Quantized topological terms in weakly coupled gauge theories and their connection to symmetry protected topological phases

TL;DR

This paper systematically constructs quantized topological terms in weakly coupled gauge theories across dimensions, revealing their connection to gapped topological phases and symmetry protected topological phases, and establishing dualities with SPT phases.

Contribution

It provides a comprehensive classification of topological terms in gauge theories using cohomology, linking them to gapped phases and SPT phases, and explores dualities and lattice models.

Findings

Topological terms classified by $H^{d+1}(BG, ext{Z})$ in gauge theories.

Duality between topological gauge theories and bosonic SPT phases.

Construction of gapped and gapless phases via topological terms.

Abstract

We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge "symmetry") and the field strength fluctuates only weakly. We ask what kind of topological terms can be added to the Lagrangian of such a weakly coupled gauge theory. In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and $F\wedge F$ terms, in space-time dimensions $d$ and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes $H^{d+1}(BG,\Z)$ on the classifying space $BG$ of the gauge group $G$. In 3$d$ or for finite gauge groups above 3$d$, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group $G$ are also classified by the same $H^{d+1}(BG,\Z)$. (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group $G$ and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.