On the Hilbert Space of the Chern-Simons Matrix Model, Deformed Double Current Algebra Action, and the Conformal Limit

TL;DR

This paper explores the geometric and algebraic structure of the Hilbert space in a Chern-Simons matrix model related to quantum Hall effects, revealing connections to conformal field theory, algebraic actions, and integrable modules.

Contribution

It provides a geometric quantization perspective, establishes the action of deformed double current algebra, and proves the conjecture relating matrix model operators to affine algebra generators.

Findings

Hilbert space identified with sections of line bundle on quiver variety

Ground states are flat sections of conformal blocks solving KZ equations

Constructs Yangian action and characterizes eigenvectors in the conformal limit

Abstract

A Chern-Simons matrix model was proposed by Dorey, Tong, and Turner to describe non-Abelian fractional quantum Hall effect. In this paper we study the Hilbert space of the Chern-Simons matrix model from a geometric quantization point of view. We show that the Hilbert space of the Chern-Simons matrix model can be identified with the space of sections of a line bundle on the quiver variety associated to a framed Jordan quiver. We compute the character of the Hilbert space using localization technique. Using a natural isomorphism between vortex moduli space and a Beilinson-Drinfeld Schubert variety, we prove that the ground states wave functions are flat sections of a bundle of conformal blocks associated to a WZW model. In particular they solve a Knizhnik-Zamolodchikov equation. We show that there exists a natural action of the deformed double current algebra (DDCA) on the Hilbert space, moreover the action is irreducible. We define and study the conformal limit of the Chern-Simons matrix model. We show that the conformal limit of the Hilbert space is an irreducible integrable module of $\widehat{\mathfrak{gl}}(n)$ with level identified with the matrix model level. Moreover, we prove that $\widehat{\mathfrak{gl}}(n)$ generators can be obtained from scaling limits of matrix model operators, which settles a conjecture of Dorey-Tong-Turner. The key to the proof is the construction of a Yangian $Y(\mathfrak{gl}_n)$ action on the conformal limit of the Hilbert space, which we expect to be equivalent to the $Y(\mathfrak{gl}_n)$ action on the integrable $\widehat{\mathfrak{gl}}(n)$ modules constructed by Uglov. We also characterize eigenvectors and eigenvalues of the matrix model Hilbert space with respect to a maximal commutative subalgebra of Yangian.