Kontsevich's Star-Product up to Order Seven for Affine Poisson Brackets

Explore the intricacies of Kontsevich's star-product for affine Poisson brackets in this 48-minute lecture by Arthemy Kiselev from the Institut des Hautes Etudes Scientifiques (IHES). Delve into the world of noncommutative associative star-products as deformations of usual function products on smooth manifolds. Examine how Kontsevich's star-products on finite-dimensional affine Poisson manifolds are encoded using weighted graphs with ordered directed edges. Discover the challenges in finding real coefficients of graphs in the star-product expansion and the conjectured appearance of irrational Riemann zeta values from the fifth order onwards. Learn about the joint work with R.Buring that yields the seventh-order formula of Kontsevich's star-product for affine Poisson brackets, revealing rational coefficients for every affine Poisson bracket. Investigate the mechanism of associativity for Kontsevich's star-product, comparing its workings up to order six with orders seven and higher in terms of graphs.