Topological field theories and symmetry protected topological phases with fusion category symmetries

TL;DR

This paper classifies 1+1d bosonic SPT phases with fusion category symmetries using topological field theories, incorporating time-reversal symmetry and self-duality considerations.

Contribution

It introduces a classification scheme for SPT phases with fusion category symmetries via equivalence classes of quintuples, extending topological field theory methods to unoriented cases.

Findings

SPT phases classified by quintuples (Z, M, i, s, φ)

Formulation of unoriented 2D topological field theories

Application to Kramers-Wannier-like self-duality cases

Abstract

Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples $(Z, M, i, s, \phi)$ where $(Z, M, i)$ is a fiber functor, $s$ is a sign, and $\phi$ is the action of orientation-reversing symmetry that is compatible with the fiber functor $(Z, M, i)$. We apply this classification to SPT phases with Kramers-Wannier-like self-duality.