Quantum Algebra of Chern-Simons Matrix Model and Large $N$ Limit

TL;DR

This paper investigates the quantum algebra of the Chern-Simons matrix model, establishing its large N limit algebra and connecting it to affine Lie algebras, thereby providing a rigorous foundation for edge excitations in fractional quantum Hall systems.

Contribution

It rigorously derives the large N limit algebra of the Chern-Simons matrix model and links it to affine Lie algebras, confirming the emergence of u(p) current algebra.

Findings

Large N limit algebra is isomorphic to the algebra studied by Costello.

Under scaling, the algebra degenerates to a Lie algebra with a surjective map to affine u(p).

Provides a rigorous derivation of edge excitations in fractional quantum Hall droplets.

Abstract

In this paper we study the algebra of quantum observables of the Chern-Simons matrix model which was originally proposed by Susskind and Polychronakos to describe electrons in fractional quantum Hall effects. We establish the commutation relations for its generators and study the large $N$ limit of its representation. We show that the large $N$ limit algebra is isomorphic to the uniform in $N$ algebra studied by Costello, which is isomorphic to the deformed double current algebra studied by Guay. Under appropriate scaling limit, we show that the large $N$ limit algebra degenerates to a Lie algebra which admits a surjective map to the affine Lie algebra of $\mathfrak{u}(p)$. This leads to a complete proof of the large $N$ emergence of the $\mathfrak{u}(p)$ current algebra as proposed by Dorey, Tong and Turner. This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet.