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

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.