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 construct a function associated germ any local symplectic groupoid provide an explicit (smooth, non-formal) universal formula $$S_\pi $$ integrating structure $$\pi on coordinate space. The second result involves semiclassical We that formal Taylor expansion $$S_{t\pi }$$ around $$t=0$$ yields extract Kontsevich’s star product based tree-graphs, recovering family introduced by Cattaneo, Dherin Felder in [6]. third aspects Sigma model. can be obtained non-perturbative functional methods, evaluating certain families solutions PDE disk, which existence classification.