A path integral approach to the Kontsevich quantization formula

TL;DR

The work presents a quantum field theory realization of Kontsevich's deformation quantization by a simple bosonic topological sigma model on a disc, whose BV quantization yields a superconformal theory and whose perturbative expansion reproduces Kontsevich’s star product. It uses BRST/BV methods to establish gauge consistency, renormalization of tadpoles, and a path-integral realization of a broader map from poly-vector fields to poly-differential operators, culminating in an L_infty morphism that implements formality. The results explain why Kontsevich’s construction works, connect boundary correlators to the star product and its center, and provide a framework to generalize deformation quantization to arbitrary poly-vector fields and higher-genus settings. The approach highlights the field-theoretic underpinnings of associativity and formality, and it points to future extensions involving nontrivial classical solutions and richer geometric structures.

Abstract

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.

Figures (2)

• Figure 1: The expectation value (\ref{['e-99']}) in the limit as $u$ approaches the boundary reduces to a path integral on this surface
• Figure 2: A component of the boundary of $B_m$. The point $\infty$ is $t_{m+1}$