Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Details Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

2009-09-29

arxiv.org/abs/0909.5339v2

Physics

Mathematical Physics

39 pages, minor changes

Scientific paper

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.

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