Gauge theory and mirror symmetry

TL;DR

This paper explores the mathematical framework of gauging 2D topological quantum field theories with compact Lie group actions, linking physical concepts to mirror symmetry and providing new insights into the geometry of flag varieties.

Contribution

It formalizes the process of gauging in extended TQFTs, connecting physical gauge theories with mirror symmetry and Langlands duality, and applies this to flag varieties.

Findings

Describes data and methods for gauging TQFTs with Lie group actions.

Provides a mirror symmetry interpretation involving holomorphic symplectic manifolds.

Recovers complex mirrors of flag varieties as an application.

Abstract

Outlined in this paper is a description of \emph{equivariance} in the world of 2-dimensional extended topological quantum field theories, under a topological action of compactLie groups. In physics language, I am gauging the theories --- coupling them to a principal bundle on the surface world-sheet. I describe the data needed to gauge the theory, as well as the computation of the gauged theory, the result of integrating over all bundles. The relevant theories are A-models, such as arise from the Gromov-Witten theory of a symplectic manifold with Hamiltonian group action, and the mathematical description starts with a group action on the generating category (the Fukaya category, in this example) which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. An application recovers the complex mirrors of flag varieties proposed by Rietsch.