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Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases

Ling-Yan Hung, Xiao-Gang Wen

TL;DR

The authors develop a general framework to classify quantized topological terms in symmetric weak-coupling gauge theories by mapping them to cohomology classes ${\rm H}^d(G,\mathbb{R}/\mathbb{Z})$ with $G$ an extension of the symmetry group $G_s$ by the gauge group $G_g$. They show that when $d=3$ or $G_g$ is finite, these terms describe gapped SET phases, with the full classification captured by the cohomology ${\rm H}^d(G,\mathbb{R}/\mathbb{Z})$ where $G/G_g=G_s$, and the extension $G$ corresponds to the PSG. The paper provides multiple formalisms—simple formal reasoning, exact lattice constructions, and a classifying-space approach—culminating in a concrete $G_s=G_g=\mathbb{Z}_2$ example that yields 12 distinct SET phases and a K-matrix description that matches group cohomology predictions. It also clarifies how Chern-Simons terms arise in odd dimensions and how the decomposition of cohomology under Künneth and universal coefficient theorems encodes mixed gauge-symmetry topological terms, offering a unified route to study both gapped and gapless symmetric gauge theories and their fractionalized excitations. The work connects SET classifications to PSG language, provides explicit computational tools (lattice cocycles, classifying-space cocycles, and K-matrix data), and extends prior group-cohomology results to a broader set of gauge-symmetry configurations.

Abstract

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,ν_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $ν_d$ an element in group cohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by $\cH^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).

Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases

TL;DR

The authors develop a general framework to classify quantized topological terms in symmetric weak-coupling gauge theories by mapping them to cohomology classes with an extension of the symmetry group by the gauge group . They show that when or is finite, these terms describe gapped SET phases, with the full classification captured by the cohomology where , and the extension corresponds to the PSG. The paper provides multiple formalisms—simple formal reasoning, exact lattice constructions, and a classifying-space approach—culminating in a concrete example that yields 12 distinct SET phases and a K-matrix description that matches group cohomology predictions. It also clarifies how Chern-Simons terms arise in odd dimensions and how the decomposition of cohomology under Künneth and universal coefficient theorems encodes mixed gauge-symmetry topological terms, offering a unified route to study both gapped and gapless symmetric gauge theories and their fractionalized excitations. The work connects SET classifications to PSG language, provides explicit computational tools (lattice cocycles, classifying-space cocycles, and K-matrix data), and extends prior group-cohomology results to a broader set of gauge-symmetry configurations.

Abstract

We study the quantized topological terms in a weak-coupling gauge theory with gauge group and a global symmetry in space-time dimensions. We show that the quantized topological terms are classified by a pair , where is an extension of by and an element in group cohomology . When and/or when is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by , where . We also apply our theory to a simple case , which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry , which may lead to different fractionalizations of quantum numbers and different fractional statistics (if in 2+1D).

Paper Structure

This paper contains 16 sections, 81 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. A simple formal approach
  3. An exact approach for finite $G_g$
  4. Discretize space-time
  5. Gauged non-linear $\sigma$-model on space-time lattice
  6. An approach based on classifying space
  7. Motivations and results
  8. Gauge theory as a non-linear $\sigma$-model with classifying space as the target space
  9. Symmetric weak-coupling gauge theory as the non-linear $\sigma$-model of $G_s\times BG_g$
  10. Chern-Simons form
  11. An example: $G_s=Z_2$ and $G_g=Z_2$
  12. A $K$-matrix approach
  13. Comparison with group cohomology construction
  14. Summary
  15. Calculate $H^*(X,\mathbb{R}/\mathbb{Z})$ from $H^*(X,\mathbb{Z})$
  16. ...and 1 more sections

Figures (2)

  • Figure 1: (Color online) (a) The possible gapped phases for a class of Hamiltonians $H(g_1,g_2)$ without any symmetry. (b) The possible gapped phases for the class of Hamiltonians $H_\text{symm}(g_1,g_2)$ with a symmetry. The yellow regions in (a) and (b) represent the phases with long range entanglement. Each phase is labeled by its entanglement properties and symmetry breaking properties. SRE stands for short range entanglement, LRE for long range entanglement, SB for symmetry breaking, SY for no symmetry breaking. SB-SRE phases are the Landau symmetry breaking phases. The SY-SRE phases are the SPT phases. The SY-LRE phases are the SET phases.
  • Figure 2: (Color online) Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.