K-theory of the moduli space of bundles on a surface and deformations of the Verlinde algebra

Abstract

Abstract We conjecture that index formulas for K -theory classes on the moduli of holomorphic G -bundles over a compact Riemann surface ∑ are controlled, in a precise way, by Frobenius algebra deformations of the Verlinde algebra of G . The Frobenius algebras in question are twisted K -theories of G , equivariant under the conjugation action, and the controlling device is the equivariant Gysin map along the ‘product of commutators’ from G 2 g to G . The conjecture is compatible with naive virtual localization of holomorphic bundles, from G to its maximal torus; this follows by localization in twisted K -theory. Introduction Let G be a compact Lie group and let M be the moduli space of flat G -bundles on a closed Riemann surface ∑ of genus g . By well-known results of Narasimhan, Seshadri and Ramanathan [NS], [R], this is also the moduli space of stable holomorphic principal bundles over ∑ for the complexified group G ; as a complex variety, it carries a fundamental class in complex K -homology. This paper is concerned with index formulas for vector bundles over M . The analogous problem in cohomology – integration formulas over M for top degree polynomials in the tautological generators – has been extensively studied [N], [K], [D], [Th], [W], and, for the smooth versions of M , the moduli of vector bundles of fixed degree co-prime to the rank, it was completely solved in [JK]. In that situation, the tautological classes generate the rational cohomology ring H *( M ; ℚ).