The large N limit of quiver matrix models and Sasaki-Einstein manifolds

TL;DR

The paper investigates how localization of the $S^3$ partition function for a broad class of ${\cal N}=2$ Chern-Simons-matter theories yields matrix models whose large-$N$ (M-theory) limit reproduces the expected $N^{3/2}$ scaling governed by Sasaki-Einstein volumes. It establishes a general conjecture linking the leading large-$N$ free energy, as a function of trial R-charges, to the volume of Sasakian metrics on links of Calabi–Yau four-fold singularities, and provides explicit checks for nontrivial non-chiral quivers: the $U(N)^2$ ${\cal A}_{n-1}$ theories and the $U(N)^3$ SPP theory. By performing saddle-point analyses and exploiting symmetry considerations, the authors derive explicit free-energy expressions, extremize over R-charges, and show agreement with geometric volumes $\mathrm{Vol}(Y)[\xi]$ obtained from Sasakian volume minimization. These results strengthen the AdS$_4$/CFT$_3$ dictionary for a wider class of ${\cal N}=2$ theories and offer a practical route to determine IR R-charges from partition-function data, with potential extensions to toric Calabi–Yau four-folds and beyond.

Abstract

We study the matrix models that result from localization of the partition functions of N=2 Chern-Simons-matter theories on the three-sphere. A large class of such theories are conjectured to be holographically dual to M-theory on Sasaki-Einstein seven-manifolds. We study the M-theory limit (large N and fixed Chern-Simons levels) of these matrix models for various examples, and show that in this limit the free energy reproduces the expected AdS/CFT result of N^{3/2}/Vol(Y)^{1/2}, where Vol(Y) is the volume of the corresponding Sasaki-Einstein metric. More generally we conjecture a relation between the large N limit of the partition function, interpreted as a function of trial R-charges, and the volumes of Sasakian metrics on links of Calabi-Yau four-fold singularities. We verify this conjecture for a family of U(N)^2 Chern-Simons quivers based on M2 branes at hypersurface singularities, and for a U(N)^3 theory based on M2 branes at a toric singularity.

Figures (4)

• Figure 1: The $\mathcal{A}_{n-1}$ Chern-Simons quiver.
• Figure 2: Sketch of $\rho(x)$, which is zero for $|x|>x_*$ and takes the constant value $\rho=\frac{1}{2x_*}$ for $|x|<x_*$. The function $y(x)$, shown in red, is linear in this latter region.
• Figure 3: The SPP Chern-Simons quiver.
• Figure 4: Sketch of the piecewise-linear functions $u(x)$, $\rho(x)$.