Topological field theories and symmetry protected topological phases with fusion category symmetries

Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, phi) where (Z, M, i) is a fiber functor, s is a sign, and phi is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

Automated summary

This study classifies 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry using topological field theories, showing that SPT phases are classified by equivalence classes of quintuples.