Fusion category symmetries of 1+1d bosonic and fermionic topological field theories

Abstract

Generalized symmetries in two-dimensional spacetime are described by the algebraic structure of topological line defects. In particular, when the number of topological lines is finite, their algebraic structure is described by a fusion category or a superfusion category depending on whether the system is bosonic or fermionic. For this reason, finite generalized symmetries in 1+1 dimensions are called fusion category symmetries. In this talk, we will construct 1+1d bosonic and fermionic topological field theories (TQFTs) with fusion category symmetries using the state sum construction of TQFTs and the pullback of fusion category symmetries. These TQFTs can be realized by the ground states of commuting projector Hamiltonians on the lattice. We will also derive the fermionization formula of fusion category symmetries by comparing the symmetries of bosonic TQFTs with those of fermionic TQFTs. Examples include the fermionization of finite group symmetries and Kramers-Wannier-like self-dualities. This talk is based on arXiv:2110.12882 and 2206.13159.