TQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D

Abstract

We derive a canonical form for 2-group gauge theory in 3+1D which shows they are either equivalent to Dijkgraaf-Witten theory or to the so-called "EF1" topological order of Lan-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 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 Córdova-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.

13+4 pages, references added

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• The Branes Behind Generalized Symmetry Operators
• Higher-form symmetries and 3-group in axion electrodynamics
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Global 3-group symmetry and 't Hooft anomalies in axion electrodynamics
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Instantons, Symmetries and Anomalies in Five Dimensions
• The anomaly that was not meant IIB
• Global 4-group symmetry and 't Hooft anomalies in topological axion electrodynamics
• Generalized 't Hooft anomalies on non-spin manifolds
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• Modified Villain formulation of abelian Chern-Simons theory
• Non-Liquid Cellular States
• Higher-group structure in $2n$-dimensional axion-electrodynamics
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Study of gapped phases of 4d gauge theories using temporal gauging of the ${\mathbb{Z}}_N$ 1-form symmetry
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints