  Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold
are discussed. The partition functions of such theories can be written as a
function of modular transformation matrices summed over different integrable
representations of affine Lie algebra $u(N)_k$ associated with boundary
Wess-Zumino-Witten (WZW) model. Using properties of modular transform matrices
we express the partition functions of these theories as a unitary matrix model.
We show that, the eigenvalues of unitary matrices are discrete and proportional
to hook lengths of the corresponding integrable Young diagram. As a result, in
the large $N$ limit, the eigenvalue density develops an upper cap. We consider
CS theory on $S^2\times S^1$ coupled with fundamental matters and express the
partition functions in terms of modular transformation matrices. Solving this
model at large $N$ we find the dominant integrable representations and show how
large $N$ representations are related to each other by transposition of Young
diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory
on $S^3$ and observed that in Seifert framing the dominant representation is no
longer an integrable representation after a critical value of 't Hooft
coupling. We also show that CS on $S^3$ admits multiple (two-gap phase) large
$N$ phases with the same free energy.
