Dimers on graphs in non-orientable surfaces (Q1013634)

A dimer configuration on a graph \(\Gamma\) is a choice of a family of edges of \(\Gamma\), called dimers, such that each vertex of \(\Gamma\) is adjacent to exactly one dimer. Assigning weights to the edges of \(\Gamma\) allows to define a probability measure on the set of dimer configurations. The main result of this article is a Pfaffian formula for the partition function of the dimer model on any graph \(\Gamma\) embedded in a closed, possibly non-orientable surface \(\Sigma\). The relation between Kasteleyn orientations and spin structures derived in a previous article [\textit{D. Cimasoni} and \textit{N. Reshetikhin}, Commun. Math. Phys. 275, No. 1, 187--208 (2007; Zbl 1135.82006)] does not hold for graphs embedded in non-orientable surfaces, as neither spin-structures nor Kasteleyn orientations make sense in this setting. The purpose here is to extend the geometric approach of the dimer model to any graph embedded in (possibly) non-orientable surfaces. The main idea is to replace spin structures by \(\text{pin}^{ - }\) structures on \(\Sigma\), and to find a natural correspondence between these \(\text{pin}^{ - }\) structures and some orientations on \(\Gamma\), that are also called here Kasteleyn orientations. Then \(\text{pin}^{ - }\) structures on \(\Gamma\) are identified with quadratic enhancements of the intersection form on \(H_1(\Gamma ; \mathbb{Z}_2)\) to obtain the Pfaffian formulae. The first Pfaffian formula is mostly interesting from a theoretical point of view, whereas the second one is more usable for computational purposes.