  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.
