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

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.


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
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