2025/08/15 by Iglesias-Zemmour, Patrick
#55R65 #58A05 #58A10 #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 53D50 #Secondary 22A22
paper · doi:10.48550/arxiv.2508.11337
For any connected and simply connected parasymplectic space (X,ω) with group of periods Pω\subsetneq R, we construct a prequantum groupoid \pmbTω as a diffeological quotient of the space Paths(X) of paths in X. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid \pmbTω has X as its objects, and its space of morphisms Y carries a canonical left-right invariant 1-form \pmbλ whose curvature encodes ω. A key property is that the isotropy group \pmbTω,x at any point x, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods Tω= R/Pω. Furthermore, the entire symmetry group Diff(X, ω) acts as faithful automorphisms of (\pmbTω, \pmbλ) 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.