Geometric Quantization by Paths -- Part I: The Simply Connected Case

TL;DR

This paper introduces a diffeological groupoid construction for prequantization of simply connected parasymplectic spaces, generalizing classical methods to broader, possibly singular or infinite-dimensional, settings.

Contribution

It constructs a prequantum groupoid from path spaces using diffeology, extending classical geometric quantization to more general spaces with complex period structures.

Findings

The groupoid encodes the classical geometry and symmetries.

Isotropy groups are isomorphic to tori of periods.

Symmetry groups act faithfully without central extensions.

Abstract

For any connected and simply connected parasymplectic space $(\mathrm{X},\omega)$ with group of periods $\mathrm{P}_\omega \subsetneq \mathbf{R}$, we construct a prequantum groupoid $\pmb{\mathrm{T}}_\omega$ as a diffeological quotient of the space $\mathrm{Paths}(\mathrm{X})$ of paths in $\mathrm{X}$. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid $\pmb{\mathrm{T}}_\omega$ has $\mathrm{X}$ as its objects, and its space of morphisms $\mathcal{Y}$ carries a canonical left-right invariant $1$-form $\pmb{\lambda}$ whose curvature encodes $\omega$. A key property is that the isotropy group $\pmb{\mathrm{T}}_{\omega,x}$ at any point $x$, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods $\mathrm{T}_\omega = \mathbf{R}/\mathrm{P}_\omega$. Furthermore, the entire symmetry group $\mathrm{Diff}(\mathrm{X}, \omega)$ acts as faithful automorphisms of $(\pmb{\mathrm{T}}_\omega, \pmb{\lambda})$ without central extensions at this level. Built within the framework of diffeology, this construction generalizes classical prequantization by applying to broad classes of spaces, including those with singularities or infinite-dimensional aspects, and by accommodating generalized (e.g., irrational) tori of periods. This paper focuses on the simply connected case; the construction will be extended to general diffeological spaces in a subsequent publication.