The index formula for the moduli of<i>G</i>-bundles on a curve

Teleman, Constantin; Woodward, Chris

doi:10.4007/annals.2009.170.495

Cited by 32 publications

(33 citation statements)

References 30 publications

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“…In fact, the partition function of the gauged WZW-matter model appeared in the mathematical literature even earlier! It can be identified with an index formula for the moduli stack of algebraic G C -bundles over Σ first conjectured by Teleman [53] and later proved by Teleman and Woodward [54]. As we mentioned earlier, considering the index associated to the prequantum line bundle L over Bun G C (Σ) -which is basically M flat (Σ; G) away from stacky points -gives the Verlinde formula.…”

Section: R = 0 and Gauged Wzw-matter Modelmentioning

confidence: 75%

Equivariant Verlinde Formula from Fivebranes and Vortices

Gukov

Pei

2017

Commun. Math. Phys.

196

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We study complex Chern-Simons theory on a Seifert manifold M 3 by embedding it into string theory. We show that complex Chern-Simons theory on M 3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) Chern-Simons theory on Σ × S 1 and 4) index of a spin c Dirac operator on the moduli space of flat connections to a new set of relations between 1) the "equivariant Verlinde algebra" for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) complex Chern-Simons theory on Σ × S 1 and 4) the equivariant index of a spin c Dirac operator on the moduli space of Higgs bundles.

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Section: R = 0 and Gauged Wzw-matter Modelmentioning

confidence: 75%

Equivariant Verlinde Formula from Fivebranes and Vortices

Gukov

Pei

2017

Commun. Math. Phys.

196

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“…[11]; its elements classify rank-one reflexive sheaves on M , and Verlinde's formula then expresses their holomorphic Euler characteristic. † A simple proof and generalisation of the formula, inspired by these ideas, is given in [35]. ‡ Its topological meaning will be discussed in [19, I], in connection with the Frobenius algebra structure.…”

Section: Twisted K Meaning Of the Right-hand Sidementioning

confidence: 99%

Twisted equivariantK-theory with complex coefficients

Freed

Hopkins

Teleman

2007

Journal of Topology

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137

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Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q = 1. Verlinde's formula is also discussed in this context. IntroductionLet X be a locally compact topological space acted upon by a compact Lie group G. The equivariant K-theory K * G (X) was defined by Atiyah and Segal; some foundational papers are [3,30]. Twisted versions of K-theory, both equivariant and not equivariant, have recently attracted some attention. The equivariant twistings that we consider are classified up to isomorphism by an equivariant cohomology class(A twisting is a representative cocycle for such a class in some model of equivariant cohomology.) For torsion, non-equivariant twistings, the relevant K-theory was first introduced in [14]; subsequent treatments ([28] and, more recently, [1, 5, 8]) remove the torsion assumption. We recall for convenience the topologists' definition in the case ε = 0. Because the projective unitary group PU has classifying space K(Z; 3), a class [τ ] ∈ H 3 (X; Z) defines a principal PU-bundle over X up to isomorphism. Let F X be the associated bundle of Fredholm endomorphisms. The negative τ K(X) groups are the homotopy groups of the space of sections of F X ; the others are determined by Bott periodicity. In the presence of a group action, the equivariant τ K-groups arise from the space of invariant sections (a technical variation is needed in the equivariant case, pertaining to the topology on Fredholm operators [5, 31]). In this paper, we shall implicitly assume the basic topological properties of twisted K-theory; their justification is found in the papers mentioned earlier. However, since H 1 twistings get less coverage in the literature, we discuss them in more detail in § 4.One of the basic results [30] of the equivariant theory expresses, in terms of fixed-point data, the localisation of K * G (X) at prime ideals in the representation ring R G of G. The situation simplifies considerably after complexification, when the maximal ideals in R G correspond to (complex semi-simple) conjugacy classes. Recall that, in the non-equivariant case, the Chern character maps complex K-theory isomorphically onto complex cohomology. The localisation results can be assembled into a description of complex equivariant K-theory by a globalised Chern character [4,29,32] supported over the entire group. Part I of our paper generalises these results to the twisted case: the main result, Theorem 3.9, describes τ K * G (X), via the twisted Chern character ( § 2), in terms of (twisted) equivariant cohomology of fixed-point sets, with coefficients in certain equivariantly flat complex line bundles. For orbifolds, this is Vafa's discrete torsion [36,37]. We give a more detailed outline of Part I in § 1.In Part II, we apply our main result to the ...

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“…[18,19,21,25]. Of course the physical derivation skips various subtleties, which are by now thoroughly addressed in [26,27].…”

Section: Topological Metric and The Handle Gluing Operatormentioning

confidence: 99%

Bethe/gauge correspondence on curved spaces

Nekrasov

Shatashvili

2015

J. High Energ. Phys.

200

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Bethe/gauge correspondence identifies supersymmetric vacua of massive gauge theories invariant under the two dimensional N = 2 Poincare supersymmetry with the stationary states of some quantum integrable system. The supersymmetric theory can be twisted in a number of ways, producing a topological field theory. For these theories we compute the handle gluing operator H. We also discuss the Gaudin conjecture on the norm of Bethe states and its connection to H.

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The index formula for the moduli ofG-bundles on a curve

Cited by 32 publications

References 30 publications

Equivariant Verlinde Formula from Fivebranes and Vortices

Equivariant Verlinde Formula from Fivebranes and Vortices

Twisted equivariantK-theory with complex coefficients

Bethe/gauge correspondence on curved spaces

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