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Ibrahima Bah - One of the best experts on this subject based on the ideXlab platform.
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holographic duals of five dimensional scfts on a Riemann Surface
Journal of High Energy Physics, 2019Co-Authors: Ibrahima Bah, Achilleas Passias, Peter WeckAbstract:We study the twisted compactifications of five-dimensional Seiberg SCFTs, with $$ {\mathrm{SU}}_{\mathrm{\mathcal{M}}}(2)\times {E}_{N_f}+1 $$ flavor symmetry, on a generic Riemann Surface that preserves four supercharges. The five-dimensional SCFTs are obtained from the decoupling limit of N D4-branes probing a geometry of Nf < 8 D8-branes and an O8-plane. In addition to the R-symmetry, we can also twist the flavor symmetry by turning on background flux on the Riemann Surface. In particular, in the string theory construction of the five-dimensional SCFTs, the background flux for the SUℳ(2) has a geometric origin, similar to the topological twist of the R-symmetry. We argue that the resulting low-energy three-dimensional theories describe the dynamics on the world-volume of the N D4-branes wrapped on the Riemann Surface in the O8/D8 background. The Riemann Surface can be described as a curve in a Calabi-Yau three-fold that is a sum of two line bundles over it. This allows for an explicit construction of AdS4 solutions in massive IIA supergravity dual to the world-volume theories, thereby providing strong evidence that the three-dimensional SCFTs exist in the low-energy limit of the compactification of the five-dimensional SCFTs. We compute observables such as the free energy and the scaling dimensions of operators dual to D2-brane probes; these have non-trivial dependence on the twist parameter for the U(1) in SUℳ(2). The free energy exhibits the N5/2 scaling that is emblematic of five-dimensional SCFTs.
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Holographic duals of five-dimensional SCFTs on a Riemann Surface
SpringerOpen, 2019Co-Authors: Ibrahima Bah, Achilleas Passias, Peter WeckAbstract:Abstract We study the twisted compactifications of five-dimensional Seiberg SCFTs, with S U ℳ 2 × E N f + 1 $$ {\mathrm{SU}}_{\mathrm{\mathcal{M}}}(2)\times {E}_{N_f}+1 $$ flavor symmetry, on a generic Riemann Surface that preserves four supercharges. The five-dimensional SCFTs are obtained from the decoupling limit of N D4-branes probing a geometry of N f < 8 D8-branes and an O8-plane. In addition to the R-symmetry, we can also twist the flavor symmetry by turning on background flux on the Riemann Surface. In particular, in the string theory construction of the five-dimensional SCFTs, the background flux for the SUℳ(2) has a geometric origin, similar to the topological twist of the R-symmetry. We argue that the resulting low-energy three-dimensional theories describe the dynamics on the world-volume of the N D4-branes wrapped on the Riemann Surface in the O8/D8 background. The Riemann Surface can be described as a curve in a Calabi-Yau three-fold that is a sum of two line bundles over it. This allows for an explicit construction of AdS4 solutions in massive IIA supergravity dual to the world-volume theories, thereby providing strong evidence that the three-dimensional SCFTs exist in the low-energy limit of the compactification of the five-dimensional SCFTs. We compute observables such as the free energy and the scaling dimensions of operators dual to D2-brane probes; these have non-trivial dependence on the twist parameter for the U(1) in SUℳ(2). The free energy exhibits the N 5/2 scaling that is emblematic of five-dimensional SCFTs
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ads5 solutions from m5 branes on Riemann Surface and d6 branes sources
Journal of High Energy Physics, 2015Co-Authors: Ibrahima BahAbstract:We describe the gravity duals of four-dimensional $$ \mathcal{N}=1 $$ superconformal field theories obtained by wrapping M5-branes on a punctured Riemann Surface. The internal geometry, normal to the AdS 5 factor, generically preserves two U(1)s, with generators (J +, J −), that are fibered over the Riemann Surface. The metric is governed by a single potential that satisfies a version of the Monge-Ampere equation. The spectrum of $$ \mathcal{N}=1 $$ punctures is given by the set of supersymmetric sources of the potential that are localized on the Riemann Surface and lead to regular metrics near a puncture. We use this system to study a class of punctures where the geometry near the sources corresponds to M-theory description of D6-branes. These carry a natural (p, q) label associated to the circle dual to the killing vector pJ + + qJ − which shrinks near the source. In the generic case the world volume of the D6-branes is AdS 5 × S 2 and they locally preserve $$ \mathcal{N}=2 $$ supersymmetry. When p = −q, the shrinking circle is dual to a flavor U(1). The metric in this case is non-degenerate only when there are co-dimension one sources obtained by smearing M5-branes that wrap the AdS 5 factor and the circle dual the superconformal R-symmetry. The D6-branes are extended along the AdS 5 and on cups that end on the co-dimension one branes. In the special case when the shrinking circle is dual to the R-symmetry, the D6-branes are extended along the AdS 5 and wrap an auxiliary Riemann Surface with an arbitrary genus. When the Riemann Surface is compact with constant curvature, the system is governed by a Monge-Ampere equation.
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ads5 solutions from m5 branes on Riemann Surface and d6 branes sources
arXiv: High Energy Physics - Theory, 2015Co-Authors: Ibrahima BahAbstract:We describe the gravity duals of four-dimensional N=1 superconformal field theories obtained by wrapping M5-branes on a punctured Riemann Surface. The internal geometry, normal to the AdS5 factor, generically preserves two U(1)s, with generators (J+,J-), that are fibered over the Riemann Surface. The metric is governed by a single potential that satisfies a version of the Monge-Ampere equation. The spectrum of N=1 punctures is given by the set of supersymmetric sources of the potential that are localized on the Riemann Surface and lead to regular metrics near a puncture. We use this system to study a class of punctures where the geometry near the sources corresponds to M-theory description of D6-branes. These carry a natural (p,q) label associated to the circle dual to the killing vector (p J+ + q J-) which shrinks near the source. In the generic case the world volume of the D6-branes is AdS5 X S^2 and they locally preserve N=2 supersymmetry. When p=-q, the shrinking circle is dual to a flavor U(1). The metric in this case is non-degenerate only when there are co-dimension one sources, M9-branes, obtained by smearing M5-branes that wrap the AdS5 factor and the circle dual the superconformal R-symmetry. In the IIA limit, they can interpreted as D8-branes. The D6-branes are extended along the AdS5 and on cups that end on the co-dimension one branes. In the special case when the shrinking circle is dual to the R-symmetry, the D6-branes are extended along the AdS5 and wrap an auxiliary Riemann Surface with an arbitrary genus. When the Riemann Surface is compact with constant curvature, the system is governed by a Monge-Ampere equation.
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quarter bps ads 5 solutions in m theory with a t 2 bundle over a Riemann Surface
Journal of High Energy Physics, 2013Co-Authors: Ibrahima BahAbstract:We study and classify quarter-BPS AdS 5 systems in M-theory, whose internal six-dimensional geometry is a T 2 bundle over a Riemann Surface and two interval directions. The general system presented, provides a unified description of all known AdS 5 solutions in M-theory. These systems are governed by two functions, one that corresponds to the conformal factor of the Riemann Surface and another that describes the T 2 fibration. We find a special set of solutions that can be organized into two classes. In the first one, solutions are specified by the conformal factor of the Riemann Surface which satisfies a warped generalization of the SU(∞) Toda equation. The system in the second class requires the Riemann Surface to be S 2, H 2 or T 2. Class one contains the M-theory AdS 5 solutions of Lin, Lunin and Maldacena; the solutions of Maldacena and Nunez; the solutions of Gauntlett, Martelli, Sparks and Waldram; and the eleven-dimensional uplift of the Y p,q metrics. The second includes the recently found solutions of Beem, Bobev, Wecht and the author. Within each class there are new solutions that will be studied in a companion paper.
Indranil Biswas - One of the best experts on this subject based on the ideXlab platform.
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deformation quantization of moduli spaces of higgs bundles on a Riemann Surface with translation structure
Journal of Mathematical Physics, 2021Co-Authors: Indranil BiswasAbstract:Let X be a compact connected Riemann Surface of genus g ≥ 1 equipped with a nonzero holomorphic 1-form. Let MX(r) denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g − 1) + 1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of MX(r).
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projective structures on Riemann Surface and natural differential operators
Differential Geometry and Its Applications, 2020Co-Authors: Indranil Biswas, Sorin DumitrescuAbstract:Abstract We investigate the holomorphic differential operators on a Riemann Surface M. This is done by endowing M with a projective structure. Let L be a theta characteristic on M. We explicitly describe the jet bundle J k ( E ⊗ L ⊗ n ) , where E is a holomorphic vector bundle over M equipped with a holomorphic connection, for all k and n. This provides a description of global holomorphic differential operators from E ⊗ L ⊗ n to another holomorphic vector bundle F using the natural isomorphism Diff k ( E ⊗ L ⊗ n , F ) = F ⊗ ( J k ( E ⊗ L ⊗ n ) ) ⁎ .
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on the moduli space of holomorphic g connections on a compact Riemann Surface
European Journal of Mathematics, 2020Co-Authors: Indranil BiswasAbstract:Let X be a compact connected Riemann Surface of genus at least two and G a connected reductive complex affine algebraic group. The Riemann–Hilbert correspondence produces a biholomorphism between the moduli space $${{\mathscr {M}}}_X(G)$$ parametrizing holomorphic G-connections on X and the G-character variety While $${{\mathscr {R}}}(G)$$ is known to be affine, we show that $${{\mathscr {M}}}_X(G)$$ is not affine. The scheme $${{\mathscr {R}}}(G)$$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on $${{\mathscr {M}}}_X(G)$$ with the property that the Riemann–Hilbert correspondence pulls back the Goldman symplectic form to it. Therefore, despite the Riemann–Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann–Hilbert correspondence nevertheless continues to be algebraic.
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projective structures on Riemann Surface and natural differential operators
arXiv: Algebraic Geometry, 2019Co-Authors: Indranil Biswas, Sorin DumitrescuAbstract:We investigate the holomorphic differential operators on a Riemann Surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otimes n}, F)= F\otimes (J^k(E\otimes {\mathcal L}^{\otimes n}))^*$.
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on the moduli space of holomorphic g connections on a compact Riemann Surface
arXiv: Algebraic Geometry, 2019Co-Authors: Indranil BiswasAbstract:Let $X$ be a compact connected Riemann Surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(\pi_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on ${\mathcal M}_X(G)$ with the property that the Riemann--Hilbert correspondence pulls back to the Goldman symplectic form to it. Therefore, despite the Riemann--Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann--Hilbert correspondence nevertheless continues to be algebraic.
Lisa C Jeffrey - One of the best experts on this subject based on the ideXlab platform.
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The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann Surface and the homotopy of the large N limit
Letters in Mathematical Physics, 2017Co-Authors: Lisa C Jeffrey, Daniel A. Ramras, Jonathan WeitsmanAbstract:We show that the prequantum line bundle on the moduli space of flat SU (2) connections on a closed Riemann Surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat SU ( n ) connections, in the limit as n tends to infinity, and $$ {\mathbb C}P^\infty $$ C P ∞ . Applications to the stable moduli space of flat unitary connections are also discussed.
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intersection pairings on singular moduli spaces of bundles over a Riemann Surface and their partial desingularisations
Transformation Groups, 2006Co-Authors: Lisa C Jeffrey, Frances Kirwan, Younghoon Kiem, Jonathan WoolfAbstract:This paper studies intersection theory on the compactified moduli space \({\mbox{$\cal M$}} (n,d)\) of holomorphic bundles of rank n and degree d over a fixed compact Riemann Surface \(\Sigma\) of genus \(g \geq 2\) where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups \(I\!H^*({\mbox{$\cal M$}} (n,d))\) defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities \(\widetilde{{\mbox{$\cal M$}}} (n,d)\) of \({\mbox{$\cal M$}} (n,d).\) Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on \(\widetilde{{\mbox{$\cal M$}}}(n,d).\) The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.
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intersection pairings on singular moduli spaces of bundles over a Riemann Surface and their partial desingularisations
arXiv: Algebraic Geometry, 2005Co-Authors: Lisa C Jeffrey, Frances Kirwan, Younghoon Kiem, Jonathan WoolfAbstract:This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann Surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of singularities of M(n,d). The case when n=2 is discussed in detail. Finally Witten's integral is considered for this singular case.
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intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann Surface
Annals of Mathematics, 1998Co-Authors: Lisa C Jeffrey, Frances KirwanAbstract:Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann Surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann Surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by
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intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann Surface
arXiv: Algebraic Geometry, 1996Co-Authors: Lisa C Jeffrey, Frances KirwanAbstract:We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann Surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
Jonathan Weitsman - One of the best experts on this subject based on the ideXlab platform.
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relations in the cohomology ring of the moduli space of flat so 2 n 1 connections on a Riemann Surface
Philosophical Transactions of the Royal Society A, 2018Co-Authors: Elisheva Adina Gamse, Jonathan WeitsmanAbstract:We consider the moduli space of flat SO(2n1)-connections (up to gauge transformations) on a Riemann Surface, with fixed holonomy around a marked point. There are natural line bundles over this modu...
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The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann Surface and the homotopy of the large N limit
Letters in Mathematical Physics, 2017Co-Authors: Lisa C Jeffrey, Daniel A. Ramras, Jonathan WeitsmanAbstract:We show that the prequantum line bundle on the moduli space of flat SU (2) connections on a closed Riemann Surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat SU ( n ) connections, in the limit as n tends to infinity, and $$ {\mathbb C}P^\infty $$ C P ∞ . Applications to the stable moduli space of flat unitary connections are also discussed.
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toric structures on the moduli space of flat connections on a Riemann Surface volumes and the moment map
Advances in Mathematics, 1994Co-Authors: Lisa C Jeffrey, Jonathan WeitsmanAbstract:Abstract In earlier papers we constructed a Hamiltonian torus action on an open dense set in the moduli space of flat SU (2) connections on a compact Riemann Surface, where the dimension of the torus is half the dimension of the moduli space. This torus action shows that this set can be viewed symplectically as a (noncompact) toric variety. The number of integral points of the moment map for the torus action turns out to be identical to the Verlinde dimension D ( g, k ). As an application, we furnish a new proof of the relation between the large- k limit of D( g, k ) and the volume of the moduli space. From our point of view, this relation follows from the equality between the symplectic volume of a toric variety and the Euclidean volume of the image of the moment map. Similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a Riemann Surface. Knowledge of these volumes has been shown to allow a proof of the Verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space.
Frances Kirwan - One of the best experts on this subject based on the ideXlab platform.
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intersection pairings on singular moduli spaces of bundles over a Riemann Surface and their partial desingularisations
Transformation Groups, 2006Co-Authors: Lisa C Jeffrey, Frances Kirwan, Younghoon Kiem, Jonathan WoolfAbstract:This paper studies intersection theory on the compactified moduli space \({\mbox{$\cal M$}} (n,d)\) of holomorphic bundles of rank n and degree d over a fixed compact Riemann Surface \(\Sigma\) of genus \(g \geq 2\) where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups \(I\!H^*({\mbox{$\cal M$}} (n,d))\) defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities \(\widetilde{{\mbox{$\cal M$}}} (n,d)\) of \({\mbox{$\cal M$}} (n,d).\) Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on \(\widetilde{{\mbox{$\cal M$}}}(n,d).\) The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.
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intersection pairings on singular moduli spaces of bundles over a Riemann Surface and their partial desingularisations
arXiv: Algebraic Geometry, 2005Co-Authors: Lisa C Jeffrey, Frances Kirwan, Younghoon Kiem, Jonathan WoolfAbstract:This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann Surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of singularities of M(n,d). The case when n=2 is discussed in detail. Finally Witten's integral is considered for this singular case.
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the hodge numbers of the moduli spaces of vector bundles over a Riemann Surface
arXiv: Algebraic Geometry, 2000Co-Authors: Richard Earl, Frances KirwanAbstract:Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann Surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory. In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces.
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intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann Surface
Annals of Mathematics, 1998Co-Authors: Lisa C Jeffrey, Frances KirwanAbstract:Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann Surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann Surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by
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intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann Surface
arXiv: Algebraic Geometry, 1996Co-Authors: Lisa C Jeffrey, Frances KirwanAbstract:We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann Surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
