The Stable Symplectic Category and Quantization

TL;DR

This paper introduces a stabilized version of the symplectic category that overcomes composition issues, enriching it over spectra and enhancing its relevance to deformation and geometric quantization.

Contribution

It proposes a stabilization method for Weinstein's symplectic category, enabling well-defined composition and spectral enrichment, advancing the framework for geometric quantization.

Findings

The stabilized symplectic category has well-defined composition.

Morphisms are described as infinite loop spaces.

The category is enriched over spectra, facilitating deformation analysis.

Abstract

We study a stabilization of the symplectic category introduced by A. Weinstein as a domain for the geometric quantization functor. The symplectic category is a topological category with objects given by symplectic manifolds, and morphisms being suitable lagrangian correspondences. The main drawback of Weinstein's symplectic category is that composition of morphisms cannot always be defined. Our stabilization procedure rectifies this problem while remaining faithful to the original notion of composition. The stable symplectic category is enriched over the category of spectra (in particular, its morphisms can be described as infinite loop spaces representing the space of immersed lagrangians), and it possesses several appealing properties that are relevant to deformation, and geometric quantization.