The expansion $\star$ mod $\bar{o}(\hbar^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization

TL;DR

This paper develops software tools to compute and verify the Kontsevich star product up to order four in deformation quantization, simplifying the complex relations between graph weights and outlining a computer-assisted proof scheme.

Contribution

It introduces software modules for generating Kontsevich graphs, expanding the star product, and deriving relations, enabling explicit computation and proof of associativity up to order four.

Findings

Expressed all graph coefficients using 149 weights, with 67 known exactly.

Reduced remaining weights to 10 parameters, 6 modulo gauge equivalence.

Outlined a scheme for computer-assisted proof of associativity.

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 $\star$-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 $\star$-product up to order 4 in the deformation parameter $\hbar$. Already at this stage, the $\star$-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, modulo $\bar{o}(\hbar^4)$, for the newly built $\star$-product expansion.