Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

TL;DR

This paper introduces a systematic method to identify 2D bosonic SPT phases by analyzing their edge excitations, linking domain wall F-symbols to the bulk anomaly via cohomology classification.

Contribution

It provides a general procedure to determine the SPT phase from edge theories by computing F-symbols, applicable to various symmetries and models.

Findings

Method successfully applied to lattice and continuum models

Connects edge domain wall properties to bulk SPT classification

Enables calculation of anomalies using F-symbols

Abstract

We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\omega$ of the cohomology group $H^3(G, U_T(1))$. This element $\omega \in H^3(G, U_T(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$-symbol associated with domain walls in a symmetry broken edge theory; this domain wall $F$-symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.