| Summary: | 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( H $$ \mathcal{H} $$ u ), which is the supercategory of super representations of a weak Hopf superalgebra H $$ \mathcal{H} $$ u . The weak Hopf superalgebra H $$ \mathcal{H} $$ u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H $$ \mathcal{H} $$ u by explicitly constructing fermionic TQFTs with SRep( H $$ \mathcal{H} $$ 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.
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