A path integral approach to the Kontsevich quantization formula. (English) Zbl 1038.53088

Cattaneo, Alberto S.; Felder, Giovanni

Commun. Math. Phys. 212, No. 3, 591-611 (2000).

Summary: We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin-Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra.

MSC:

53D55	Deformation quantization, star products
58D30	Applications of manifolds of mappings to the sciences
81S40	Path integrals in quantum mechanics
81T18	Feynman diagrams
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T45	Topological field theories in quantum mechanics