Fermionization of fusion category symmetries in 1+1 dimensions

Abstract

Abstract We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep($$ \mathcal{H} $$ H u), which is the supercategory of super representations of a weak Hopf superalgebra $$ \mathcal{H} $$ H u. The weak Hopf superalgebra $$ \mathcal{H} $$ H u depends not only on H but also on a choice of a non-anomalous ℤ2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for $$ \mathcal{H} $$ H u by explicitly constructing fermionic TQFTs with SRep($$ \mathcal{H} $$ H u) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.