Two-dimensional state sum models and spin structures

Abstract

The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai are generalised by allowing algebraic data from a non-symmetric Frobenius algebra. Without any further data, this leads to a state sum model on the sphere. When the data is augmented with a crossing map, the partition function is defined for any oriented surface with a spin structure. An algebraic condition that is necessary for the state sum model to be sensitive to spin structure is determined. Some examples of state sum models that distinguish topologically-inequivalent spin structures are calculated.

43 pages. Mathematica script in ancillary file. v2: nomenclature of models and their properties changed, some proofs simplified, more detailed explanations. v3: extended introduction, presentational improvements; final version

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