Gaiotto's Lagrangian subvarieties via derived symplectic geometry

TL;DR

This paper interprets Gaiotto's construction of Lagrangian subvarieties in the moduli space of G-bundles using derived symplectic geometry, broadening the framework to more general symplectic G-manifolds with Hamiltonian actions.

Contribution

It provides a new interpretation of Gaiotto's Lagrangian subvarieties through derived symplectic geometry, extending the construction to arbitrary symplectic G-manifolds with scaling actions.

Findings

Gaiotto's construction is interpreted via derived symplectic geometry.

The framework is generalized to include symplectic G-manifolds with Hamiltonian actions.

The approach facilitates broader applications in geometric representation theory.

Abstract

Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, D. Gaiotto associated to any symplectic representation of G a Lagrangian subvariety of the cotangent bundle of Bun_G. We give a simple interpretation of (a generalization of) Gaiotto's construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.