Fermionization of fusion category symmetries in 1+1 dimensions

TL;DR

The work provides a complete framework for fermionizing fusion category symmetries in 1+1d by turning Rep$(H)$ bosonic symmetries into superfusion symmetries ${\\mathrm{SRep}}(\\mathcal{H}^u)$ in the fermionic theory, controlled by a chosen non-anomalous $\\mathbb{Z}_2$ subgroup. It develops explicit constructions of the Hopf superalgebra $\\mathcal{H}^u$, the associated boundary-category maps, and state-sum formulations that realize fermionic TQFTs with superfusion symmetries, including lattice Hamiltonians for gapped fermionic phases. The paper provides concrete examples—such as fermionization of finite groups, Rep$(H_8)$, and Tambara–Yamagami categories—demonstrating how $F$-symbols and centrality data govern the appearance of $q$-type objects and dualities, and showing how fermionization can extend beyond topological theories to non-topological QFTs. These results furnish a systematic method to study fermionic generalized symmetries, with potential implications for classifying fermionic phases and understanding dualities via fermionization.

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.

Figures (4)

• Figure 1: Bosonization and fermionization in 1+1 dimensions can be summarized by the above diagram. The Jordan-Wigner transformation is the gauging of a $\mathbb{Z}_2$ symmetry coupled to a spin structure Tachikawa2019GK2021. The dual of the gauged $\mathbb{Z}_2$ symmetry is the fermion parity symmetry. The inverse of the Jordan-Wigner transformation is the summation over spin structures, which is called the Gliozzi-Scherk-Olive (GSO) projection GSO1977. In this paper, fermionization refers to the map from bosonic theory $A$ to fermionic theory $F$. For example, the fermionization of a trivial TQFT is a trivial TQFT, while the fermionization of a $\mathbb{Z}_2$ symmetry broken TQFT is the Kitaev chain. The bosonization is the inverse map of the fermionization. We note that the fermionization in this paper is called the Jordan-Wigner transformation in HNT2021, whereas the Jordan-Wigner transformation in this paper is called the fermionization in Thorngren2020JSW2020.
• Figure 2: Markings related by the above local moves give the same spin structure. (1) Change of an edge orientation: we reverse the orientation of an edge and shift the edge index by 1. (2) Change of a marked edge: we choose the edge next to the original marked edge as the new marked edge and shift the edge index accordingly. The marked edge is represented by an edge with a small red triangle attached. (3) Leaf exchange on a triangle: we shift the edge indices of all edges on the boundary of a triangle simultaneously. We note that the leaf exchange is equivalent to changing the marked edge three times.
• Figure 3: The Pachner 2-2 move (left) and the Pachner 3-1 move (right). The edge indices are determined uniquely up to local moves. For the Pachner 2-2 move, we have $s^{\prime} = s, s^{\prime}_A = s_A, s^{\prime}_B = s_B + s + 1, s^{\prime}_C = s_C + 1, s^{\prime}_D = s_D + s + 1$. For the Pachner 3-1 move, we have $s^{\prime}_A = s_A, s^{\prime}_B = s_B + s_1, s^{\prime}_C = s_C + s_1 + s_2$, $s_3 = s_1 + s_2 + 1$.
• Figure 4: A triangulation of a cylinder with topological defect $X$ wrapping around a spatial cycle. The outer triangle is the in-boundary, and the inner triangle is the out-boundary. The black dots represent marked vertices. The edge indices are determined by the admissibility condition uniquely up to local moves as eq. \ref{['cylinder indices']}.