The formulae of Kontsevich and Verlinde from the perspective of the Drinfeld double

TL;DR

This paper explores the connection between 2D gauge theories derived from Drinfeld doubles and their relation to the Kontsevich formula and Verlinde formula, including explicit calculations of partition functions and their q-deformations.

Contribution

It demonstrates how certain 2D gauge theories correspond to well-known models and computes the genus partition function of the Poisson sigma-model, revealing its q-deformation and relation to the Verlinde formula.

Findings

Partition function of the Poisson sigma-model is a q-deformation of Yang-Mills.

At roots of unity, the series exhibits affine Weyl symmetry.

The truncated series matches the Verlinde formula.

Abstract

A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang-Mills theory, the G/G gauged WZNW model or the Poisson $\sigma$-model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the $q$-deformation of the ordinary Yang-Mills partition function in the sense that the series over the weights is replaced by the same series over the $q$-weights. For $q$ equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the Verlinde formula.