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It has been recently demonstrated that the thermal partition function of anylarge $N$ Chern-Simons gauge theories on $S^2$, coupled to fundamental matter,reduces to a capped unitary matrix model. The matrix models corresponding toseveral specific matter Chern-Simons theories at temperature $T$ weredetermined in arXiv:1301.6169. The large $N$ saddle point equations for thesetheories were determined in the same paper, and were solved in the lowtemperature phase. In this paper we find exact solutions for these saddle pointequations in three other phases of these theories and thereby explicitlydetermine the free energy of the corresponding theories at all values of$T^2/N$. As anticipated on general grounds in arXiv:1301.6169, our results arein perfect agreement with conjectured level rank type bosonization dualitiesbetween pairs of such theories.

Duality and Higher Temperature Phases of Large $N$ Chern-Simons Matter Theories on $S^2 \times S^1$