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Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices. (English) Zbl 1243.82017

Summary: 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.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
57M15 Relations of low-dimensional topology with graph theory
52C99 Discrete geometry
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