Geodesic
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices
Journal of the European Mathematical Society, Tome 14 (2012) no. 4, pp. 1209-1244

Voir la notice de l'article provenant de la source EMS Press

Let Σ be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in Σ, we define discrete analogs of the 22g Dirac operators on Σ. 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 Γ⊂Σ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ. As a consequence, if these conditions are met, the partition function of the dimer model on Γ can be explicitly written as an alternating sum of the determinants of these 22g discrete Dirac operators.
DOI : 10.4171/jems/331
Classification : 82-XX, 52-XX, 57-XX, 00-XX
Keywords: Perfect matching, dimer model, discrete complex analysis, isoradial graph, Dirac operator, Kasteleyn matrices 
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David Cimasoni. Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices. Journal of the European Mathematical Society, Tome 14 (2012) no. 4, pp. 1209-1244. doi: 10.4171/jems/331

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