  This paper discusses the relationships between gauge theories defined by
gauge groups with finite trivially-acting centers, and theories with
restrictions on nonperturbative sectors, in two and four dimensions. In two
dimensions, these notions seem to coincide. Generalizing old results on
orbifolds and abelian gauge theories, we propose a decomposition of
two-dimensional nonabelian gauge theories with center-invariant matter into
disjoint sums of theories with rotating discrete theta angles; for example,
schematically, SU(2) = SO(3)_+ + SO(3)_-. We verify that decomposition directly
in pure nonsupersymmetric two-dimensional Yang-Mills as well as in
supersymmetric theories. In four dimensions, by contrast, these notions do not
coincide. To clarify the relationship, we discuss theories obtained by
restricting nonperturbative sectors. These theories violate cluster
decomposition, but we illustrate how they may at least in special cases be
understood as disjoint sums of well-behaved quantum field theories, and how
dyon spectra can be used to distinguish, for example, an SO(3) theory with a
restriction on instantons from an SU(2) theory. We also briefly discuss how
coupling various analogues of Dijkgraaf-Witten theory, as part of a description
of instanton restriction via coupling TQFT's to QFT's, may modify these
results.
