Matrix model of Chern-Simons matter theories beyond the spherical limit

A class of matrix models that arises as a partition function in U($N$) Chern-Simons matter theories on the three-sphere is investigated. Employing the standard technique of $1/N$ expansion we solve the system beyond the planar limit. In particular, we study a case where the matrix model potential has $1/N$ correction and give a general solution thereof up to the order of $1/N^2$. We confirm that the general solution correctly reproduces the past exact result of the free energy up to the order in the case of pure Chern-Simons theory. We also apply to the matrix model of $N=2$ Chern-Simons theory with arbitrary numbers of fundamental chiral multiplets and anti-fundamental ones, which does not admit Fermi gas analysis in general.