  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.
