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.