From phase space to integrable representations and level-rank duality

Abstract

We explicitly find representations for different large N phases of Chern-Simons matter theory on S2 × S1. 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 [1] 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.