Gaiotto's Lagrangian subvarieties via loop groups

TL;DR

This paper provides a straightforward proof that a specific substack of the Higgs bundle moduli stack has a Lagrangian structure, linking Higgs bundles, symplectic varieties, and group actions.

Contribution

It offers a simple proof of the Lagrangian property of a substack related to Higgs bundles and symplectic varieties, clarifying geometric structures in moduli spaces.

Findings

The substack has a Lagrangian structure.

The substack is related to the moment map images of sections.

Connections between Higgs bundles and symplectic geometry are established.

Abstract

The purpose of this note is to give a simple proof of the fact that a certain substack, defined in [2], of the moduli stack $T^{\ast}Bun_G(\Sigma)$ of Higgs bundles over a curve $\Sigma$, for a connected, simply connected semisimple group $G$, possesses a Lagrangian structure. The substack, roughly speaking, consists of images under the moment map of global sections of principal $G$-bundles over $\Sigma$ twisted by a smooth symplectic variety with a Hamiltonian $G$-action.