Characterize spin cobordism invariants in dimer models

Question

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a graph embedded in the surface.

In particular, there is a 1-1 correspondence between spin structures on a surface and Kasteleyn orientation on a surface graph with dimer configuration. The number of non-equivalent Kasteleyn orientations of a surface graph of genus $g$ is $2^{2g}$ and is equal to the number of non-equivalent spin structures on the surface. Kasteleyn operator can be naturally identified with a discrete version of the Dirac operator. And the partition function of the dimer model is equal to the sum of $2^{2g}$ Pfaffians, reminiscent of the partition function of free fermions on a Riemann surface of genus $g$, which is a linear combination of $2^{2g}$ Pfaffians of Dirac operators.

My question is, given the above combinatorial description of spin structure, is there a way to write a local combinatorial description for the spin cobordism invariants in 2d, e.g. the Arf invariant? (see Wikipedia http://en.wikipedia.org/wiki/Arf_invariant for a definition of the Arf invariant, in particular, the section "The Arf invariant in topology".)

This post imported from StackExchange MathOverflow at 2014-10-04 15:29 (UTC), posted by SE-user Zitao Wang