Brane Geometry and Dimer Models

TL;DR

This paper investigates the geometric origin of dimer models in gauge theories within AdS_5/CFT_4, comparing the complex structures derived from brane constructions and field theory, revealing close but not identical values.

Contribution

It demonstrates that the complex structure of the dimer's torus from brane geometry closely matches the field theory value in simple cases, highlighting a new geometric perspective.

Findings

For C^3 and the conifold, tau_R equals tau_G.

In more complex theories, tau_R and tau_G differ slightly, by only a few percent.

The near match suggests a deep geometric connection, yet the precise reason remains unexplained.

Abstract

The field content and interactions of almost all known gauge theories in AdS_5/CFT_4 can be expressed in terms of dimer models or bipartite graphs drawn on a torus. Associated with the fundamental cell is a complex structure parameter tau_R. Based on the brane realization of these theories, we can specify a special Lagrangian (SLag) torus fibration that is the natural candidate to be identified as the torus on which the dimer lives. Using the metrics known in the literature, we compute the complex structure tau_G of this torus. For the theories on C^3 and the conifold and for orbifolds thereof tau_R = tau_G. However, for more complicated examples, we show that the two complex structures cannot be equal and yet, remarkably, differ only by a few percent. We leave the explanation for this extraordinary proximity as an open challenge.