Geometric quantization of symplectic and Poisson manifolds

Abstract

The first part of this thesis provides an introduction to recent development in geometric quantization of symplectic and Poisson manifolds, including modern refinements involving Lie groupoid theory and index theory/K-theory. We start by giving a detailed treatment of traditional geometric quantization of symplectic manifolds, where we cover both the quantization scheme via polarization and via push-forward in K-theory. A different approach is needed for more general Poisson manifolds, which we treat by the geometric quantization of Poisson manifolds via the geometric quantization of their associated symplectic groupoids, due to Weinstein, Xu, Hawkins, et al. In the second part of the thesis we show that this geometric quantization via symplectic groupoids can naturally be understood as an instance of higher geometric quantization in higher geometry, namely as the boundary theory of the 2d Poisson sigma-model. This thesis closes with an outlook on the implications of this change of perspective.