Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond

TL;DR

This paper develops a continuum field theory framework using gauge fields to represent symmetry twists, enabling the classification and analysis of symmetry-protected topological states, including those beyond group cohomology, across various dimensions.

Contribution

It introduces a gauge field approach to express SPT invariants in continuum theories, capturing both group cohomology and beyond-group-cohomology SPTs, and provides new examples in 4+1D.

Findings

SPT invariants described by pure gauge actions match group cohomology predictions.

Mixed gauge-gravity actions reveal beyond-group-cohomology SPTs, such as in 4+1D.

Application to topological insulators with U(1) and time-reversal symmetries.

Abstract

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are the universal SPT invariants defining topological probe responses, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs, recently observed by Kapustin. We find new examples of mixed gauge-gravity actions for U(1) SPTs in 4+1D via mixing the gauge first Chern class with a gravitational Chern-Simons term, or viewed as a 5+1D Wess-Zumino-Witten term with a Pontryagin class. We rule out U(1) SPTs in 3+1D mixed with a Stiefel-Whitney class. We also apply our approach to the bosonic/fermionic topological insulators protected by U(1) charge and $\mathbb{Z}_2^T$ time-reversal symmetries whose pure gauge action is the axion $\theta$-term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context.