Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Abstract

Let \Sigma be a flat surface of genus g with cone type singularities. Given a bipartite graph \Gamma isoradially embedded in \Sigma , we define discrete analogs of the 2^{2g} Dirac operators on \Sigma . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair \Gamma\subset\Sigma for these discrete Dirac operators to be Kasteleyn matrices of the graph \Gamma . As a consequence, if these conditions are met, the partition function of the dimer model on \Gamma can be explicitly written as an alternating sum of the determinants of these 2^{2g} discrete Dirac operators.