On the Kontsevich ★-product associativity mechanism

Abstract

The deformation quantization by Kontsevich is a way to construct an associative noncommutative star-product ⋆ = × + { , } P +ō( ) in the algebra of formal power series in on a given finite-dimensional affine Poisson manifold: here × is the usual multiplication, { , } P = 0 is the Poisson bracket, and is the deformation parameter. The product ⋆ is assembled at all powers k 0 via summation over a certain set of weighted graphs with k+2 vertices; for each k>0, every such graph connects the two co-multiples of ⋆ using k copies of { , } P . Cattaneo and Felder interpreted these topological portraits as genuine Feynman diagrams in the Ikeda-Izawa model for quantum gravity.By expanding the star-product up toō( 3 ), i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = ♦ (Poisson) that converts the Jacobi identity for the bracket { , } P into the associativity of ⋆.

The Kontsevich tetrahedral flow revisited

“…For the balanced flow we have: # All 3-dimensional Poisson bi-vectors are of the following form. [ [1,2,3], 0] This reasoning hints us that the condition a : b = 1 : 6 could be sufficient for equation (2) to hold for all Poisson structures on all finite dimensional affine real manifolds. A rigorous proof of the respective claim in Theorem 3 is provided in section 2.…”

“…The proof is explicit: in section 2 we reveal the mechanism of factorization -via the Jacobi identity -in (2) at a : b = 1 : 6. On the left-hand side of factorization problem (2) we expand the Poisson differential of the Kontsevich tetrahedral flow at the balance 1 : 6 into the sum of 39 graphs (see Figure 3 on page 8 and Table 2 in Appendix A). On the other side of that factorization, we take the sum that runs with undetermined coefficients over all those fragments of differential consequences of the Jacobi identity [[P, P]] = 0 which are known to vanish independently.…”

“…On the other side of that factorization, we take the sum that runs with undetermined coefficients over all those fragments of differential consequences of the Jacobi identity [[P, P]] = 0 which are known to vanish independently. We then find a linear polydifferential operator ♦(P, ·) that acts on the filtered components of the Jacobiator Jac (P) := [[P, P]] for the bi-vector P; the operator ♦ provides the factorization [[P, Q 1:6 (P)]](f, g, h) = ♦ P, Jac (P)(·, ·, ·) (f, g, h) of the ∂ P -cocycle condition, see (2), through the Jacobi identity Jac (P) = 0. To describe the differential operators that produce such consequences of the Jacobi identity, we use the pictorial language of graphs: every internal vertex contains a copy of the bi-vector P and the operators are reduced by using its skew-symmetry.…”

“…In Appendix B we outline a different method to tackle the factorization problem, namely, by making the Jacobi identity visible in (2) by perturbing the original structure P → P in such a way that P is not Poisson and Q 1:6 ( P) = 0. Hence P contributes to the right-hand side of (2) such that the respectively perturbed bi-vector Q 1:6 ( P) stops being compatible with the perturbed Poisson structure P. The first-order balance of both sides of perturbed equation (2) then suggests the coefficients of those differential consequences of the Jacobiatior which are actually involved in the factorization mechanism.…”

“…Hence P contributes to the right-hand side of (2) such that the respectively perturbed bi-vector Q 1:6 ( P) stops being compatible with the perturbed Poisson structure P. The first-order balance of both sides of perturbed equation (2) then suggests the coefficients of those differential consequences of the Jacobiatior which are actually involved in the factorization mechanism. The coefficients of operators realized by graphs which were found by following this scheme are reproduced in the full run-through that gave us the solution ♦ in section 2.…”