Reflection Structures and Spin Statistics in Low Dimensions

Abstract

We give a complete classification of topological field theories with reflection structure and spinstatistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group G which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of G. In spacetime dimension two we give a classification in terms strongly G-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

Generalized symmetries of topological field theories

“…A key point is that we in fact need both a structure and a property (corresponding respectively to reflection and postivity, respectively, in the euclidean context [FH21,MS23]), which we can implement here by means of 0-and 1-form symmetries, respectively.…”