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

Date: Wed, Jun 9, 2021

Location: Zoom, Online

Conference: CMS Scientific Session on Quantum Mathematics

Subject: Mathematics, Physics, Condensed Matter and Statistical Mechanics, Quantum Physics, Quantum Information, Quantum Computing

Class: Scientific

### Abstract:

I will describe a way to compute anomalies in general (2+1)D fermionic topological phases. First, a mathematical characterization of symmetry fractionalization for (2+1)D fermionic topological phases is presented, and then this data will be used to define a (3+1)D state sum for a topologically invariant path integral that depends on a generalized spin structure and G bundle on a 4-manifold. This path integral is a cobordism invariant and describes a (3+1)D fermion symmetry-protected topological state (SPT). The special case of time-reversal symmetry with 𝑇2=−1𝐹 gives a ℤ16 invariant of the 4D Pin+ smooth bordism group, and gives an example of a state sum that can distinguish exotic smooth structure.

**Please note, the last 3 minutes of the talk are missing from the video**