# Kontsevich quantization formula

In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.

## Deformation quantization of a Poisson algebra

Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization is an associative unital product $\star$ on the algebra of formal power series in ħ, A[[ħ]], subject to the following two axioms,
    ${\begin{aligned}f\star g&=fg+{\mathcal {O}}(\hbar )\\[f,g]&=f\star g-g\star f=i\hbar \{f,g\}+{\mathcal {O}}(\hbar ^{2})\end{aligned}}$
If one were given a Poisson manifold (M, {⋅, ⋅}), one could ask, in addition, that
    $f\star g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),$
where the B$_k$ are linear bidifferential operators of degree at most k.
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
    ${\begin{cases}D:A[[\hbar ]]\to A[[\hbar ]]\\\sum _{k=0}^{\infty }\hbar ^{k}f_{k}\mapsto \sum _{k=0}^{\infty }\hbar ^{k}f_{k}+\sum _{n\geq 1,k\geq 0}D_{n}(f_{k})\hbar ^{n+k}\end{cases}}$
where D$_n$ are differential operators of order at most n. The corresponding induced $\star$-product, $\star '$, is then
    $f\,{\star }'\,g=D\left(\left(D^{-1}f\right)\star \left(D^{-1}g\right)\right).$
For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" $\star$-product.

## Kontsevich graphs

A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and n internal vertices, labeled Π. From each internal vertex originate two edges. All (equivalence classes of) graphs with n internal vertices are accumulated in the set G$_n$(2).
An example on two internal vertices is the following graph,

### Associated bidifferential operator

Associated to each graph Γ, there is a bidifferential operator B$_Γ$( f, g) defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding index from the source symbol. The term for the graph Γ is the product of all its symbols together with their partial derivatives. Here f and g stand for smooth functions on the manifold, and Π is the Poisson bivector of the Poisson manifold.
The term for the example graph is
    $\Pi ^{i_{2}j_{2}}\partial _{i_{2}}\Pi ^{i_{1}j_{1}}\partial _{i_{1}}f\,\partial _{j_{1}}\partial _{j_{2}}g.$

### Associated weight

For adding up these bidifferential operators there are the weights w$_Γ$ of the graph Γ. First of all, to each graph there is a multiplicity m(Γ) which counts how many equivalent configurations there are for one graph. The rule is that the sum of the multiplicities for all graphs with n internal vertices is (n(n + 1))$^n$. The sample graph above has the multiplicity m(Γ) = 8. For this, it is helpful to enumerate the internal vertices from 1 to n.
In order to compute the weight we have to integrate products of the angle in the upper half-plane, H, as follows. The upper half-plane is H ⊂ $\mathbb {C}$, endowed with the Poincaré metric
    $ds^{2}={\frac {dx^{2}+dy^{2}}{y^{2}}};$
and, for two points z, w ∈ H with z ≠ w, we measure the angle φ between the geodesic from z to i∞ and from z to w counterclockwise. This is
    $\phi (z,w)={\frac {1}{2i}}\log {\frac {(z-w)(z-{\bar {w}})}{({\bar {z}}-w)({\bar {z}}-{\bar {w}})}}.$
The integration domain is C$_n$(H) the space
    $C_{n}(H):=\{(u_{1},\dots ,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq j\}.$
The formula amounts
    $w_{\Gamma }:={\frac {m(\Gamma )}{(2\pi )^{2n}n!}}\int _{C_{n}(H)}\bigwedge _{j=1}^{n}\mathrm {d} \phi (u_{j},u_{t1(j)})\wedge \mathrm {d} \phi (u_{j},u_{t2(j)})$,
where t1(j) and t2(j) are the first and second target vertex of the internal vertex j. The vertices f and g are at the fixed positions 0 and 1 in H.

## The formula

Given the above three definitions, the Kontsevich formula for a star product is now
    $f\star g=fg+\sum _{n=1}^{\infty }\left({\frac {i\hbar }{2}}\right)^{n}\sum _{\Gamma \in G_{n}(2)}w_{\Gamma }B_{\Gamma }(f\otimes g).$

### Explicit formula up to second order

Enforcing associativity of the $\star$-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in ħ, to just
    ${\begin{aligned}f\star g&=fg+{\tfrac {i\hbar }{2}}\Pi ^{ij}\partial _{i}f\,\partial _{j}g-{\tfrac {\hbar ^{2}}{8}}\Pi ^{i_{1}j_{1}}\Pi ^{i_{2}j_{2}}\partial _{i_{1}}\,\partial _{i_{2}}f\partial _{j_{1}}\,\partial _{j_{2}}g\\&-{\tfrac {\hbar ^{2}}{12}}\Pi ^{i_{1}j_{1}}\partial _{j_{1}}\Pi ^{i_{2}j_{2}}(\partial _{i_{1}}\partial _{i_{2}}f\,\partial _{j_{2}}g-\partial _{i_{2}}f\,\partial _{i_{1}}\partial _{j_{2}}g)+{\mathcal {O}}(\hbar ^{3})\end{aligned}}$