Gaiotto-Witten superpotential and Whittaker D-modules on monopoles

Abstract

Let $G$ be an almost simple simply connected group over complex numbers. For a positive element $\alpha$ of the coroot lattice of $G$ let $Z^{\alpha }$ denote the space of based maps from the projective line to the flag variety of $G$ of degree $\alpha$. This space is known to be isomorphic to the space of framed euclidean $G$-monopoles with maximal symmetry breaking at infinity of charge $\alpha$. In [Finkelberg-Kuznetsov-Markarian-Mirković] a system of (étale, rational) coordinates on $Z^{\alpha }$ is introduced. In this note we compute various known structures on $Z^{\alpha }$ in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules introduced by D.Gaitsgory.

19 pages, to appear in Advances in Mathematics