Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond

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 universal SPT invariants, 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. We find new examples of mixed gauge-gravity actions for U(1) SPTs in $(4+1)\mathrm{D}$ via the gravitational Chern-Simons 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.

Figure 1

On a spacetime manifold, the 1-form probe-field $A$ can be implemented on a codimension-1 symmetry twist [60, 61] (with flat $dA=0$) 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 $\text{---}▹$ would obtain an Aharonov-Bohm phase ${\mathrm{e}}^{ig}$ with $g\in G$ by crossing the branch cut [Fig. (a) for two dimensions, Fig. (d) for three dimensions]. However, if the symmetry twist ends, its ends are monodromy defects with $dA\ne 0$, effectively with a gauge flux insertion. Monodromy defects in (b) of two dimensions act like 0D point particles carrying flux [26, 44, 60, 63, 64], in (e) of three dimensions act like 1D line strings carrying flux [65, 66, 67, 68]. The nonflat monodromy defects with $dA\ne 0$ are essential to realize $\int A_{u}d{A}_v$ and $\int A_u{A}_{v}d{A}_w$ for two and three dimensions, while the flat connections ($dA=0$) 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 (c),(f) in $(2+1)\mathrm{D}$, $(3+1)\mathrm{D}$] renders a nontrivial element for Eq. (2).