Branes and quantization

Abstract

The problem of quantizing a symplectic manifold (M,ω) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of (M,ω) is the space of (Bcc, B1) strings, where Bcc and B1 are two A-branes; B1 is an ordinary Lagrangian A-brane, and Bcc is a space-filling coisotropic A-brane. B1 is supported on M, and the choice of ω is encoded in the choice of Bcc. As an example, we describe from this point of view the representations of the group SL(2, R{double-struck}). Another application is to Chern-Simons gauge theory. © 2009 International Press.