Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization

Abstract: This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_\pi$ for integrating any Poisson structure $\pi$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{t\pi}$ around $t=0$ yields an extract of Kontsevich's star product formula based on tree-graphs, recovering the formal family introduced by Cattaneo, Dherin and Felder in [6]. The third result involves the relation to semiclassical aspects of the Poisson Sigma model. We show that $S_\pi$ can be obtained by non-perturbative functional methods, evaluating a certain functional on families of solutions of a PDE on a disk, for which we show existence and classification.