Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

作者：Cimasoni David^*

来源：Journal of the European Mathematical Society, 2012, 14(4): 1209-1244.

DOI：10.4171/JEMS/331

• 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 of 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.

• 出版日期2012