Open Access
October 2009 Dimer models, free fermions and super quantum mechanics
Robbert Dijkgraaf, Domenico Orlando, Susanne Reffert
Adv. Theor. Math. Phys. 13(5): 1255-1315 (October 2009).


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 reinterpret 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.


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Robbert Dijkgraaf. Domenico Orlando. Susanne Reffert. "Dimer models, free fermions and super quantum mechanics." Adv. Theor. Math. Phys. 13 (5) 1255 - 1315, October 2009.


Published: October 2009
First available in Project Euclid: 17 August 2010

zbMATH: 1214.82013
MathSciNet: MR2672463

Rights: Copyright © 2009 International Press of Boston

Vol.13 • No. 5 • October 2009
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