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• W4310854375 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, which includes the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction proceeds by using 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 condensing 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 $${textbf{T}}^2 = (-1)^F$$ . Mathematically, with some standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $$mathbb Z_{16}$$ $$textrm{Pin}^+$$ 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." @default.
• W4310854375 created "2022-12-19" @default.
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• W4310854375 date "2022-12-05" @default.
• W4310854375 modified "2023-10-18" @default.
• W4310854375 title "Anomalies in (2+1)D Fermionic Topological Phases and (3+1)D Path Integral State Sums for Fermionic SPTs" @default.
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• W4310854375 doi "https://doi.org/10.1007/s00220-022-04484-w" @default.
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