Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs

Abstract

Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form ${\mathbb{Z}}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the ${\mathbb{Z}}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\mathbf{T}}^2=(-1{)}^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all ${\mathbb{Z}}_{16}$ ${\mathrm{Pin}}^{+}$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin${}^{+}$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.

arXiv comment44+68 pages, 11+51 figures. v2: updated references, added Appendix F

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• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Enforced symmetry breaking by invertible topological order
• Classification of Fermionic Topological Orders from Congruence Representations
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory