  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.
