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Arthemy Kiselev: Kontsevich’s star-product up to order seven for affine Poisson brackets, or: Where are the Riemann zeta values?
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 differential 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 onward.
In a joint work with R.Buring (arXiv:2209.14438 [q-alg]) we obtain the seventh 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 consequences 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.