Anomalies in bosonic symmetry-protected topological edge theories: Connection to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> symbols and a method of calculation
Kyle Kawagoe, Michael Levin
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 $\ensuremath{\omega}$ of the cohomology group ${H}^{3}(G,{U}_{T}(1))$. This element $\ensuremath{\omega}\ensuremath{\in}{H}^{3}(G,{U}_{T}(1))$ 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.