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

TL;DR

The paper addresses how to realize and classify anomalies in $3+1$D finite gauge theories without spontaneous symmetry breaking, using Crane–Yetter/2-group gauge theories to derive a canonical reduction to a $1$-form theory. It provides a comprehensive anomaly classification via cobordism and group cohomology, identifies simple gapped realizations for many cases, and establishes dualities that connect 2-group gauge theories to ordinary $1$-form theories, resonating with the EF1/EF2 framework of Lan and Wen. A key result is the fermionic canonical form $S=\omega(a)+\tfrac12(\gamma(a)+b)\cup b$ with $db=\beta(a)+sw_3(TX)$, which captures both anomaly matching and the simplest possible TQFT realizations in many cases. The work advances understanding of symmetry protected gaplessness and the landscape of 3+1D TQFTs, with implications for lattice spin liquids and related topological phases, while outlining open questions regarding EF2 theories and the full anomaly set.

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.

Figures (1)

• Figure 1: A typical symmetry fractionalization pattern where a 3-fold junction of 0-form symmetry defects corresponding to the group elements $g_1,g_2,g_1g_2$ (blue lines) acts on a line operator $W$. In the case that $W$ is a Wilson line, we understand this in terms of the global symmetry being nontrivially extended by the gauge symmetry. In some situations, the line $W$ changes type as it passes through the symmetry defects. This complicates the description of symmetry fractionalization Barkeshli_2019Etingof_2010. However, in 3+1D gauge theory, such "anyon-permuting" symmetries are highly constrained, and amount to an action of the global symmetry on the gauge symmetry by group automorphisms.