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