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 G2g to G. The conjecture is compatible with naïve 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).