State sum construction of two-dimensional topological quantum field theories on spin surfaces

Abstract

We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behavior of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Δ-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.