Multi-Matrix Models and Tri-Sasaki Einstein Spaces

TL;DR

This paper develops a method to evaluate large N eigenvalue densities and free energies for multi-matrix models in supersymmetric Chern-Simons theories, confirming dualities with tri-Sasaki Einstein spaces in M-theory.

Contribution

It introduces a simple evaluation technique for p-matrix integrals in N=3 Chern-Simons theories, linking free energies to tri-Sasaki Einstein space volumes and testing AdS4/CFT3 dualities.

Findings

Derived a general formula for p-matrix free energies.

Confirmed the inverse square root volume scaling of dual spaces.

Validated the formula against known tri-Sasaki Einstein space volumes.

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.