Kontsevich quantization formula

In mathematics, the Kontsevich quantization formula describes how to construct an generalized ∗-product operator algebra from a given Poisson manifold. This operator algebra amounts to the deformation quantization of the 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 ∗ on the algebra of formal power series in ħ, A[ [ ħ ] ], subject to the following two axioms,

$f*g=fg+\mathcal{O}(\hbar),$

and

$[f,g]:=f*g-g*f=i\hbar\{f,g\}+\mathcal{O}(\hbar^2).$

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

$f*g=fg+\sum_{k=1}^\infty \hbar^kB_k(f\otimes g),$

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,

$D\colon A[[\hbar]]\mapsto A[[\hbar]]: \sum_{k=0}^\infty \hbar^k f_k \mapsto \sum_{k=0}^\infty \hbar^k f_k +\sum_{n\ge1, k\ge0} D_n(f_k)\hbar^{n+k} ~,$

where D_n are differential operators of order at most n. The corresponding induced ∗-product, ∗', is

$f\,{*}'\,g = D((D^{-1}f)*(D^{-1}g)).$

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 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 $\Pi^{i_2j_2}\partial_{i_2}\Pi^{i_1j_1}\partial_{i_1}f\,\partial_{j_1}\partial_{j_2}g$.

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 weight 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⊂ℂ, endowed with a metric $ds^2=(dx^2+dy^2)/y^2$; 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

$C_n(H):=\{(u_1,\dots,u_n)\in H^n: u_i\ne u_j\forall i\ne j\}.$

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

$f*g = fg+\sum_{n=1}^\infty\left(\frac{i\hbar}{2}\right)^n \sum_{\Gamma \in G_n(2)} w_\Gamma B_\Gamma(f\otimes g).$

Formula up to second order

Enforcing associativity of the ∗-product, it is easy to check directly that the Kontsevich formula must reduce to second order in ħ to just

$\begin{align} f*g & = fg +\frac{i\hbar}{2}\Pi^{ij}\partial_i f\,\partial_j g +\frac{-\hbar^2}{8}\Pi^{i_1j_1}\Pi^{i_2j_2}\partial_{i_1}\,\partial_{i_2}f \partial_{j_1}\,\partial_{j_2}g \\ & {} + \frac{-\hbar^2}{12}\Pi^{i_1j_1}\partial_{j_1}\Pi^{i_2j_2}(\partial_{i_1}\partial_{i_2}f \,\partial_{j_2}g -\partial_{i_2}f\,\partial_{i_1}\partial_{j_2}g) +\mathcal{O}(\hbar^3).\end{align}$

References

• A. Cattaneo and G. Felder (2000), Path Integral Approach to the Kontsevich Quantization Formula, Communications in Mathematical Physics 212.