The Poisson -model is a 2-dimensional sigma-model quantum field theory whose target space is a Poisson Lie algebroid. This may be thought of as encoding the quantum mechanics of a string propagating on the phase space? of a system in classical mechanics.
In his solution of the problem of deformation quantization? Maxim Kontsevich showed that correlators for the 2-string interaction (the correlator on the worldsheet that is a disk with three marked points on its boundary) describe a product operation which is a deformation of the Poisson bracket on the target space.
The principal variant of the nonlinear Poisson sigma model is sometimes called Cattaneo-Felder model. The graphical expansion used in Kontsevich’s approach to the deformation quantization is explained via a Feynman diagram expansion in this model.
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A. Cattaneo, G. Felder, On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56, 163–179 (2001) math.QA/0102108.
A. Cattaneo, G. Felder, Poisson sigma models and deformation quantization, Mod. Phys. Lett. A 16, 179–190 (2001) hep-th/0102208.
A. Cattaneo, G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591–611 (2000) doi, math.QA/9902090.