Geometric Quantization by Paths Part II: The General Case

Abstract

In Part I, we constructed the prequantum groupoid for connected and simply connected parasymplectic spaces. This second part extends the construction to arbitrary connected parasymplectic diffeological spaces $(X,\omega )$: the obstruction to the additivity of the action form on loops is measured by a surfacic cocycle on ${\pi }_1(X)$, and a choice of rectifying cochain $c$ yields a prequantum groupoid $T_c$ whose isotropy group is the torus of periods $\mathbf{R}/P_{\omega }$ and which carries a canonical invariant $1$-form of curvature ${\mathrm{trg}}^{\ast }\omega -{\mathrm{src}}^{\ast }\omega$. The resulting family $\{T_c{\}}_c$ is an affine space directed by the character group $\mathrm{Hom}({\pi }_1(X),\mathbf{R}/P_{\omega })$.

arXiv commentRevised version. This version presents the construction of the prequantum groupoids ${\mathbf{T}}_c$ only, for arbitrary connected parasymplectic diffeological spaces. The incorrect claim of the previous version, concerning the group of automorphisms of ${\mathbf{T}}_{\omega }$, has been removed. The symmetry analysis, is deferred to a subsequent work