Decomposition in Chern-Simons theories in three dimensions

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.