Equivalence of A-Maximization and Volume Minimization
Richard Eager
TL;DR
The paper proves the equivalence between a-maximization and horizon-volume minimization for AdS$_5$×L$^5$ backgrounds by recasting the geometric volume in terms of the Hilbert series of Calabi–Yau algebras associated with quiver gauge theories. It develops a framework linking mesonic R-symmetries and baryonic charges via non-commutative crepant resolutions, then derives a perturbative expansion showing the leading pole in the Hilbert series matches the a-anomaly: $\lambda(s) = \frac{32}{27} a \, s^3 + \cdots$, yielding ${\rm Vol}[L^5] = \frac{\pi^3 N^2}{4 a}$. The method is illustrated with explicit examples ($\mathbb{C}^3$ and the conifold) and extended beyond toric cases, suggesting a broad applicability to general Calabi–Yau singularities. Together, these results reinforce the holographic link between geometric volume minimization and field-theoretic a-maximization, with Hilbert-series techniques providing a concrete computational bridge. The work thus enhances our understanding of the AdS/CFT dictionary for a wide class of four-dimensional SCFTs and offers a practical route to compute horizon volumes directly from quiver data.
Abstract
The low energy effective theory on a stack of D3-branes at a Calabi-Yau singularity is an $\mathcal{N} = 1$ quiver gauge theory. The AdS/CFT correspondence predicts that the strong coupling dynamics of the gauge theory is described by weakly coupled type IIB supergravity on $AdS_5 \times L^5,$ where $L^5$ is a Sasaki-Einstein manifold. Recent results on Calabi-Yau algebras efficiently determine the Hilbert series of any superconformal quiver gauge theory. We use the Hilbert series to determine the volume of the horizon manifold in terms of the fields of the quiver gauge theory. One corollary of the AdS/CFT conjecture is that the volume of the horizon manifold $L^5$ is inversely proportional to the a-central charge of the gauge theory. By direct comparison of the volume determined from the Hilbert series and the a-central charge, this prediction is proved independently of the AdS/CFT conjecture.