Apparaît dans la collection : Combinatorics and Arithmetic for Physics : Special Days

Noncommutative associative star-products are deformations of the usual product of functions on smooth manifolds; in every star-product, its leading deformation term is a Poisson bracket. Kontsevich’s star-products on finite-dimensional affine Poisson manifolds are encoded using weighted graphs with ordering of directed edges. The associativity is then obstructed only by the Jacobiator (and its differen- tial consequences) for the bi-vector which starts the deformation. Finding the real coefficients of graphs in Kontsevich’s star-product expansion is hard in practice; conjecturally irrational Riemann zeta values appear from the firth order onwards. In a joint work with R.Buring (arXiv:2209.14438 [q-alg]) we obtain the sev- enth order formula of Kontsevich’s star-product for affine Poisson brackets (in particular, for linear brackets on the duals of Lie algebras). We discover that all the graphs near the Riemann ”zetas of concern” assimilate into differential conse- quences of the Jacobi identity, so that all the coefficient in the star-product formula are rational for every affine Poisson bracket. Thirdly, we explore the mechanism of associativity for Kontsevich’s star-product for generic or affine Poisson brackets (and with harmonic propagators from the original formula for the graph weights ): here, we contrast the work of this mechanism up to order six with the way associativity works in terms of graphs for orders seven and higher.