Fermionization of fusion category symmetries in 1+1 dimensions

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 $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}({\mathcal{H}}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra ${\mathcal{H}}^u$. The weak Hopf superalgebra ${\mathcal{H}}^u$ depends not only on $H$ but also on a choice of a non-anomalous ${\mathbb{Z}}_2$ subgroup of $\mathrm{Rep}(H)$ that is used for the fermionization. We derive a general formula for ${\mathcal{H}}^u$ by explicitly constructing fermionic TQFTs with $\mathrm{SRep}({\mathcal{H}}^u)$ 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.

63 pages; v2: references added; v3: Appendices A and B added, many small changes throughout

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