# Characterize spin cobordism invariants in dimer models

+ 8 like - 0 dislike
160 views

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
asked May 4, 2014
retagged Oct 31, 2014

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.