From Toric Geometry to Quiver Gauge Theory: the Equivalence of a-maximization and Z-minimization

TL;DR

This paper reviews the verification of the AdS/CFT correspondence relating central charges and operator dimensions in D3-brane theories at toric singularities, using geometric and combinatorial methods like dimers.

Contribution

It demonstrates the equivalence of a-maximization and Z-minimization for all toric singularities, integrating recent advances in tilings and dimer models.

Findings

Confirmed the relation between central charge and geometric volumes for toric singularities.

Connected dimer models with R-charge distributions in quiver gauge theories.

Extended previous results with new computational techniques.

Abstract

AdS/CFT predicts a precise relation between the central charge a, the scaling dimensions of some operators in the CFT on D3-branes at conical singularities and the volumes of the horizon and of certain cycles in the supergravity dual. We review how a quantitative check of this relation can be performed for all toric singularities. In addition to the results presented in hep-th/0506232, we also discuss the relation with the recently discovered map between toric singularities and tilings; in particular, we discuss how to find the precise distribution of R-charges in the quiver gauge theory using dimers technology.