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

TL;DR

<3-5 sentence high-level summary> Quantized topological terms in weakly coupled gauge theories are systematically classified by the cohomology group $H^{d+1}(BG,\mathbb{Z})$, with the classification matched across three complementary formalisms: group cohomology cocycles, nonlinear sigma-models with target space $BG$, and differential-character/Chern-Simons perspectives. For finite gauge groups, these terms yield gapped topological orders in low dimensions and extend to gapless regimes in higher dimensions; for continuous groups, the discrete (torsion) part of the same cohomology governs the quantized terms via $\text{Tor}[H^{d+1}(BG,\mathbb{Z})]$. A central result is a duality between such topological gauge theories and bosonic SPT phases, generalizing Levin-Gu, and a parallel connection to Levin-Wen string-net models. The work provides explicit lattice formulations, fixed-point constructions, and concrete examples (notably $\mathbb{Z}_N$ in 3D and 2D cases), clarifying how topological data from $BG$ shapes both gapped and gapless phases. This framework advances the systematic classification of quantum phases in gauge theories and bridges topological order, SPT physics, and lattice realizations.

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.

Figures (9)

• Figure 1: (Color online) Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.
• Figure 2: Complices with spherical topology: (a) a tetrahedron -- a (1+1)D sphere, (b) a pentachoron -- a (2+1)D sphere. The arrows on the link represent the branching structure.
• Figure 3: A torus represented by a rectangle with opposite sides identified. Each independent loop of the torus is assigned a group element $g_{l_1}$ and $g_{l_2}$ respectively.
• Figure 4: String net on a honeycomb lattice and the dual triangular lattice on which the lattice gauge theory is defined.
• Figure 5: The string net dual of a tetrahedron which is still a tetrahedron.