2024/01/26 by Yau, YuTung · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2401.14574
In their physical proposal for quantization [20], Gukov-Witten suggested that, given a symplectic manifold M with a complexification X, the A-model morphism spaces Hom(Bcc, Bcc) and Hom(B, Bcc) should recover holomorphic deformation quantization of X and geometric quantization of M respectively, where Bcc is a canonical coisotropic A-brane on X and B is a Lagrangian A-brane supported on M. Assuming M is spin and Kähler with a prequantum line bundle L, Chan-Leung-Li [10] constructed a subsheaf Oqu(k) of smooth functions on M with a non-formal star product and a left Oqu(k)-module structure on the sheaf of holomorphic sections of L⊗ k ⊗ √(K). In this paper, we give a careful treatment of the relation between (holomorphic) deformation quantizations of M and X. As a result, Chan-Leung-Li's work [10] provides a mathematical realization of the action of Hom(Bcc, Bcc) on Hom(B, Bcc). By Fedosov's gluing arguments, we also construct a Oqu(k)-Oqu(k)-bimodule structure on the sheaf of smooth sections of L⊗ 2k to realize the actions of Hom(Bcc, Bcc) and Hom(Bcc, Bcc) on Hom(Bcc, Bcc), which is related to the analytic geometric Langlands program.