Combinatorics

Physics

Fusion

Topology (electrical circuits)

Invariant (physics)

Pure mathematics

Symmetry (geometry)

Mathematics

Topological order

Symmetry protected topological order

Spontaneous symmetry breaking

Philosophy

Political science

Functor

Global symmetry

Symmetry breaking

Geometry

Duality (order theory)

Theoretical physics

Linguistics

Quantum mechanics

Homogeneous space

Unitary state

Quantum

Journal of High Energy Physics

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.

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