Kontsevich quantization formula

Statement

Let $U \subseteq \R^d$ be open and let $\pi^{ij} = -\pi^{ji}$, $1 \le i, j \le d$, be smooth functions on $U$. Assume that $\{f, g\} = \sum_{i,j} \pi^{ij} \partial_i f \, \partial_j g$ satisfies the Jacobi identity. So $U$ is a Poisson manifold with Poisson bivector $\pi$. For $n \ge 0$, let $G_n$ be the set of admissible graphs. Such a graph has aerial vertices $1, \ldots, n$, ground vertices $L$ and $R$, and $2n$ directed edges. Each aerial vertex is the source of exactly two edges, labeled first and second. No edge goes from a vertex to itself, and no two edges have the same source and target. To $\Gamma \in G_n$ attach:

1. the bidifferential operator $B_\Gamma$. Place a copy of $(\pi^{ij})$ at each aerial vertex, $f$ at $L$ and $g$ at $R$. Let each edge act as a partial derivative $\partial_i$ on the object at its target, and sum over all indices;
2. the weight $w_\Gamma$ defined below.

Let $C_n$ be the space of configurations of $n$ distinct points in the upper half-plane $\mathbb{H}$, and put $L = 0$ and $R = 1$. Let $e_k^1$ and $e_k^2$ be the edges leaving vertex $k$. For an edge $e$, let $\phi_e$ be the angle at the source of $e$, in the hyperbolic metric of $\mathbb{H}$. It goes from the geodesic to $\infty$ to the geodesic to the target of $e$. Then $$w_\Gamma = \frac{1}{n! \, (2\pi)^{2n}} \int_{C_n} \bigwedge_{k=1}^n \left( d\phi_{e_k^1} \wedge d\phi_{e_k^2} \right).$$

Let $C^\infty(U)[\![\hbar]\!]$ be the space of formal power series in $\hbar$ with coefficients in $C^\infty(U)$. Extend the following product $\hbar$-bilinearly to it: $$f \star g = \sum_{n \ge 0} \hbar^n \sum_{\Gamma \in G_n} w_\Gamma \, B_\Gamma(f, g).$$ Then $\star$ is associative, and $f \star g = fg + \hbar \{f, g\} + O(\hbar^2)$. So $\star$ is a deformation quantization of $(U, \pi)$.