Beijing Institute of Mathematical Sciences and Applications

Veronica Pasquarella

Sen Hu

Wednesday, October 30, 2024 4:00 PM - 5:00 PM

SIMIS-1610

In this talk I shall discuss quantization of Chern-Simons matrix model proposed by Susskind, Polychronakos, Dorey, Tong, and Turner to describe fractional quantum Hall effect. The Hilbert space of the Chern-Simons matrix model can be constructed from canonical quantization or geometric quantization point of view. In canonical quantization 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 u(p). Using geometric quantization 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 the conformal limit of the Hilbert space is an irreducible integrable module of glˆ(n) with level identified with the matrix model level. Moreover, we prove that glˆ(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(gln) action on the conformal limit of the Hilbert space. The above is based on joint works with Si Li, Dongheng Ye and Yehao Zhou.