Dimer Models, Free Fermions and Super Quantum Mechanics

TL;DR

This work establishes a precise bridge between the dimer model on toric graphs and massless free Majorana fermions by identifying the discretized Dirac operator with the Kasteleyn matrix, and showing that a loop expansion of the fermion determinant yields loop states that decompose into two perfect matchings. It then leverages categorification to reinterpret the Newton polynomial as the Euler characteristic of a bi-graded cochain complex, with zero-winding loop configurations corresponding to supersymmetric ground states, effectively realizing the system as supersymmetric quantum mechanics. The authors further exploit the dimer–quiver gauge theory correspondence to reinterpret loop states as maps between perfect matchings, linking $(0,0)$ loops to Seiberg dualities and helix foundations, while winding loops relate to birational transformations between partial resolutions of toric CY singularities. The resulting dictionary connects loop states, SQM, and brane physics, offering a unified view across statistical mechanics, quantum field theory, and string-theoretic geometry with potential applications to crystal melting and black-hole configurations in toric setups.

Abstract

This note relates topics in statistical mechanics, graph theory and combinatorics, lattice quantum field theory, super quantum mechanics and string theory. We give a precise relation between the dimer model on a graph embedded on a torus and the massless free Majorana fermion living on the same lattice. A loop expansion of the fermion determinant is performed, where the loops turn out to be compositions of two perfect matchings. These loop states are sorted into co-chain groups using categorification techniques similar to the ones used for categorifying knot polynomials. The Euler characteristic of the resulting co-chain complex recovers the Newton polynomial of the dimer model. We re-interpret this system as supersymmetric quantum mechanics, where configurations with vanishing net winding number form the ground states. Finally, we make use of the quiver gauge theory - dimer model correspondence to obtain an interpretation of the loops in terms of the physics of D-branes probing a toric Calabi-Yau singularity.

Figures (15)

• Figure 1: Example of a square graph on the torus
• Figure 2: Dirac operator
• Figure 3: The diagrammatic representation for the permutation $\pi = \pi_3 \circ \pi_2 \circ \pi_1$ with $\pi_1 : (17, 18) \mapsto (18,17)$, $\pi_2 : (8,9,10,13) \mapsto (9,10,13,8)$ and $\pi_3 : (1,3,11,6) \mapsto (3,11,6,1)$.
• Figure 4: The diagrammatic representation for a permutation allowed by the action for a free massless fermion. Orientations are not represented.
• Figure 5: Decomposition of a subgraph $\mathinner{|{\Psi}\rangle}$ containing two loops as the combination $\mathinner{|{\Psi}\rangle} = |{m_1}\rangle\space\rangle \otimes |{m_2}\rangle\space\rangle$. Orientations are not represented.