标题： 关于Chern-Simons矩阵模型的Hilbert空间、变形双流代数作用以及共形极限 显示英文标题

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

摘要： 由Dorey、Tong和Turner提出了一种Chern-Simons矩阵模型，用于描述非阿贝尔分数量子霍尔效应。 在本文中，我们从几何量子化角度研究了Chern-Simons矩阵模型的希尔伯特空间。 我们证明Chern-Simons矩阵模型的希尔伯特空间可以与关联到一个带框架的Jordan箭图的箭图变体上的线丛的截面空间相识别。 我们使用局部化技术计算了希尔伯特空间的特征。 利用涡旋模空间与Beilinson-Drinfeld Schubert流形之间的自然同构，我们证明了基态波函数是与WZW模型相关的共形块丛的平坦截面。 特别是它们解了一个Knizhnik-Zamolodchikov方程。 我们证明希尔伯特空间上存在一个变形双电流代数（DDCA）的自然作用，而且该作用是不可约的。 我们定义并研究了Chern-Simons矩阵模型的共形极限。 我们证明希尔伯特空间的共形极限是$\widehat{\mathfrak{gl}}(n)$的一个不可约可积模，其等级与矩阵模型等级一致。 此外，我们证明$\widehat{\mathfrak{gl}}(n)$生成元可以通过矩阵模型算子的尺度极限得到，这解决了Dorey-Tong-Turner的猜想。 证明的关键是构造了希尔伯特空间共形极限上的Yangian$Y(\mathfrak{gl}_n)$作用，我们期望该作用与Uglov构造的可积$\widehat{\mathfrak{gl}}(n)$模上的$Y(\mathfrak{gl}_n)$作用等价。 我们还针对Yangian的一个极大交换子代数，表征了矩阵模型希尔伯特空间的本征向量和本征值。 显示英文摘要

摘要： 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.