Generating Functions for Local Symplectic Groupoids and Non-perturbative Semiclassical Quantization

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 π for integrating any Poisson structure π on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of S t π 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 π 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.