Symmetry defects and orbifolds of two-dimensional Yang-Mills theory

TL;DR

This paper explores the discrete symmetries and orbifold constructions in two-dimensional Yang-Mills theory, providing exact calculations of partition functions with defects and linking them to moduli space volumes, using both lattice and functorial methods.

Contribution

It introduces a comprehensive framework for understanding symmetry defects and orbifolds in 2D Yang-Mills theory, including exact partition function computations and the construction of orbifold theories with extended gauge groups.

Findings

Exact computation of partition functions with defects

Connection between weak-coupling limit and moduli space volume

Construction of orbifold theories via defect networks

Abstract

We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to outer automorphisms of $G$, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted $G$-bundles, and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted $G$-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang-Mills theory but with gauge group given by an extension of $G$ by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang-Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.