Gaiotto-Witten superpotential and Whittaker D-modules on monopoles

TL;DR

This paper explores the geometric and algebraic structures of monopole moduli spaces related to complex simple groups, connecting Gaiotto-Witten superpotentials with Whittaker D-modules through coordinate systems.

Contribution

It provides a new interpretation of the Gaiotto-Witten superpotential and links it to Whittaker D-modules using explicit coordinate descriptions of monopole spaces.

Findings

Explicit coordinate systems on monopole spaces are used to interpret superpotentials.

The Gaiotto-Witten superpotential is related to Whittaker D-modules.

Structural insights into monopole moduli spaces are achieved.

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-Mirkovi\'c] a system of (\'etale, 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.