Advances in Theoretical and Mathematical Physics

Dimer models, free fermions and super quantum mechanics

Robbert Dijkgraaf, Domenico Orlando, and Susanne Reffert

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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 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.

Article information

Source
Adv. Theor. Math. Phys., Volume 13, Number 5 (2009), 1255-1315.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1282054095

Mathematical Reviews number (MathSciNet)
MR2672463

Zentralblatt MATH identifier
1214.82013

Citation

Dijkgraaf, Robbert; Orlando, Domenico; Reffert, Susanne. Dimer models, free fermions and super quantum mechanics. Adv. Theor. Math. Phys. 13 (2009), no. 5, 1255--1315. https://projecteuclid.org/euclid.atmp/1282054095


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