In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold.
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 Bk 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, where Dn are differential operators of order at most n. The corresponding induced
All (equivalence classes of) graphs with n internal vertices are accumulated in the set Gn(2).
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.
, endowed with the Poincaré metric 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.
The vertices f and g are at the fixed positions 0 and 1 in H. Given the above three definitions, the Kontsevich formula for a star product is now Enforcing associativity of the
-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in ħ, to just