Multimatrix models and tri-Sasaki Einstein spaces

Localization methods reduce the path integrals in N≥2 supersymmetric Chern-Simons gauge theories on S³ to multimatrix integrals. A recent evaluation of such a two-matrix integral for the N=6 superconformal U(N)×U(N) Aharony-Bergman-Jafferis-Maldacena theory produced detailed agreement with the AdS/CFT correspondence, explaining, in particular, the N3/2 scaling of the free energy. We study a class of p-matrix integrals describing 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 kᵢ fixed. The dual M-theory backgrounds are AdS₄×Y, where Y are seven-dimensional tri-Sasaki Einstein spaces specified by the kᵢ. 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 AdS₄/CFT₃ dualities. This formula is consistent with the Seiberg duality conjectured for Chern-Simons gauge theories.