From Phase Space to Integrable Representations and Level-Rank Duality

TL;DR

This paper explores the connection between large N phases of Chern-Simons matter theory, integrable representations, and level-rank duality, using phase space and Young diagram descriptions to unify these concepts.

Contribution

It explicitly characterizes large N phases with Young diagrams and links them to integrable representations and level-rank duality using phase space methods.

Findings

Lower-gap and lower-cap phases correspond to integrable representations.

Young diagrams for dual phases are related by transposition.

Phase space droplets encode eigenvalue and Young diagram information.

Abstract

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.