R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization

TL;DR

This work establishes a universal toric-data–driven framework to compute R-charges and multiplicities of chiral fields for gauge theories on branes at toric Calabi–Yau singularities. It proves that a-maximization in field theory and Z-minimization (volume minimization) in geometry are completely equivalent, providing a broad, metric-free check of AdS/CFT for toric horizons. The authors give explicit algorithms to extract charges from toric diagrams, test them across multiple families (Y^{p,q}, L^{p,q;r,s}, X^{p,q}, dP3, orbifolds), and discuss the role of perfect matchings in dimers as a mechanism to assign charges. They also extend the framework to non-smooth horizons and provide detailed proofs ensuring anomaly constraints and decoupling of baryonic directions, thereby reinforcing the deep link between toric geometry and superconformal gauge theories.

Abstract

We conjecture a general formula for assigning R-charges and multiplicities for the chiral fields of all gauge theories living on branes at toric singularities. We check that the central charge and the dimensions of all the chiral fields agree with the information on volumes that can be extracted from toric geometry. We also analytically check the equivalence between the volume minimization procedure discovered in hep-th/0503183 and a-maximization, for the most general toric diagram. Our results can be considered as a very general check of the AdS/CFT correspondence, valid for all superconformal theories associated with toric singularities.

Figures (17)

• Figure 1: The convex polygon $P$.
• Figure 2: The $(p,q)$ web for $P$.
• Figure 3: The Reeb vector as a point $B$ inside the polygon $P$.
• Figure 4: The toric diagram and $(p,q)$ web for $L^{p,q;r,s}$
• Figure 5: Toric diagram for $X^{p,q}$. We draw the case $X^{4,2}$.