2022/06/27 by Kansei Inamura, Inamura, Kansei · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Strongly Correlated Electrons (cond-mat.str-el)
paper · pdf · doi:10.48550/arxiv.2206.13159
openalex publication_date 2022/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(Hu), which is the supercategory of super representations of a weak Hopf superalgebra Hu. The weak Hopf superalgebra Hu 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 Hu by explicitly constructing fermionic TQFTs with SRep(Hu) 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.