Dimers on surface graphs and spin structures. I

TL;DR

This work establishes a precise bridge between dimer models on surface graphs and discrete spin geometry by showing that Kasteleyn orientations correspond to spin structures via quadratic forms on $H_1(\Sigma;\mathbb Z_2)$. It proves a Pfaffian partition function formula $Z=\frac{1}{2^{g}}\sum_{\phi} \mathrm{Arf}(q^{K_\phi}_{D_0})\varepsilon^{K_\phi}(D_0)\mathrm{Pf}(A^{K_\phi})$ and provides explicit expressions for local dimer correlations in terms of Pfaffians and Grassmann integrals, linking the combinatorics to discrete Dirac-type operators. The paper also offers an explicit algorithm to construct the $2^{2g}$ non-equivalent Kasteleyn orientations and clarifies how dimer configurations determine spin-structure data via $q^K_D$, with Johnson’s theorem ensuring the spin-structure–quadratic-form correspondence. Overall, it advances the geometric understanding of dimer models on high-genus surfaces and sets the stage for fermionic discretizations and topological interpretations in statistical mechanics.

Abstract

Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

Figures (8)

• Figure 1: A dimer configuration on a graph.
• Figure 2: An example of composition cycles: dimers from $D$ are in solid, and dimers from $D'$ are in traced lines. On this example, there is one $(D,D')$-composition cycle of length 2, one of length 4, and one of length 6.
• Figure 3: A Kasteleyn orientation on the boundary edges of a face.
• Figure 4: Orientation change at a vertex.
• Figure 5: An example of the explicit construction of a Kasteleyn orientation.

Theorems & Definitions (17)

• Theorem 3.1
• proof
• Theorem 3.2
• Corollary 3.3
• proof : Proof of the theorem.
• Theorem 4.1
• Proposition 4.2
• proof
• Corollary 4.3
• proof