  We explicitly find representations for different large $N$ phases of
Chern-Simons matter theory on $S^2\times S^1$. These representations are
characterised by Young diagrams. We show that no-gap and lower-gap phase of
Chern-Simons-matter theory correspond to integrable representations of
$SU(N)_k$ affine Lie algebra, where as upper-cap phase corresponds to
integrable representations of $SU(k-N)_k$ affine Lie algebra. We use phase
space description of arXiv:0711.0133 to obtain these representations and argue
how putting a cap on eigenvalue distribution forces corresponding
representations to be integrable. We also prove that the Young diagrams
corresponding to lower-gap and upper-cap representations are related to each
other by transposition under level-rank duality. Finally we draw phase space
droplets for these phases and show how information about eigenvalue and Young
diagram descriptions can be captured in topologies of these droplets in a
unified way.
