The Expansion ⋆ mod ō (ℏ⁴) and Computer-Assisted Proof Schemes in the Kontsevich Deformation Quantization

Abstract

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter . Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, moduloō( 4 ), for the newly built ⋆-product expansion.Assoc ⋆ (f, g, h) = ♦ P, Jac(P), Jac(P) modō( 4 ), is quadratic and has differential order two with respect to the Jacobiator. For all Poisson brackets {·, ·} P on finite-dimensional affine manifolds N n our ten-parameter expression of the ⋆-product does agree up toō( 4 ) with previously known results about the values of Kontsevich graph weights at some fixed values of the ten master-parameters and about the linear relations between those weights at all values of the master-parameters. 4The software implementation [5] consists of a C++ library and a set of command-line programs. The latter are specified in what follows; a full list of new C++ subroutines and their call syntaxis is contained in Appendix B. Whenever a command-line program refers to just one particular function in C++, we indicate that in the text. The current text refers to version 0.16 of the software.

Multiple zeta values in deformation quantization

“…We checked that the coefficients in our database satisfy all of these constraints. Furthermore, our results agree with the 1640 additional linear relations between harmonic coefficients at order 5 that were communicated to us by the authors of [19].…”

The orientation morphism: from graph cocycles to deformations of Poisson structures

“…By construction, the right-to-left ordering of the operators ∆ ij is inherited from the wedge ordering of edges E(γ) in the graph γ: the operator corresponding to the firstmost edge acts first. 4 The operator mult k , acting at the end of the day, is the ordered multiplication of the resulting terms in (…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…Example 1. The Kontsevich tetrahedral flow ( [22,24] and [2,3]) perturbs the Poisson bracket by the terms starting at 4−1 = 3 , which shows up nontrivially at 4 in the expansion ⋆ modō( 4 ), now available from [6].…”