Multi-Matrix Models and Tri-Sasaki Einstein Spaces

TL;DR

This paper generalizes localization-derived matrix models from ABJM to a broad class of ${\rm N}=3$ necklace quivers, yielding $p$-matrix integrals whose large-$N$ saddle points produce an $N^{3/2}$ free-energy scaling. The authors derive explicit eigenvalue densities and free energies for various quivers, and connect these gauge-theory results to dual M-theory backgrounds $AdS_4\times Y$ with tri-Sasaki Einstein spaces $Y$, whose volumes determine the gravitational action and thus the free energy. A key contribution is a conjectured general formula for ${\rm Vol}(Y)/{\rm Vol}(S^7)$ in terms of tree graphs over the quiver nodes, which passes checks against known Eschenburg volumes and Yee’s integral in several cases and is argued to be Seiberg-duality invariant. The work thus provides a precise quantitative bridge between large-$N$ matrix models, geometry of compact spaces, and dualities in ${\rm AdS}_4/{\rm CFT}_3$, with potential extensions to 1/N corrections and broader brane constructions.

Abstract

Localization methods reduce the path integrals in {\cal N} >= 2 supersymmetric Chern-Simons gauge theories on S^3 to multi-matrix integrals. A recent evaluation of such a two-matrix integral for the {\cal N}=6 superconformal U(N) x U(N) ABJM theory produced detailed agreement with the AdS/CFT correspondence, explaining, in particular the N^{3/2} scaling of the free energy. We study a class of p-matrix integrals describing {\cal N}=3 superconformal U(N)^p Chern-Simons gauge theories. We present a simple method that allows us to evaluate the eigenvalue densities and the free energies in the large N limit keeping the Chern-Simons levels k_i fixed. The dual M-theory backgrounds are AdS_4 x Y, where Y are seven-dimensional tri-Sasaki Einstein spaces specified by the k_i. The gravitational free energy scales inversely with the square root of the volume of Y. We find a general formula for the p-matrix free energies that agrees with the available results for volumes of the tri-Sasaki Einstein spaces Y, thus providing a thorough test of the corresponding AdS_4/CFT_3 dualities. This formula is consistent with the Seiberg duality conjectured for Chern-Simons gauge theories.

Figures (8)

• Figure 1: Numerical saddle points for the ABJM matrix model. The eigenvalues for $N=25$ are plotted in black and those for $N=100$ are plotted in orange. The plot has been obtained with $\tau_\lambda = \tau_{\tilde{\lambda}} = 1$. As mentioned in the text, the real parts of the eigenvalues grow with $\sqrt N$.
• Figure 2: Comparison between analytical prediction and numerical results for the density of eigenvalues $\rho$ defined in eq. \ref{['rhoDef']}. The dotted black line represents the analytical calculation, and the numerical result is shown in orange dots.
• Figure 3: Necklace quiver diagrams for $U(N)^p$ Chern-Simons gauge theories.
• Figure 4: Four-node quiver diagram obtained as a particular case of the general quivers presented in figure \ref{['NecklaceQuiver']}.
• Figure 5: Comparison between numerics and analytical prediction for the four-node quiver with $k = \{1, 1, -1, -1\}$. The dotted black lines represent the large $N$ analytical prediction, and the orange dots represent numerical results.