LargeNlimit of quiver matrix models and Sasaki-Einstein manifolds

Abstract

We study the matrix models that result from the localization of the partition functions of $\mathcal{N}=2$ 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 ${N}^{3/2}/\mathrm{Vol}(Y{)}^{1/2}$, where $\mathrm{Vol}(Y)$ 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 fourfold 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.