Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Understand

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.