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The challenge of identifying symmetry-protected topological states (SPTs) isdue to their lack of symmetry-breaking order parameters and intrinsictopological orders. For this reason, it is impossible to formulate SPTs underGinzburg-Landau theory or probe SPTs via fractionalized bulk excitations andtopology-dependent ground state degeneracy. However, the partition functionsfrom path integrals with various symmetry twists are the universal SPTinvariants defining topological probe responses, fully characterizing SPTs. Inthis work, we use gauge fields to represent those symmetry twists in closedspacetimes of any dimensionality and arbitrary topology. This allows us toexpress the SPT invariants in terms of continuum field theory. We show that SPTinvariants of pure gauge actions describe the SPTs predicted by groupcohomology, while the mixed gauge-gravity actions describe thebeyond-group-cohomology SPTs, recently observed by Kapustin. We find newexamples of mixed gauge-gravity actions for U(1) SPTs in 4+1D via mixing thegauge first Chern class with a gravitational Chern-Simons term, or viewed as a5+1D Wess-Zumino-Witten term with a Pontryagin class. We rule out U(1) SPTs in3+1D mixed with a Stiefel-Whitney class. We also apply our approach to thebosonic/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 astools for classifying SPTs, but also guide us in designing physical probes forthem. In addition, our field theory representations are independently powerfulfor studying group cohomology within the mathematical context.

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