Speaker: Ryohei Kobayashi (ISSP, University of Tokyo)

Date & Time: 3pm, Jul. 8, 2021.

Title: 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, 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 Z2 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 Z16 anomaly indicator for time-reversal symmetric topological superconductors with 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 Z16 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.