Topological Phase Transitions and Holonomies in the Dimer Model

We demonstrate that the classical dimer model defined on a toroidal hexagonal lattice acquires holonomy phases in the thermodynamic limit. When all activities are equal the lattice sizes must be considered mod 6 in which case the finite size corrections to the bulk partition function correspond to a massless Dirac Fermion in the presence of a flat connection with nontrivial holonomy. For general bond activities we find that the phase transition in this model is a topological one, where the torus degenerates and its modular parameter becomes real at the critical temperature. We argue that these features are generic to bipartite dimer models and we present a more general lattice whose continuum partition function is that of a massive Dirac Fermion.Comment: 7 pages, 4 figures. Minor corrections with additional figure

On the Renormalizability of Horava-Lifshitz-type Gravities

In this note, we discuss the renormalizability of Horava-Lifshitz-type gravity theories. Using the fact that Horava-Lifshitz gravity is very closely related to the stochastic quantization of topologically massive gravity, we show that the renormalizability of HL gravity only depends on the renormalizability of topologically massive gravity. This is a consequence of the BRST and time-reversal symmetries pertinent to theories satisfying the detailed balance condition.Comment: 13 pages, references added, typos fixe

The Omega Deformation From String and M-Theory

We present a string theory construction of Omega-deformed four-dimensional gauge theories with generic values of \epsilon_1 and \epsilon_2. Our solution gives an explicit description of the geometry in the core of Nekrasov and Witten's realization of the instanton partition function, far from the asymptotic region of their background. This construction lifts naturally to M-theory and corresponds to an M5-brane wrapped on a Riemann surface with a selfdual flux. Via a 9-11 flip, we finally reinterpret the Omega deformation in terms of non-commutative geometry. Our solution generates all modified couplings of the \Omega-deformed gauge theory, and also yields a geometric origin for the quantum spectral curve of the associated quantum integrable system.Comment: LaTeX, 35 pages, 1 figure. Appendix on couplings of hypermultiplets in N=4 SYM adde

Dimer models, free fermions and super quantum mechanics

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