Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

TL;DR

This paper introduces discrete Dirac operators on bipartite graphs embedded in flat Riemann surfaces with cone singularities, showing their convergence to continuous operators and linking them to Kasteleyn matrices for dimer model computations.

Contribution

It defines discrete Dirac operators on such surfaces, establishes their convergence to continuous operators, and characterizes when they serve as Kasteleyn matrices, enabling explicit partition function calculations.

Findings

Discrete Dirac operators are constructed and shown to converge to continuous counterparts.

Necessary and sufficient conditions are identified for these operators to be Kasteleyn matrices.

Partition functions of the dimer model are expressed as sums of determinants of these operators.

Abstract

Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. 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 (S,G) for these discrete Dirac operators to be Kasteleyn matrices of the graph G. As a consequence, if these conditions are met, the partition function of the dimer model on G can be explicitly written as an alternating sum of the determinants of these 2^{2g} discrete Dirac operators.