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

TL;DR

The paper addresses how to compute the central charge and exact R-charges of SCFTs on D3-branes at toric Calabi–Yau cones by linking field-theory a-maximization to geometric Z-minimization. It proves their equivalence, showing that the two-parameter volume minimization in MSY corresponds to maximizing a over a toric-determined subspace via a_i = f_i(x,y) with f_i(x,y)=2 l_i(x,y)/sum l_j(x,y). The authors derive explicit toric formulas for R-charges and multiplicities from the toric diagram and connect these to dimer/zig-zag constructions for recovering full quivers and charge distributions from toric data. These results enable metric-free AdS/CFT checks for all toric manifolds and provide practical algorithms to extract quivers and R-charge distributions from toric data alone, unifying a-maximization, Z-minimization, and brane tiling 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.

Figures (4)

• Figure 1: The convex polygon $P$.
• Figure 2: The $(p,q)$ web for $P$.
• Figure 3: The correspondence for $Y^{2,1}$.
• Figure 4: Equivalence of the two algorithm for charge distribution.