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Phases of large $N$ vector Chern-Simons theories on $S^2 \times S^1$

We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to\infty$, with $λ= N/k$ and $\frac{T^2 V_{2}}{N}$ held fixed where $T$ is the temperature and $V_{2}$ the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of $λ$. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of ${2π\over k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{2 πλ}$. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of $λ$. At low temperatures our partition function smoothly matches onto the $N$ and $λ$ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large $N$ Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.