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

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

2003-06-24

arxiv.org/abs/math/0306347v1

Mathematics

Algebraic Geometry

To appear in the Segalfest proceedings

Scientific paper

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.

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