor the parallel configuration, adding polymer to a junction of CNTs separated by 6.0 \\AA\\ with initial conductance of 3.6$\\cdot 10^{-13}$ S \nproduces conductance mean value equal to 1.8$\\cdot 10^{-6}$ S. This gives the factor 0.5$\\cdot 10^7$; the value of the analogous factor for the perpendicular configuration is 0.33$\\cdot 10^6$.\n\nThe probable reason for the more effective conductance increase, when polymer is added, for the configurations with smaller angles between CNTs axes, is\nthat the smaller is an intersection angle, the larger is the overlap area between CNTs where polymer can penetrate and, thus, create tunneling bridges.\nThe higher fluctuation of conductance over time for the parallel configuration can be explained by the same reason: a larger CNTs overlap area gives\nmore freedom for polymer atoms to adjust their positions.\n\nThe dependence of the calculated composite conductivity $\\sigma$ on CNTs volume fraction $\\eta$ $\\sigma(\\eta)$ is presented in Fig. \\ref{fig_cond_pol}. \nThe value of the percolation threshold $\\eta_{thresh}$ is estimated in this work as $\\eta_{thresh} = 0.007$. To test our realization of the percolation algorithm against the previous results of \\cite{Yu_2010} we calculated the composite conductivity using\nthe fixed CNTs junction conductance equal to 1 M$\\Omega$ for all junctions in a percolation network. Our results presented in\nFig. \\ref{fig_cond_pol} by the red circles coincide within graphical accuracy to the results of \\cite{Yu_2010} shown by the red squares.\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/fig9\/cnt_9.pdf}\n\\caption{ The conductivity of CNT enhanced nanocomposites above the percolation threshold obtained in this work. The symbols of different shapes and colors are used to designate the following results. The red circles: the fixed CNTs tunneling junctions resistance of $R=1$ M$\\Omega$ is used, the red squares: the conductivity results for the fixed 1 M$\\Omega$ tunneling resistance from \\cite{Yu_2010}, the black triangles: the same as the red circles but for $R=0.54$ M$\\Omega$ corresponding to the mean value of the tunneling junction resistance for the parallel configuration from Table \\ref{tab_cond}, the blue rhombi: the angle dependence of the CNTs junctions resistance is taken into account, the green pentagons: CNTs agglomeration is considered in addition to the angle dependence. The red line is a guide for an eye.}\n\\label{fig_cond_pol}\n\\end{figure}\nThe 1 M$\\Omega$, used in various sources, for example, \\cite{Yu_2010}, is not an arbitrary value, but rather a typical contact resistance of CNTs junctions filled with polymer for simple geometries. In this work, we obtained for the parallel configurations $1\/\\langle G \\rangle =0.54$ M$\\Omega$. The $\\sigma(\\eta)$ dependence for the fixed tunneling resistance of $0.54$ M$\\Omega$ is shown in Fig. \\ref{fig_cond_pol} by the black triangles.\n\nTaking into account the angle dependence of CNTs junctions conductances with the statistical parameters according to Table\n\\ref{tab_cond}, leads to the lowering of composite conductivity just above the percolation threshold by the factor of about 30. This number correlates with the ratio of the mean conductances for the parallel, $\\langle G \\rangle_{par}$, and 45$^\\circ$, $\\langle G \\rangle_{45}$, configurations: $f_G = \\langle G \\rangle_{par}\/ \\langle G \\rangle_{45} = 53$, but is higher than $f_G$ due to the presence of junctions with $\\varphi < \\pi\/4$. \n\nIf agglomeration of CNTs, modeled by the inhomogeneity of their distribution according to formula (\\ref{agglo}) and the parameter values discussed in section \\ref{sec_percolation}, is taken into account in addition to the angle dependence of conductance, the composite conductivity is further reduced above a percolation threshold by the factor of 2.5. Lowering of conductivity of composites with agglomerated CNTs above a percolation threshold was also mentioned in \\cite{Bao_2011}. The calculated results for the conductivity of a percolation network of agglomerated CNTs are shown in Fig. \\ref{fig_cond_pol} by the green pentagons.\n\nWe believe that in this work we have identified some of the key factors that influence nanocomposites electrical conductivity: the geometry of tunneling junctions and changes of atomic configurations due to thermal fluctuations. Among other causes that may affect conductivity, but are not considered in this work, are the presence of defects in CNTs, a distribution of CNTs over chiralities, lengths, aspect ratios, different separations between CNTs. \n\n\nUntil the specific experiments on conductivity for R-BAPB polyimide are not available, we can make a preliminary comparison of our modeling results to the available experimental results for different composites. The calculated conductivity of composite just above the percolation threshold at $\\eta=0.0075$ is equal to $3.6\\cdot10^{-3}$ S\/m.\nThis is a reasonable value that falls into the range of experimentally observed composites conductivities (for the comprehensive compilation of experimental results see Table 1 of \\cite{Eletskii}). To make a quantitative comparison of modeling results with experiments the full details of nanocomposites structure are necessary, including the CNTs parameters mentioned in the previous paragraph. All these factors can be easily incorporated into the approach proposed in this work if sufficient computational resources are available.\n\n\\section*{Conclusions}\n\nWe have proposed a physically consistent, computationally simple, and at the same time precise, multi-scale method for calculations of electrical conductivity of CNT enhanced\nnanocomposites. The method starts with the atomistic determination of the positions of polymer atoms intercalated between CNTs junctions, proceeds with the fully first-principles calculations of polymer-filled CNTs junctions conductance at the microscale and finally performs\nmodeling of percolation through an ensemble of CNTs junctions by the Monte-Carlo technique. \n\nThe developed approach has been applied to the modeling of electrical conductivity of polyimide R-BAPB + single wall (5,5) CNTs nanocomposite.\n\nOur major contributions to the field are the following. We have proposed a straightforward\nmethod to calculate a contact resistance and conductance for polymer-filled CNTs junctions with arbitrary atomic configurations without resorting to any simplifying assumptions. We have demonstrated that a consistent multiscale approach, based on solid microscopic physical methods can give reasonable results, lying within the experimental range, for the conductivity of composites and suggested a corresponding work-flow. \n \n\nIt is shown that a contact resistance\nand nanocomposite conductivity is highly sensitive to the geometry of junctions, including an angle between CNTs axes and subtle thermal shifts of polymer atoms in an inter-CNT's gap.\nThus, we argue that for the precision calculations of nanocomposites electrical properties rigorous atomistic quantum-mechanical approaches are indispensable.\n\nWe have to admit though, that we have not considered all possible factors that may influence CNT junctions conductivity on the micro-level. We concentrated on the CNTs crossing angle factor because it seems to be the\nmost influential. The additional factors may include, for example, defects in CNTs,\nCNTs overlap lengths and others. On the other hand, the proposed approach may be used to take all those factors into account, provided sufficient computational resources are available.\n\n\\section*{Acknowledgments}\n\nThis work was supported by the State Program \"Organization of\nScientific Research\" (project 1001140) and by the Research Center \"Kurchatov Institute\" (order No.~1878~of~ 08\/22\/2019).\n\nThis work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC \"Kurchatov Institute\", \\url{http:\/\/ckp.nrcki.ru\/}.\n\n\n\n\\section*{Introduction}\nPolymer materials, while possessing some unique and attractive qualities, such as low weight, high strength, resistance to chemicals, ease of processing, are for the most part\ninsulators. If methods could be devised to turn common insulating polymers into conductors, that would open great prospects for using such materials in many more areas than they are\ncurrently used. These areas may include organic solar cells, printing electronic circuits, light-emitting diodes, actuators, supercapacitors, chemical sensors, and biosensors \\cite{Long_2011}.\n\n\nSince the reliable methods for carbon nanotubes (CNT) fabrication had been developed in the 1990s, growing attention has been paid to\nthe possibility of dispersing CNTs in polymers, where CNTs junctions may form a percolation network and turn insulating polymer into a good conductor when a percolation threshold is overcome.\nAn additional benefit of using such polymer\/CNTs nanocomposites instead of intrinsically conducting polymers, such as polyaniline \\cite{Polyaniline}\nfor example, is that dispersed CNTs, besides providing electrical conductivity, enhance\npolymer mechanical properties as well.\n\nCNTs enhanced polymer nanocomposites have been intensively investigated experimentally, including composites conductivity \\cite{Eletskii}. As for the theoretical research in this area, the results are more modest. \nIf one is concerned with nanocomposite conductivity, its value \ndepends on many factors, among which are the polymer type, CNTs density, nanocomposite preparation technique, CNTs and their junctions geometry, a possible presence of defects\nin CNTs and others. Taking all these factors into\naccount and obtaining quantitatively correct results in modeling is a very challenging task since the resulting conductivity is formed at different length scales: at the microscopic level it is influenced by the\nCNTs junctions contact resistance and at the mesoscopic level it is determined by percolation through a network of CNTs junctions. Thus a consistent multi-scale method for the modeling of conductivity,\nstarting from atomistic first-principles calculations of electron transport through CNTs junctions is necessary.\n\nDue to the complexity of this multi-scale task, the majority of investigations in the area are carried out in some simplified forms, this is especially true for the underlying part of the modeling: determination\nof CNTs junction contact resistance. For the contact resistance either experimental values as in \\cite{Soto_2015} or the results of phenomenological Simmons model as in \\cite{Xu_2013, \nYu_2010, Jang_2015, Pal_2016} are usually taken, or even an arbitrary value of contact resistance reasonable by an order of magnitude may be set \\cite{Wescott_2007}. In \\cite{Bao_2011, Grabowski_2017} the tunneling probability through a CNT junction is modeled using a rectangular potential barrier and the quasi-classical approximation.\n\nThe authors of \\cite{Castellino_2016} employed an oversimplified two-parameter expression for contact resistance, with these parameters fitted to the experimental data.\nThe best microscopic attempt, that we are aware of, is using the semi-phenomenological tight-binding approximation for the calculations of contact resistance \\cite{Penazzi_2013}. But in \\cite{Penazzi_2013} just\nthe microscopic part of the nanocomposites conductivity problem is addressed, and the conductivity of nanocomposite is not calculated. Moreover, in \\cite{Penazzi_2013} the coaxial CNTs configuration is only considered,\nwhich is hardly realistic for real polymers.\n\nThus, the majority of investigations are concentrated on the mesoscopic part of the task:\nrefining a percolation model or phenomenologically taking into account different geometry peculiarities of CNTs junctions. Moreover, comparison with experiments is missing in some publications on this topic.\nThus, a truly multi-scale research, capable of providing quantitative results comparable with experiments, combining fully first-principles calculations of contact resistance on the microscopic level with a percolation model on \nthe mesoscopic level seems to be missing. \n\nIn our previous research \\cite{comp_no_pol}, we proposed an efficient and precise method for fully first-principles calculations of CNTs contact resistance and combined it with a Monte-Carlo statistical percolation model to\ncalculate the conductivity of a simplified example network of CNTs junctions without polymer filling. \nIn the current paper, we are applying the developed approach to the modeling of conductivity of the CNTs enhanced polymer polyimide R-BAPB. \n\nR-BAPB (Fig. \\ref{fig_RBAPB-struct}) is a novel polyetherimide synthesized using 1,3-bis-(3$'$,4-dicarboxyphenoxy)-benzene (dianhidride R) and 4,4$'$-bis-(4$''$-aminophenoxy)diphenyl (diamine BAPB). It is thermostable polymer with extremely high thermomechanical properties (glass transition temperature $T_g= 453-463$~K, melting temperature $T_m= 588$~K, Young's modulus $E= 3.2$~GPa) \\cite{Yudin_JAPS}. This polyetherimide could be used as a binder to produce composite and nanocomposite materials demanded in shipbuilding, aerospace, and other fields of industry. The two main advantages of the R-BAPB among other thermostable polymers are thermoplasticity and crystallinity. R-BAPB-based composites could be produced and processed using convenient melt technologies.\n\nCrystallinity of R-BAPB in composites leads to improved mechanical properties of the materials, including bulk composites and nanocomposite fibers. It is well known that carbon nanofillers could act as nucleating agents for R-BAPB, increasing the degree of crystallinity of the polymer matrix in composites. As it was shown in experimental and theoretical studies\n\\cite{Yudin_MRM05, Larin_RSCADV14, Falkovich_RSCADV14,Yudin_CST07}, the degree of crystallinity of \ncarbon nanofiller enhanced R-BAPB may be comparable to that of bulk polymers.\n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_Larin1\/RBAPB.png}\n\\caption{The chemical structure of R-BAPB polyimide.} \n\\label{fig_RBAPB-struct}\n\\end{figure}\n\nOrdering of polymer chains relative to nanotube axes could certainly influence a conductance of the polymer filled nanoparticle junctions. However, it is expected that such influence will depend on many parameters, including the structure of a junction, position, and orientation of chain fragments on the nanotube surface close to a junction, and others. Taking into account all of these parameters is a rather complex task that requires high computational resources for atomistic modeling and ab-initio calculations, as well as complex analysis procedures. Thus, on the current stage of the study, we consider only systems where the polymer matrix was in an amorphous state, i.e. no sufficient polymer chains ordering relative to nanotubes were observed.\n\n\n\n\n\n\\section*{Description of the multiscale procedure} \n\nThe modeling of polymer nanocomposite electrical conductivity is based on a multi-scale approach, in which different simulation models are used at different scales. For the electron transport in polymer composites with a conducting filler, the lowest scale corresponds to the contact resistance between tubes. The contact resistance is determined at the atomistic scale by tunneling of electrons between the filler particles via a polymer matrix, and hence, analysis of contact resistance requires knowledge of the atomistic structure of a contact. Therefore, at the first step, we develop an atomistic model of the contact between carbon nanotubes in a polyimide matrix using the molecular dynamics (MD) method. This method gives us the structure of the intercalated polymer molecules between carbon nanotubes for different intersection angles between the nanotubes. One should mention, that since a polymer matrix is soft, the contact structure varies with time and, therefore, we use molecular dynamics to sample these structures. \n\nBased on the determined atomistic structures of the contacts between nanotubes in the polymer matrix we calculate electron transport through the junction using electronic structure calculations and the formalism of the Green's matrix. Since this analysis requires first-principles methods, one has to reduce the size of the atomistic structure of a contact to acceptable values for the first-principles methods, and we developed a special procedure for cutting the contact structure from MD results. First-principles calculations of contact resistance should be performed for all snapshots of an atomistic contact structure of MD simulations, and an average value and a standard deviation should be extracted. In this way, one can get the dependence of a contact resistance on the intersection angle and contact distance.\n\nUsing information about contact resistances we estimate the macroscopic conductivity of a composite with nanotube fillers. For this, we used a percolation model based on the Monte Carlo method to construct a nanotube network in a polymer matrix. In this model, we used distributions of contact resistances, obtained from the first-principles calculations for the given angle between nanotubes. Using this Monte Carlo percolation model one can investigate the influence of non-uniformities of a nanotube distribution on macroscopic electrical conductivity. \n\nIn the A section, we will describe the details of molecular dynamics modeling of the atomistic structure of contacts between nanotubes. In the B section, we present the details of first-principles calculations of electron transport for estimates of contact resistance. Finally, in the C section, we present the details of the Monte Carlo percolation model.\n\n\\subsection*{A. Preparation of the composite atomic configurations}\n\nInitially, two metallic CNTs with chirality (5,5) were constructed and separated by 6~\\AA. The CNTs consisted of 20 periods along the axis, and each one had the total length of 4.92~nm.\nThe broken bonds at the ends of the CNTs were saturated with Hydrogen atoms.\nThe distance 6 \\AA\\ was chosen, because starting with this distance polymer molecules are able to penetrate the space between CNTs. The three configurations of CNTs junctions were prepared: the first one with parallel CNTs axes (angle between nanotube axes $\\varphi=0^\\circ$),\nthe second one with the axes crossing at 45 degrees ($\\varphi=45^\\circ$), and the third one with perpendicular axes ($\\varphi=90^\\circ$).\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_em1.png}&\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_compress1.png}\\\\\n\\end{tabular}\n\\caption{The snapshots of the the nanocomposite system with the parallel orientation of carbon nanotubes at the initial state (left picture) and after the compression procedure (right picture). The black lines represent the periodic simulation cell.} \\label{fig_Larin-conf}\n\\end{figure}\n\nTo produce the polymer filled samples, we used \nthe procedure similar to that employed for the simulations of the thermoplastic polyimides and polyimide-based nanocomposites in the previous works\n\\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14, Nazarychev_MA16, Larin_RSCADV15}. First, partially coiled R-BAPB chains with the polymerization degree $N_p=8$, which corresponds to the polymer regime onset \\cite{Lyulin_MA13, Lyulin_SM14}, were added to the simulation box at random positions avoiding overlapping of polymer chains. This results in the initial configuration of samples with a rather low overall density ($\\rho\\sim 100~$ kg\/m$^3$) (Fig. \\ref{fig_Larin-conf}). Then the molecular dynamics simulations were performed to compress the systems generated, equilibrate them and perform production runs.\n\nThe molecular dynamics simulations were carried out using Gromacs simulation package \\cite{gromacs1, gromacs2}. The atomistic models used to represent both the R-BAPB polyimide and CNTs were parameterized using the Gromos53a6 forcefield \\cite{gromos}. Partial charges were calculated using the Hartree-Fock quantum-mechanical method with the 6-31G* basis set, and the Mulliken method was applied to estimate the values of the particle charges from an electron density distribution. As it was shown recently, this combination of the force field and particle charges parameterization method allows one to reproduce qualitatively and quantitatively the thermophysical properties of thermoplastic polyimides \\cite{Nazarychev_EI}. The model used in the present work was successfully utilized to study structural, thermophysical and mechanical properties of the R-BAPB polyimide and R-BAPB-based nanocomposites \\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14}.\n\nAll simulations were performed using the NpT ensemble at temperature $T=600$~K, which is higher than the glass transition temperature of R-BAPB. The temperature and pressure values were maintained using Berendsen thermostat and barostat \\cite{Berendses_1, Berendsen_2} with relaxation times $\\tau_T= 0.1$~ps and $\\tau_p=0.5$~ps respectively. The electrostatic interactions were taken into account using the particle-mesh Ewald summation (PME) method \\cite{PME1, PME2}. \n\nThe step-wise compression procedure allows one to obtain dense samples with an overall density close to the experimental polyimide density value ($\\rho \\approx 1250-1300$~kg\/m$^3$), as shown in Fig. \\ref{fig_Larin-conf}. The system pressure $p$ during compression was increased in a step-wise manner up to $p=1000$~bar and decreased then to $p=1$~bar. After compression and equilibration, the production runs were performed to obtain the set of polymer filled CNT junction configurations.\n\nAs the conductance of polymer filled CNT junctions is influenced by the density and structure of a polymer matrix in the nearest vicinity of a contact between CNTs, the relaxation of the overall system density was used as the system equilibration criterion. To estimate the equilibration time, the time dependence of the system density was calculated as well as the density autocorrelation function $C_\\rho(t)$:\n\\begin{equation}\nC_\\rho(t)=\\frac{\\langle \\rho(0) \\rho(t)\\rangle}{\\langle \\rho^2 \\rangle},\n\\end{equation}\nwhere $\\rho(t)$ is the density of the system at time $t$ and $\\langle \\rho^2 \\rangle$ is the average density of the system during the simulation.\n\n\nAs shown in Fig. \\ref{fig_Larin-density}a, the system density does not change sufficiently during simulation after the compression procedure. At the same time, the analysis of the density auto-correlation functions shows some difference in the relaxation processes in the systems studied (see Fig. \\ref{fig_Larin-density}b). In the case of the system where CNTs were placed parallel to each other ($\\varphi=0^\\circ$), $C_\\rho(t)$ could be approximated by the exponential decay function $C_\\rho(t)=\\exp(-t\/\\tau)$ with relaxation time $\\tau = 4$~ps. The density relaxation in the systems with crossed CNTs ($\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) was found to be slower. For these two systems density the auto-correlation functions could be approximated by a double exponential function $C_\\rho(t)=A\\exp(-t\/\\tau_1) + (1-A)\\exp(-t\/\\tau_2)$, and the relaxation times determined using this fitting were $\\tau_1=2.7$~ps and $\\tau_2=12.2$~ns (for $\\varphi=90^\\circ$), and $\\tau_1=9.5$~ps and $\\tau_2=24.6$~ns (in case of $\\varphi=45^\\circ$).\n\nNevertheless, the results obtained after the analysis of the system density relaxation allow us to choose the system equilibration time to be 100 ns, which is higher than the longest system density relaxation times determined by the density autocorrelation function analysis. \nThe same simulation time was used in our previous works to equilibrate the nanocomposite structure after switching on electrostatic interactions \\cite{Larin_RSCADV14, Nazarychev_EI, Larin_RSCADV15}. The equilibration was followed by the 150 ns long production run. To analyze the polymer filled CNT junction conductance, 31 configurations of each simulated system, separated by 5 ns intervals, were taken from the production run trajectory.\n\n\\begin{figure}[h]\n\\begin{tabular}{ccc}\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_left.pdf}\\\\\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_right.pdf}\\\\\n\\end{tabular}\n\\caption{The time dependence of the system density $\\rho$ (a) and the density auto-correlation functions $C_\\rho(t)$ (b) for the systems with various angles between nanotube axes $\\varphi$. The dots correspond to the calculated data. The solid lines correspond to the fitting of $C_\\rho(t)$ with the exponential (in case of $\\varphi=0^\\circ$) or double exponential (in case of of $\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) functions.} \n\\label{fig_Larin-density}\n\\end{figure}\n\nAfter the configurational relaxation is finished, we have to prepare polymer filled CNT junctions configuration for the first-principles calculations of contact resistance. \nThe method we used for the calculations of a contact resistance is based on the solution of the ballistic electronic transport problem, finding the Volt-Ampere characteristic $I(V)$ of a device and\nderiving the contact resistance from the linear part of $I(V)$ corresponding to the low voltages. For this purpose, we employed the Green's function method for \nsolving the scattering problem and the Landauer-Buttiker approach to find the current through a scattering region coupled to two semi-infinite leads, as described in \\cite{Datta}.\nSpecific details of how these techniques are applied in the case of crossed CNTs can be found in \\cite{comp_no_pol}.\n\n\\subsection*{B. The first-principles calculations of the contact resistance of CNTs junctions filled with polymer} \\label{sec_meth_fp}\n\nFor the preparation of a device for the electronic transport calculations, we first form that part of the device which consists of the atoms belonging to the CNTs used in\nthe CNTs+polymer relaxation. Regions with the same geometry as in \\cite{comp_no_pol}\nare cut from the initial 20-period long CNTs, and the rest of the atoms belonging to the CNTs are discarded. This is done to make possible a direct comparison of the results obtained\nfor the polymer filled CNTs junctions with the results for CNTs junctions without polymer reported in \\cite{comp_no_pol} for the same separation of CNTs equal to 6 \\AA. \n\nNote that\nthe CNTs parts of the scattering device contain atoms shifted from their positions in ideal CNTs due to the influence of the adjacent polymer molecules, and these shifts are time-dependent\nas a result of thermal fluctuations. \n\nThe cut regions contain two fragments of CNTs each 9 periods long, and in the case of the CNTs parallel configuration, the CNTs overlap by 7 periods.\nIn the nonparallel configurations, one of the CNTs is rotated around the axis perpendicular to the CNTs axes in the parallel configuration and passing through the\ngeometrical center of a device in the parallel configuration. \n\nAfter the construction of the CNTs part of the scattering region, we attach to it leads that consist of 5 period long fragments of an ideal CNT. The CNTs parts of the scattering regions with the attached leads for the three considered configurations are shown in Fig. \\ref{fig_CNT_conf}.\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig1\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The CNTs parts of the junctions. Left: the parallel configuration, center: CNTs axes are crossing at 45 degrees,\nright: the perpendicular configuration. The leads atoms are colored by green. } \\label{fig_CNT_conf}\n\\end{figure}\n\nAfter the preparation of the CNT parts of the junctions, we still have 17766 atoms in a device. A system with such a large number of atoms cannot be treated by fully first-principles\natomistic methods. On the other hand, keeping all those atoms for a precision calculation of the contact resistance of polymer filled CNTs junctions is not necessary,\nas only those polymer atoms which are close enough to a CNT will serve as tunneling bridges and give a contribution to the junctions conductivity. Thus,\nfor the calculations of the contact resistance, only those atoms were kept which are closer to the CNTs than a certain distance $d$. It has been established by numerical experiments\nthat if the value of $d$ is taken equal to the CNTs separation $d=6$ \\AA\\ this is quite sufficient, and taking into account more distant atoms does not change the contact resistance significantly.\n\nThe procedure of sorting the polymer atoms is as follows. In our molecular dynamics simulations we used 27 separate polymer molecules, consisting of 8 monomers each.\nIf at least one of the atoms of a polymer molecule was closer to the CNTs part of a junction than $d=6$ \\AA, the whole molecule was kept for a while, and discarded \notherwise. Having applied this first part of the procedure, we kept 4 polymer molecules for the parallel configuration, 8 molecules for the perpendicular configuration and\n11 molecules for the 45 degrees configuration. \n\nThen we looked at the monomers of each molecule that survived the first round of selection. The same procedure was applied\nto monomers as the one used earlier for molecules: if at least one of the monomer atoms was closer to the CNTs part of a junction than $d=6$ \\AA, \nthe whole monomer was kept for a while, and \ndiscarded otherwise. \n\nAfter the second round of selection with monomers was over, we dealt in the same manner with the individual residues comprising a monomer. The broken\nbonds that appeared in the second and third stages were saturated with hydrogen atoms. The described procedure resulted in the following numbers of atoms in the whole\ndevice, including the central scattering region and the leads: 881 for the parallel configuration, 1150 for the perpendicular configuration, and 1074 for the 45 degrees configuration.\nThe atomic configurations obtained using the described procedure for the first time steps in the corresponding series are presented in Fig. \\ref{fig_junction_conf}.\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig2\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The atomic configurations for the first-principles calculations of the polymer filled CNTs junctions contact resistance. The configurations are for the first time steps in the corresponding series.\nLeft: the parallel configuration, center: CNTs axes are crossed at 45 degrees,\nright: the perpendicular configuration. The Carbon atoms are gray, the Nitrogen atoms are blue, the Oxygen atoms are red, and the Hydrogen atoms are light gray. The leads atoms are colored by green. } \n\\label{fig_junction_conf}\n\\end{figure}\n\nA fully ab-initio method for electronic structure investigations utilizing a localized pseudo-atomic basis set, as described in \\cite{Ozaki_DFT} and implemented in \\cite{OpenMX}, \nwas used for the calculations of the electronic structures of the whole device and the leads.\nWe used basis set s2p2d1, the Pseudo Atomic Orbitals (PAO) cutoff radius equal to 6.0 a. u., and the cutoff energy of 150 Ry.\nThe pseudo-potentials generated according to Morrison, Bylander, and Kleinman scheme \\cite{pseudopot} were used. For the \ndensity functional calculations, the exchange-correlation functional was used in PBE96 form \\cite{PBE96}.\n\nUsing the electronic structures of the whole device and the leads we calculated energy the dependent transmission function\n through the device. Then the dependence $I(V)$ of the current $I$ on the voltage $V$ between the leads was determined\nwith the Green's function approach as described in detail in \\cite{Datta}. Finally, the Landauer-Buttiker approach was used to find the current through a polymer filled CNTs junction.\n\nSolving a scattering problem for a nano-device at arbitrary voltages is a computationally very complex task since it requires achieving self-consistency for both\nelectron density and induced electrostatic potential simultaneously. Fortunately, for contact resistance calculations one can take\nadvantage of the fact that the required voltages are very low. \n\nAccording to the experimental evidence, the size of a nanocomposite specimen used in conductivity experiments is about 10 mm \\cite{exp_CNT_length_1}, and the typical voltages applied across such specimen do not exceed 100 V \\cite{CNT_book}. Then for the size of a central scattering region about 1 nm, the voltage drop is about 10$^{-5}$ V, which is well within the range where the simplified approach is applicable.\n\nThe question of the modeling of quantum transport in the limit of low voltages was discussed in detail in \\cite{comp_no_pol}, where it was demonstrated that in the case of moderate voltages between leads,\nthe scattering probability $T(E)$ is not sensitive to the details of the electrostatic potential distribution $V({\\mathbf r})$ in the central scattering region, and\nsome physically reasonable approximation may be chosen for $V({\\mathbf r})$. \n\nThis is due to the fact that the difference of the Fermi functions\n$f(\\varepsilon - \\mu_L)$ and $f(\\varepsilon - \\mu_R)$ \nfor the left and right leads with corresponding the chemical potentials $\\mu_L$ and $\\mu_R$, present in the original Landauer-Buttiker formula:\n\\begin{equation}\nI=\\frac{2e}{h} \\int T(\\varepsilon) \\left (f(\\varepsilon - \\mu_L) - f(\\varepsilon - \\mu_R) \\right )d\\varepsilon ,\n\\label{LB_formula}\n\\end{equation}\nwhere $e$ is the elementary charge, $h$ is the Planck constant, $\\varepsilon$ is the electron energy and $T(\\varepsilon)$ \nis the energy dependent transmission probability,\n is reduced in this case to a very narrow and sharp peak centered at the Fermi level\nof the device. \n\nIn addition to the analysis performed in \\cite{comp_no_pol}, in this article, to verify the accuracy of our approach, we made contact resistance calculations for a simple test CNT junction in a coaxial configuration, using both the simplified method we suggest and the full NEGF method, where not only electron charge density but the electric potential was converged as well. The interlead voltage used in those test calculations was\nset to $10^{-4}$ V, and the gap between the CNTs tips was 0.94 \\AA. The atomic configurations for the test calculations are presented in Fig. \\ref{fig_NEGF_compare}. The consistent NEGF calculations produced 1.71~$\\cdot 10^{-5}$ S for the conductance of the junctions shown in Fig. \\ref{fig_NEGF_compare}, while modeling without searching for convergence of potential yielded 1.72 $\\cdot 10^{-5}$ S. \n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_conf_compare\/fig_compare.pdf}\n\\caption{The atomic configuration used in this work to validate the method of calculating quantum transport by comparing it againts the consistent NEGF approach.} \\label{fig_NEGF_compare}\n\\end{figure}\n\nThus, in our case, a very complex task of finding the $I(V)$ characteristic of a nano-device can be significantly simplified without the loss\nof precision. For the $I(V)$ calculations in the current paper we employed the abrupt potential model introduced in \\cite{comp_no_pol}: the potentials $V_L$ for the\nleft lead and $V_R$ for the right lead were set and were used for all atoms of the corresponding CNT to which that lead belonged. As for the polymer atoms, both $V_L$ and\n$V_R$ can be safely used for them, and at the considered voltages, adopting these two options, as we have checked by direct calculations, leads to the differences in\ncurrent not exceeding 0.1\\%.\n\n\\subsection*{C. The percolation model} \\label{sec_percolation}\nDetermination of the conductivity of a polymer-CNT system can be implemented in 2 stages. First, a percolation cluster is formed, and the second stage implies solving the matrix problem for a random resistor circuit (network). \n\nAt the first stage, the modeling area -- a cube of the linear size $L$ -- is filled with CNTs. For this task, permeable capsules (cylinders with hemispheres at the ends) with a fixed length and diameter were chosen as filling objects corresponding to CNTs. The cube is filled by the successive addition of CNTs until a fixed bulk density of CNTs \n\\begin{equation}\n\\eta = \\frac{((4\/3)\\pi R^3 + \\pi R^2 h)N}{L^3} \\label{cnt_vol_frac}\n\\end{equation}\nin the cube is reached, where $R$ is the radius of the cylinder and hemisphere, $h$ is the height of the cylinder, and $N$ is the number of CNTs in the cube. The percolation problem for permeable capsules was previously solved in \\cite{Xu_16}, and in \\cite{Schilling_15} capsules with a semipermeable shell were considered.\n\nIn the percolation problem, we use periodic boundary conditions as, for example, in \\cite{Bao_2011}. \nWe use the method of finding a percolation threshold based on the Newman and Ziff algorithm \\cite{Newman_2001}, where the identification of a percolation cluster is made at the stage of its formation.\n\n\nWhen a percolation cluster is formed, the obtained CNT configuration is transformed into a resistor circuit (2nd stage). \n\nThe contributions to a conductance matrix resulting from inner resistance of CNTs and the junctions tunneling resistance are usually discussed in connection with constructing conductivity percolation algorithms. Direct measurements of CNTs resistance per unit length are available. In \\cite{Chiodarelli_11}, the inner resistance of CNTs is estimated as $15 \\cdot 10^3$ $\\Omega\/\\mu$m. The results of \\cite{Ebbesen_96} give specific CNTs resistance in the range $(12-86) \\cdot 10^3$ $\\Omega\/\\mu$m. Taking into account that the characteristic CNTs lengths in nanocomposites are about several $\\mu$m \\cite{exp_CNT_length_1,CNT_book}, this results in the inner CNTs resistance approximately $10^4-10^5$ $\\Omega$, which is at least one order of magnitude less than the tunneling resistance obtained in this work. The specific results on tunneling resistance will be discussed below in section \\ref{RND}. Thus, in our percolation model, the inner resistance of CNTs is neglected, and only tunneling resistance of CNTs junctions is taken into account. This can significantly reduce the requirements for computational time. \n\nWhen contact resistance is determined only by tunneling, the principle of compiling the matrix for the percolation problem will be as follows. \nFirst, the matrix (N, N) is compiled from the bonds of percolation elements, where N is the number of CNTs participating in percolation. \nThen this matrix is transformed into the conductivity matrix according to the second Kirchhoff law \n\\begin{equation}\n\\sum_j G_{ij} (V_i-V_j) = 0 \\label {second_Kirchoff}\n\\end{equation}\n- the sum of currents for all internal elements of a percolation network is zero -- where $G_{ij}$ are the elements of the conductance matrix ${\\mathbf G}$ and $V_i$ is the component of the voltage vector ${\\mathbf V}$ corresponding to the $i$-th contact point in a percolation network \\cite{Kirkpatrick_73}. \nThe voltages on the left and right borders of a simulation volume are set to $V_L=1$ V and $V_R=0$ V, respectively. \n\nNow finding the conductivity of the system is reduced to the problem $\\mathbf{GV}=\\mathbf{I}$, where\n$\\mathbf{I}$ is the vector of the currents between the contact points. \n\nThe dimension of the matrix problem can be further reduced to \n(N-2, N-2) by excluding boundary elements. \n\nAfter solving the equation (\\ref{second_Kirchoff}), with the elements of the $\\mathbf{G}$ matrix obtained by the first-principles calculations, we obtain the voltage vector for all internal elements. \nThen, knowing this vector, we sum up all the currents on each of the boundaries. The currents on the left $I_L$ and right $I_R$ \nboundaries of a simulation volume are equal in magnitude and opposite in sign $I_L= - I_R$. Knowing these currents, we determine the conductance of the simulation system as \n$G=|I_L|\/(V_L-V_R) = |I_R|\/(V_L-V_R)$. \nAnd then the conductivity of the composite is calculated as $\\sigma=GL\/S$, where $L$ is the distance between the faces of a simulation volume where voltage is applied, and $S$ is the area of that kind of face. In our case, for the simulation volume of a cubic shape, $S=L^2$, and $\\sigma=G\/L$.\n\nTo calculate the conductivity, the following system parameters were selected: the length of a nanotube is $l=3$ $\\mu$m, the diameter of a CNT is $D=30$ nm, the aspect ratio $l\/D=100$, and the size of the system is 4 $\\mu$m. The same values were used in \\cite{Yu_2010}. We adopted those values to test our realization of the percolation algorithm against the previously obtained results \\cite{Yu_2010}.\nThen for the given parameters for each fixed tube density, the Monte Carlo method (100 implementations of various configurations of CNT networks) was used to calculate the system conductivity. \n\nThe quality of CNTs dispersion is one of the key factors that affect the properties of nanocomposites, and a lot of efforts is taken\nto achieve a homogeneous distribution of fillers \\cite{Mital_14}. In this work, we take into consideration the effect of inhomogeneity of a CNTs distribution on composite conductivity.\nThe spatial density of nanotubes $\\rho_{CNT}$, in this case, has one peak with a Gaussian distribution:\n\\begin{equation} \n\\rho_{CNT}=\\rho_0 \\cdot exp(-(\\mathbf {r-r_0})^2\/\\rho_\\sigma^2), \\label {agglo}\n\\end{equation}\nwhere ${\\mathbf r_0}$ coincides with the geometrical center of a simulation volume, and $\\rho_\\sigma = L\/12$. The value of the $\\rho_0$ parameter is chosen so that the CNTs volume fraction in the inhomogeneous case is the same as in the homogeneous distribution.\n\n\n\\section*{Results and discussion}\\label{RND}\nTo find the contact resistance of polymer filled CNTs junctions one first needs to find their Volt-Ampere characteristics $I(V)$ and to determine the voltage range where\n$I(V)$ is linear and is not sensitive to the specific distribution of an electrostatic potential in the scattering region. In Fig. \\ref{fig_I_V} the $I(V)$ plot\nfor the first time step in the atomic geometry series for the parallel configuration is shown.\n\\begin{figure}[b]\n\\begin{tabular}{lr}\n\\includegraphics[width=3.5cm]{figs\/I_V_test\/I_V.pdf} &\n\\includegraphics[width=3.5cm]{figs\/I_V_test_low_V\/I_V_low_V.pdf}\n\\end{tabular}\n\\caption{The Volt-Ampere characteristic for the polymer filled CNTs junction corresponding to the first time step in the series for the parallel configuration.\nLeft frame: maximum inter-lead voltage is $10^{-3}$ V, right frame: $10^{-4}$ V. The circles correspond to the results of calculations,\nthe lines are guides for an eye.} \n\\label{fig_I_V}\n\\end{figure}\n\nIt is clearly seen from Fig. \\ref{fig_I_V} that up to about $10^{-4}$ V the $I(V)$ characteristic is linear, and after that value, it starts to deviate from a\nsimple linear dependence. Thus, for the calculations of a contact resistance $R$ and its inverse, a junction conductivity $G$,\nwe used the electrical current values obtained for the inter-lead voltage equal to $10^{-4}$ V. Note, that according to our estimates in section \\ref{sec_meth_fp}, a characteristic voltage drop on the length of a CNTs tunneling junction is about $10^{-5}$ V which is well within the region where the linear $I(V)$ is observed.\n\nThe time dependences of the junctions conductances for the three considered configurations are presented in Fig. \\ref{fig_G_t}. One might expect that the \nshifts of both CNTs atoms and polymer atoms in the central scattering region due to thermal fluctuations would lead to fluctuations of junctions conductances $G$, but \nquantitative characteristics of this phenomenon such as minimum $G_{min}$, maximum $G_{max}$, mean values $\\langle G \\rangle$ and a standard deviation $G_\\sigma$ can only be captured by highly precise fully atomistic \nfirst-principles methods, like those employed in the current paper. The resulting fluctuations of conductance are very high. For the parallel CNTs configuration\nthe minimum value, $G_{min} = 2.4 \\cdot 10^{-8}$ S, and the maximum value, $G_{max} = 6.8 \\cdot 10^{-6}$ S, differ by more than two orders of magnitude, for the 45 degrees and perpendicular\nconfigurations the corresponding ratios are about 30. The same strong variations of conductance over time were reported in \\cite{Penazzi_2013} for the coaxial\nCNTs configuration, where the results were obtained using a semi-empirical tight-binding approximation. Thus, it is obvious that for the precise determination\nof a conductance of polymer filled CNTs junctions one needs to use fully atomistic approaches, and phenomenological methods taking atomic configurations into account \non the average are not reliable.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/G_t\/g_t.pdf}\n\\caption{The time dependence of the polymer filled junctions conductance $G$ in S. Red color correspond to the parallel configuration, the green lines -- to the perpendicular configuration\nand the blue lines -- the 45 degrees configuration. The results of the calculations are shown by circles, the saw-tooth lines serve as a guide for an eye.\nThe straight solid lines designate the mean values of conductance $\\langle G \\rangle$, and the dashed ones -- $\\langle G \\rangle \\pm G_\\sigma$.} \n\\label{fig_G_t}\n\\end{figure}\n\n\n\n\\begin{table*}[h]\n \\centering\n\\caption{The results of statistical analysis of the CNTs junctions conductances in S, for CNTs separation equal to 6 \\AA \\ and different values of the CNTs crossing angles $\\varphi$, \nwithout polymer from \\cite{comp_no_pol}, and with polyimide R-BAPB filling obtained in the current paper.\n}\n\\begin{tabular}{cccccc}\n\\textrm{$\\varphi$}&\n\\textrm{no polymer}&\n\\textrm{}&\n\\textrm{polymer present}&\n\\textrm{}&\n\\textrm{}\\\\\n\\textrm{}&\n\\textrm{the results of \\cite{comp_no_pol}}&\n\\textrm{$G_{min}$}&\n\\textrm{$G_{max}$}&\n\\textrm{$\\langle G \\rangle$}&\n\\textrm{$G_\\sigma$}\\\\\n\\hline\n& & & & &\\\\ \n0 & 3.6$\\cdot10^{-13}$ &2.4$\\cdot10^{-8}$ &6.8$\\cdot10^{-6}$ & 1.8$\\cdot10^{-6}$ & 1.6$\\cdot10^{-6}$\\\\\n& & & & & \\\\\n\\hline\n& & & & &\\\\ \n0.2$\\pi$ & 1.4$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.25$\\pi$ & --- &4.8$\\cdot10^{-9}$ &1.4$\\cdot10^{-7}$ &3.4$\\cdot10^{-8}$ &2.7$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.3$\\pi$ & 1.2$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.5$\\pi$ & 4.2$\\cdot10^{-14}$ &2.2$\\cdot10^{-9}$ &4.3$\\cdot10^{-8}$ & 1.4$\\cdot10^{-8}$ & 1.1$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\end{tabular}\n\\label{tab_cond}\n\\end{table*}\n\nTo assign the tunneling resistance to a polymer filled CNT junction the following algorithm was used. First, for each junction that had to be used in the percolation algorithm, a uniformly distributed random number $\\varphi$ in the range [0, $\\pi\/2$] was generated. The value of the intersection angle for that junction was assigned to the obtained random number. The mean values and standard deviations for CNTs tunneling resistances and conductances calculated for the different atomic configurations corresponding to the different time steps are known for $\\varphi = 0, \\pi\/4$, and $\\pi\/2$. Analyzing figure 4 of \\cite{comp_no_pol}, one can see that though an angle dependence of current and hence conductivity is a rather complex function, in the first approximation one can adopt a roughly piece-wise linear character for this function with the minimum located at $\\varphi=0.25\\pi$. Thus the logarithm of the mean value of conductance $\\mu_\\varphi$ for the generated $\\varphi$ was set by linear interpolation between the logarithms of the mean values of conductances for $\\varphi=0$ and $\\varphi=\\pi\/4$ or $\\varphi=\\pi\/4$ and $\\varphi=\\pi\/2$ presented in Table \\ref{tab_cond}. The same algorithm was applied to finding the standard deviation values $\\sigma_\\varphi$ for the generated $\\varphi$. After the statistical parameters for the generated $\\varphi$ are estimated, the conductivity of the junction is set to a random number generated using the normal distribution with the parameters $\\mu_\\varphi$ and $\\sigma_\\varphi$.\n\n\nIn \\cite{comp_no_pol}, the conductances were reported for the CNTs junctions with almost the same geometry as the CNTs parts of the devices considered in the current paper. The only difference \nbetween the configurations is that in this work the carbon atoms belonging to the CNTs part of the central scattering region are shifted somewhat from their equilibrium positions\ndue to the interaction with polymer. The maximum values of those shifts along the $x$, $y$, and $z$ coordinates lie in the range $0.2-0.5$ \\AA. This gives us the possibility to directly compare the current results to the data \nfrom \\cite{comp_no_pol} and,\nthus, elucidate the influence of polymer filling on the junctions conductance. The corresponding data and the results of a basic statistical\nanalysis for the case of the polymer filled junctions are provided in Tab. \\ref{tab_cond}.\n\nFirst, as was expected, filling CNTs junctions with polymer creates carrier tunneling paths and increases junctions conductance by 6-7 orders of magnitude.\nSecond, it is evident that the CNTs axes crossing angle is crucial for the junctions conductivity when polymer is present as it was the case without\npolymer \\cite{comp_no_pol}. At the same time, the sharp dependence of polymer filled junctions conductance on the CNTs crossing angle is somewhat different\nfrom the analogous dependence for junctions without polymers. While in the latter case this dependence is sharply non-monotonous, with a pronounced minimum\nat the angles around $0.25\\pi$, in the former case there is a significant difference between the conductance values for the parallel and nonparallel configurations, but\nthe configurations with the angle $\\varphi$ between CNTs angles equal to $0.25\\pi$ and $0.5\\pi$ have very close conductances, and their mean values averaged over\ntime $\\langle G \\rangle_{45}$ and $\\langle G \\rangle_{per}$ lie within the\nranges $\\langle G \\rangle \\pm G_\\sigma$ of each other. Moreover, in contrast to the geometries without polymer, for the polymer filled CNTs junctions $\\langle G \\rangle_{per}$ is lower than $\\langle G \\rangle_{45}$ by a factor of 2.4.\n\nNote also that for the parallel configuration, the polymer influence on the junction conductance is more pronounced than for the nonparallel ones.\nFor the parallel configuration, adding polymer to a junction of CNTs separated by 6.0 \\AA\\ with initial conductance of 3.6$\\cdot 10^{-13}$ S \nproduces conductance mean value equal to 1.8$\\cdot 10^{-6}$ S. This gives the factor 0.5$\\cdot 10^7$; the value of the analogous factor for the perpendicular configuration is 0.33$\\cdot 10^6$.\n\nThe probable reason for the more effective conductance increase, when polymer is added, for the configurations with smaller angles between CNTs axes, is\nthat the smaller is an intersection angle, the larger is the overlap area between CNTs where polymer can penetrate and, thus, create tunneling bridges.\nThe higher fluctuation of conductance over time for the parallel configuration can be explained by the same reason: a larger CNTs overlap area gives\nmore freedom for polymer atoms to adjust their positions.\n\nThe dependence of the calculated composite conductivity $\\sigma$ on CNTs volume fraction $\\eta$ $\\sigma(\\eta)$ is presented in Fig. \\ref{fig_cond_pol}. \nThe value of the percolation threshold $\\eta_{thresh}$ is estimated in this work as $\\eta_{thresh} = 0.007$. To test our realization of the percolation algorithm against the previous results of \\cite{Yu_2010} we calculated the composite conductivity using\nthe fixed CNTs junction conductance equal to 1 M$\\Omega$ for all junctions in a percolation network. Our results presented in\nFig. \\ref{fig_cond_pol} by the red circles coincide within graphical accuracy to the results of \\cite{Yu_2010} shown by the red squares.\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/fig9\/cnt_9.pdf}\n\\caption{ The conductivity of CNT enhanced nanocomposites above the percolation threshold obtained in this work. The symbols of different shapes and colors are used to designate the following results. The red circles: the fixed CNTs tunneling junctions resistance of $R=1$ M$\\Omega$ is used, the red squares: the conductivity results for the fixed 1 M$\\Omega$ tunneling resistance from \\cite{Yu_2010}, the black triangles: the same as the red circles but for $R=0.54$ M$\\Omega$ corresponding to the mean value of the tunneling junction resistance for the parallel configuration from Table \\ref{tab_cond}, the blue rhombi: the angle dependence of the CNTs junctions resistance is taken into account, the green pentagons: CNTs agglomeration is considered in addition to the angle dependence. The red line is a guide for an eye.}\n\\label{fig_cond_pol}\n\\end{figure}\nThe 1 M$\\Omega$, used in various sources, for example, \\cite{Yu_2010}, is not an arbitrary value, but rather a typical contact resistance of CNTs junctions filled with polymer for simple geometries. In this work, we obtained for the parallel configurations $1\/\\langle G \\rangle =0.54$ M$\\Omega$. The $\\sigma(\\eta)$ dependence for the fixed tunneling resistance of $0.54$ M$\\Omega$ is shown in Fig. \\ref{fig_cond_pol} by the black triangles.\n\nTaking into account the angle dependence of CNTs junctions conductances with the statistical parameters according to Table\n\\ref{tab_cond}, leads to the lowering of composite conductivity just above the percolation threshold by the factor of about 30. This number correlates with the ratio of the mean conductances for the parallel, $\\langle G \\rangle_{par}$, and 45$^\\circ$, $\\langle G \\rangle_{45}$, configurations: $f_G = \\langle G \\rangle_{par}\/ \\langle G \\rangle_{45} = 53$, but is higher than $f_G$ due to the presence of junctions with $\\varphi < \\pi\/4$. \n\nIf agglomeration of CNTs, modeled by the inhomogeneity of their distribution according to formula (\\ref{agglo}) and the parameter values discussed in section \\ref{sec_percolation}, is taken into account in addition to the angle dependence of conductance, the composite conductivity is further reduced above a percolation threshold by the factor of 2.5. Lowering of conductivity of composites with agglomerated CNTs above a percolation threshold was also mentioned in \\cite{Bao_2011}. The calculated results for the conductivity of a percolation network of agglomerated CNTs are shown in Fig. \\ref{fig_cond_pol} by the green pentagons.\n\nWe believe that in this work we have identified some of the key factors that influence nanocomposites electrical conductivity: the geometry of tunneling junctions and changes of atomic configurations due to thermal fluctuations. Among other causes that may affect conductivity, but are not considered in this work, are the presence of defects in CNTs, a distribution of CNTs over chiralities, lengths, aspect ratios, different separations between CNTs. \n\n\nUntil the specific experiments on conductivity for R-BAPB polyimide are not available, we can make a preliminary comparison of our modeling results to the available experimental results for different composites. The calculated conductivity of composite just above the percolation threshold at $\\eta=0.0075$ is equal to $3.6\\cdot10^{-3}$ S\/m.\nThis is a reasonable value that falls into the range of experimentally observed composites conductivities (for the comprehensive compilation of experimental results see Table 1 of \\cite{Eletskii}). To make a quantitative comparison of modeling results with experiments the full details of nanocomposites structure are necessary, including the CNTs parameters mentioned in the previous paragraph. All these factors can be easily incorporated into the approach proposed in this work if sufficient computational resources are available.\n\n\\section*{Conclusions}\n\nWe have proposed a physically consistent, computationally simple, and at the same time precise, multi-scale method for calculations of electrical conductivity of CNT enhanced\nnanocomposites. The method starts with the atomistic determination of the positions of polymer atoms intercalated between CNTs junctions, proceeds with the fully first-principles calculations of polymer-filled CNTs junctions conductance at the microscale and finally performs\nmodeling of percolation through an ensemble of CNTs junctions by the Monte-Carlo technique. \n\nThe developed approach has been applied to the modeling of electrical conductivity of polyimide R-BAPB + single wall (5,5) CNTs nanocomposite.\n\nOur major contributions to the field are the following. We have proposed a straightforward\nmethod to calculate a contact resistance and conductance for polymer-filled CNTs junctions with arbitrary atomic configurations without resorting to any simplifying assumptions. We have demonstrated that a consistent multiscale approach, based on solid microscopic physical methods can give reasonable results, lying within the experimental range, for the conductivity of composites and suggested a corresponding work-flow. \n \n\nIt is shown that a contact resistance\nand nanocomposite conductivity is highly sensitive to the geometry of junctions, including an angle between CNTs axes and subtle thermal shifts of polymer atoms in an inter-CNT's gap.\nThus, we argue that for the precision calculations of nanocomposites electrical properties rigorous atomistic quantum-mechanical approaches are indispensable.\n\nWe have to admit though, that we have not considered all possible factors that may influence CNT junctions conductivity on the micro-level. We concentrated on the CNTs crossing angle factor because it seems to be the\nmost influential. The additional factors may include, for example, defects in CNTs,\nCNTs overlap lengths and others. On the other hand, the proposed approach may be used to take all those factors into account, provided sufficient computational resources are available.\n\n\\section*{Acknowledgments}\n\nThis work was supported by the State Program \"Organization of\nScientific Research\" (project 1001140) and by the Research Center \"Kurchatov Institute\" (order No.~1878~of~ 08\/22\/2019).\n\nThis work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC \"Kurchatov Institute\", \\url{http:\/\/ckp.nrcki.ru\/}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}