K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra

TL;DR

This paper conjectures a deep connection between index formulas for K-theory classes on moduli spaces of bundles over Riemann surfaces and Frobenius algebra deformations of the Verlinde algebra, linking geometric and algebraic structures.

Contribution

It introduces a conjecture relating K-theory index formulas to Frobenius algebra deformations of the Verlinde algebra via twisted K-theories and equivariant Gysin maps.

Findings

Conjecture aligns with virtual localization techniques.

Frobenius algebra deformations control K-theory indices.

Localization in twisted K-theory supports the conjecture.

Abstract

I conjecture that index formulas for $K$-theory classes on the moduli of holomorphic $G$-bundles over a compact Riemann surface $\Sigma$ 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 $G^{2g}$ 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.