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

TL;DR

Gauge fields are used to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology and show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravityactions describe the beyond-group-cohomology SPTs.

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 $θ$-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.

Figures (2)

• Figure 1: On a spacetime manifold, the 1-form probe-field $A$ can be implemented on a codimension-1 symmetry-twistWen:2013ueHung:2013cda (with flat $\dd A=0$) modifying the Hamiltonian $H$, but the global symmetry $G$ is preserved as a whole. The symmetry-twist is analogous to a branch cut, going along the arrow - - -$\vartriangleright$ would obtain an Aharonov-Bohm phase $\ep^{ig}$ with $g \in G$ by crossing the branch cut (Fig.(a) for 2D, Fig.(d) for 3D). However if the symmetry twist ends, its ends are monodromy defects with $\dd A \neq 0$, effectively with a gauge flux insertion. Monodromy defects in Fig.(b) of 2D act like 0D point particles carrying flux,Wen:2013ueWang:2014tiaLG1209Santos:2013udaBarkeshli:2013yta in Fig.(e) of 3D act like 1D line strings carrying flux.Wang:2014oyaJMR1462WL1437Jian:2014vfa The non-flat monodromy defects with $\dd A\neq 0$ are essential to realize $\int A_u \dd A_v$ and $\int A_u A_v \dd A_w$ for 2D and 3D, while the flat connections ($\dd A=0$) are enough to realize the top Type $\int A_1 A_2 \dots A_{d+1}$ whose partition function on a spacetime $\mathbb{T}^{d+1}$ torus with $(d+1)$ codimension-1 sheets intersection (shown in Fig.(c),(f) in 2+1D, 3+1D) renders a nontrivial element for Eq.(\ref{['eq:SPTZ']}).
• Figure 2: The net sum of fluxes at monodromy defects (as punctures or holes of the spatial manifold) must be $2\pi n$ units of fluxes, with $n \in \Z$. e.g. $\sum_j \Phi_\text{B}(x_j)=\iint \dd A_v =2\pi n$.