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.
Barrett, J. W., & Tavares, S. O. G. (2014). Two-dimensional state sum models and spin structures. Communications in Mathematical Physics, 336(1), 63-100. doi:10.1007/s00220-014-2246-z