# Quantization of Polysymplectic Manifolds

## TL;DR

This paper extends geometric quantization to polysymplectic manifolds, defining prequantization and quantization procedures, and explores the implications of complex polarization, including the falsity of a conjecture.

## Contribution

It introduces a framework for quantization on polysymplectic manifolds, including new definitions of prequantum bundles and analysis of polarization effects.

## Key findings

- Prequantization extends symmetries to sections of Hermitian bundles.
- Quantization with polarization and spin^c structures is defined.
- The polysymplectic Guillemin-Sternberg conjecture is shown to be false.

## Abstract

We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in an essential way the action of the space of coefficients. We define quantization with respect to a polarization and to a spin$^\text{c}$ structure. In the presence of a complex polarization, it is shown that the polysymplectic Guillemin-Sternberg conjecture is false. We conclude with potential extensions and applications.