Geometric quantization with applications to Gromov-Witten theory

Abstract

In the physics literature, geometric quantization is an operation that arises from an attempt to make sense of the passage from a classical theory to the corresponding quantum theory. In mathematics, on the other hand, the work of Alexander Givental and others has revealed quantization to be a powerful tool for studying Gromov–Witten-type theories in higher genus. For example, if the quantization of a symplectic transformation matches two total descendent potentials, then the original symplectic transformations matches their genus-zero theories, and, at least when a semisimplicity condition is satisfied, the converse is also true. In these notes, we give a mathematically-minded presentation of quantization of symplectic vector spaces, and we illustrate how quantization appears in specific applications to Gromov–Witten theory.