# Kontsevich quantization formula

In mathematics, the **Kontsevich quantization formula** describes how to construct a generalized ★-product operator algebra from a given arbitrary Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.^{[1]}^{[2]}

## Deformation quantization of a Poisson algebra

Given a Poisson algebra (*A*, {⋅, ⋅}), a deformation quantization is an **associative** unital product ★ on the algebra of formal power series in *ħ*, *A*[[*ħ*]], subject to the following two axioms,

If one were given a Poisson manifold (*M*, {⋅, ⋅}), one could ask, in addition, that

where the Template:Mvar are linear bidifferential operators of degree at most Template:Mvar.

Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,

where Template:Mvar are differential operators of order at most Template:Mvar. The corresponding induced ★-product, ★′, is then

For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" ★-product.

## Kontsevich graphs

A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled *f* and *g*; and Template:Mvar internal vertices, labeled Π. From each internal vertex originate two edges. All (equivalence classes of) graphs with Template:Mvar 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

### 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 Template:Mvar internal vertices is (*n*(*n* + 1))^{n}. The sample graph above has the weight *m*(Γ) = 8. For this, it is helpful to enumerate the internal vertices from 1 to Template:Mvar.

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* ⊂ ℂ, endowed with a metric

and, for two points *z*, *w* ∈ *H* with *z* ≠ *w*, we measure the angle Template:Mvar between the geodesic from Template:Mvar to *i*∞ and from Template:Mvar to Template:Mvar counterclockwise. This is

The integration domain is *C*_{n}(*H*) the space

The formula amounts

where *t*1(*j*) and *t*2(*j*) are the first and second target vertex of the internal vertex Template:Mvar. The vertices *f* and *g* are at the fixed positions 0 and 1 in Template:Mvar.

## The formula

Given the above three definitions, the Kontsevich formula for a star product is now

### Explicit formula up to second order

Enforcing associativity of the ★-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in Template:Mvar, to just

## References

- ↑ M. Kontsevich (2003),
*Deformation Quantization of Poisson Manifolds*,*Letters of Mathematical Physics***66**, pp. 157–216. - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}