Gauge theories in various dimensions often admit discrete theta
angles, that arise from gauging a global symmetry with an additional
symmetry protected topological (SPT) phase. We discuss how the global
symmetry and ’t Hooft anomaly depends on the discrete theta angles by
coupling the gauge theory to a topological quantum field theory (TQFT).
We observe that gauging an Abelian subgroup symmetry, that participates
in symmetry extension, with an additional SPT phase leads to a new
theory with an emergent Abelian symmetry that also participates in a
symmetry extension. The symmetry extension of the gauge theory is
controlled by the discrete theta angle which comes from the SPT phase.
We find that discrete theta angles can lead to two-group symmetry in
4d4d
QCD with SU(N),SU(N)/\mathbb{Z}_kSU(N),SU(N)/ℤk
or SO(N)SO(N)
gauge groups as well as various 3d3d
and 2d2d
gauge theories.