# Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the Kasteleyn orientations on dimers. The partition function for dimers can be computed as the alternating sums of Pfaffians of the Kasteleyn matrices. I wonder if one can do similar things for a pin(pin+ or pin-) manifold.

In “Dimers on graphs in non-orientable surfaces”, §§4–5, David Cimasoni describes a generalization of Kastelyn orientations to pin surfaces, and proves in Theorem 5.3 that given a cell decomposition and a dimer configuration on a closed surface $\Sigma$, generalized Kastelyn orientations are equivalent to pin-structures on $\Sigma$.