Topological quantum field theory, symmetry breaking, and finite gauge theory in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mtext>1D</mml:mtext></mml:math>
Ryan Thorngren
Abstract
We derive a canonical form for 2-group gauge theory in $3+\text{1D}$ 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 $3+\text{1D}$ 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 C\'ordova 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, ${\mathbb{Z}}_{2}$ gauge theory almost always works.