A Graphical Calculus for Classical and Quantum Microformal Morphisms

We develop a graphical calculus for the microformal or thick morphisms introduced by Th. Voronov. This allows us to write the infinite series arising from pullbacks, compositions, and coordinate transformations of thick morphisms as sums over bipartite trees. The methods are inspired by those employed by Cattaneo-Dherin-Felder in their work on formal symplectic groupoids. We also extend this calculus to quantum thick morphisms, which are special types of Fourier integral operators quantizing classical thick morphisms. The relationship between the calculi for classical and quantum thick morphisms resembles the relationship between the semi-classical and full perturbative expansions over Feynman diagrams in quantum field theory.