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 $S^2\times S^1$. 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.

37 pages, 10 figures, v2 Introduction extended, References added

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• Mapping Large N Slightly Broken Higher Spin (SBHS) theory correlators to Free theory correlators
• The Hilbert Space of large $N$ Chern-Simons matter theories
• On thermal correlators and bosonization duality in Chern-Simons theories with massive fundamental matter
• New phase in Chern-Simons theory on lens space