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

TL;DR

This paper classifies quantized topological terms in weak-coupling gauge theories with symmetry, linking them to symmetry enriched topological phases and exploring their implications for gapped and gapless phases.

Contribution

It provides a classification framework for topological terms in gauge theories with symmetry, connecting them to group cohomology and symmetry enriched topological phases.

Findings

Classifies topological terms via group cohomology $H^d(G,\u211d/)$.

Identifies 12 distinct SET phases for $G_s=G_g=Z_2$ in 2+1D.

Shows different topological terms correspond to different gapped or gapless phases.

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 $(G,\nu_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ 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).