Phases of large N vector Chern-Simons theories on S ² × S ¹

Abstract

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 $\lambda ={N \left/ {k} \right.}$ 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 S¹ 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 $\frac{{2\pi }}{k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{{2\pi \lambda }}$. 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.