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

TL;DR

This paper investigates the large N limit of matrix models from N=2 Chern-Simons-matter theories, demonstrating their connection to Sasaki-Einstein manifolds and confirming the AdS/CFT correspondence through explicit volume calculations.

Contribution

It establishes a conjectural relation between the large N limit of partition functions and Sasakian volumes, verified for specific quiver theories with M2 branes at singularities.

Findings

Free energy scales as N^{3/2}/Vol(Y)^{1/2} in the large N limit.

Conjectured relation between partition function and Sasakian volumes.

Verified conjecture for specific quiver theories at hypersurface and toric singularities.

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.