Decomposition in Chern-Simons theories in three dimensions

TL;DR

This paper explores how three-dimensional Chern-Simons theories with certain symmetries decompose into simpler theories and how this relates to boundary conditions and known boundary decompositions.

Contribution

It demonstrates that Chern-Simons theories with gauged noneffectively-acting one-form symmetries decompose into disjoint theories with discrete theta angles, linking bulk and boundary decompositions.

Findings

Bulk discrete theta angles correspond to boundary discrete torsion.

Decomposition reduces to known boundary orbifold decompositions.

Provides a consistency check for the decomposition proposal.

Abstract

In this paper we discuss decomposition in the context of three-dimensional Chern-Simons theories. Specifically, we argue that a Chern-Simons theory with a gauged noneffectively-acting one-form symmetry is equivalent to a disjoint union of Chern-Simons theories, with discrete theta angles couplings to the image under a Bockstein homomorphism of a canonical degree-two characteristic class. On three-manifolds with boundary, we show that the bulk discrete theta angles (coupling to bundle characteristic classes) are mapped to choices of discrete torsion in boundary orbifolds. We use this to verify that the bulk three-dimensional Chern-Simons decomposition reduces on the boundary to known decompositions of two-dimensional (WZW) orbifolds, providing a strong consistency test of our proposal.