- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:07
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
14
2012
4
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices
David
Cimasoni
Université de Genève, GENÈVE 4, SWITZERLAND
Perfect matching, dimer model, discrete complex analysis, isoradial graph, Dirac operator, Kasteleyn matrices
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.
Statistical mechanics, structure of matter
Convex and discrete geometry
Manifolds and cell complexes
General
1209
1244
10.4171/JEMS/331
http://www.ems-ph.org/doi/10.4171/JEMS/331