2007/05/11 by R. Dijkgraaf, Robbert Dijkgraaf, Dijkgraaf, R. +6 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #hep-th #math.QA
paper · pdf · doi:10.48550/arxiv.0705.1645
minor corrections, reference added
openalex publication_date 2007/05/11 · arxiv created 2007/11/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.