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

For any connected and simply connected parasymplectic space $(\mathrm{X},\omega )$ with group of periods ${\mathrm{P}}_{\omega }⊊\mathbf{R}$, we construct a prequantum groupoid ${\mathrm{T}}_{\omega }$ as a diffeological quotient of the space $\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{s}(\mathrm{X})$ of paths in $\mathrm{X}$. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid ${\mathrm{T}}_{\omega }$ has $\mathrm{X}$ as its objects, and its space of morphisms $\mathcal{Y}$ carries a canonical left-right invariant $1$-form $\lambda$ whose curvature encodes $\omega$. A key property is that the isotropy group ${\mathrm{T}}_{\omega ,x}$ at any point $x$, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods ${\mathrm{T}}_{\omega }=\mathbf{R}\mathrm{/}{\mathrm{P}}_{\omega }$. Furthermore, the entire symmetry group $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\mathrm{X},\omega )$ acts as faithful automorphisms of $({\mathrm{T}}_{\omega },\lambda )$ 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.