The Verlinde formula for Higgs bundles

TL;DR

This paper establishes a Verlinde formula for the quantization of Higgs bundle moduli spaces for any simple, simply-connected group, extending previous results and connecting to topological quantum field theories.

Contribution

It generalizes the Verlinde formula to Higgs bundles for all simple, simply-connected groups and constructs a related one-parameter family of Frobenius algebras.

Findings

Verlinde formula for Higgs bundle quantization proved

Extension to parabolic Higgs bundles established

Dimensions form a 1+1D TQFT classified by the complex Verlinde algebra

Abstract

We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We construct this one-parameter family of Frobenius algebras as a deformation of the classical Verlinde algebra for $G$.