Topological field theories and symmetry protected topological phases with fusion category symmetries
Kansei Inamura
Abstract
A bstract 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 , ϕ ) where ( Z , M , i ) is a fiber functor, s is a sign, and ϕ 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.