JournalsjemsVol. 14, No. 4pp. 1209–1244

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

  • David Cimasoni

    Université de Genève, Switzerland
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices cover
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Abstract

Let Σ\Sigma be a flat surface of genus gg with cone type singularities. Given a bipartite graph Γ\Gamma isoradially embedded in Σ\Sigma, we define discrete analogs of the 22g2^{2g} Dirac operators on Σ\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 ΓΣ\Gamma\subset\Sigma for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ\Gamma. As a consequence, if these conditions are met, the partition function of the dimer model on Γ\Gamma can be explicitly written as an alternating sum of the determinants of these 22g2^{2g} discrete Dirac operators.

Cite this article

David Cimasoni, Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 1209–1244

DOI 10.4171/JEMS/331