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

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.

Finite symmetries of quantum character stacks

“…Factorisation homology on surfaces equipped with non-trivial Out(G)-bundles has also a natural field theoretical interpretation: The value of factorisation homology describes the coupling of the quantum character field theory to non-trivial Out(G)-background fields. In [MSS19] the topological limit of the partition function of 2-dimensional Yang-Mills theory coupled to an Out(G)-background field ϕ : Σ −→ B Out(G) was related to the symplectic volume of the moduli space of flat ϕ-twisted G-bundles M ρ (Σ) [Mei17,Zer21]. We will show the analogous statement for quantum character stacks, i.e.…”

“…We first recollect some background about twisted bundles in the differential geometric setting, see for example [Mei17] and [Zer21] for more details and [MSS19] for the non-flat version. Let Σ be an oriented surface equipped with a principal Out(G)-bundle P −→ Σ.…”

“…with D = Out(G). 1 An element κ ∈ Out(G) acts on a gauge field described by a principal G-bundle with connection by forming the associated bundle along the group homomorphism κ : G −→ G. In [MSS19] these symmetries have been studied in the context of 2-dimensional Yang-Mills theory. They restrict to an action of Out(G) on the moduli space M(Σ).…”