Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D

We derive a canonical form for 2-group gauge theory in 3+1D which shows that it is equivalent either to Dijkgraaf-Witten theory or to the so-called EF1 topological order of Lan and Wen. According to that classification, recently argued from a different point of view by Johnson-Freyd, this amounts to a very large class of all 3+1D topological quantum field theories (TQFTs). We use this canonical form to compute all possible anomalies of 2-group gauge theory which may occur without spontaneous symmetry breaking, providing a converse of the recent symmetry-enforced-gaplessness constraints of Cordova and Ohmori and also uncovering some possible new examples. On the other hand, in cases where the anomaly is matched by a TQFT, we try to provide the simplest possible such TQFT. For example, with anomalies involving time reversal, Z(2) gauge theory almost always works.