New phase in Chern-Simons theory on lens space

We consider $U(N)_k$ Chern-Simons theory on $S^3$ in Seifert framing and write down the partition function as a unitary matrix model. In the large $k$ and large $N$ limit the eigenvalue density satisfies an upper bound $\frac{1}{2\pi\lambda}$ where $\lambda=N/(k+N)$. We study the partition function under saddle point approximation and find that the saddle point equation admits a gapped solution for the eigenvalue density. The on-shell partition function on this solution matches with the partition function in the canonical framing up to a phase. However the eigenvalue density saturates the upper cap at a critical value of $\lambda$ and ceases to exist beyond that. We find a new phase (called cap-gap phase) in this theory for $\lambda$ beyond the critical value and see that the on-shell free energy for the cap-gap phase is less than that of the gapped phase. We also check the level-rank duality in the theory and observe that the level-rank dual of the gapped phase is a \emph{capped} phase whereas the cap-gap phase is level-rank dual to itself.