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 $α$ of the coroot lattice of $G$ let $Z^α$ denote the space of based maps from the projective line to the flag variety of $G$ of degree $α$. This space is known to be isomorphic to the space of framed euclidean $G$-monopoles with maximal symmetry breaking at infinity of charge $α$. In [Finkelberg-Kuznetsov-Markarian-Mirković] a system of (étale, rational) coordinates on $Z^α$ is introduced. In this note we compute various known structures on $Z^α$ 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.