From tensor categories of conformal field theories to membrane models and topological phases in three dimensions

Topological phases in two dimensions, such as quantum Hall states and chiral spin liquids, and their low-energy edge degrees of freedom are related by a bulk-boundary correspondence. It relates the topological structure of gapped anyonic quasiparticle excitations in the (2+1)D bulk to the emergent conformal field theory (CFT) that describes the gapless modes along the (1+1)D edge. There has been recent theoretical discussions on relationships, such as anyon condensation and gauging, between (2+1)D topological phases and correspondingly between their (1+1)D CFTs. A collection of CFTs, equipped with a closed tensor product structure relative to these relationships, forms a tensor category. We establish the general framework of such categories of CFTs, and discuss its prospects in describing (3+1)D topological phases.