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.