Multiple zeta values in deformation quantization

TL;DR

This paper develops a systematic approach to evaluate Kontsevich's deformation quantization integrals using polylogarithms, demonstrating they can be expressed as combinations of multiple zeta values and providing a computational tool.

Contribution

It introduces a new algebraic framework for integrating over moduli spaces and offers an algorithm and software for symbolic calculation of Kontsevich's formula.

Findings

Kontsevich's integrals are expressible as integer-linear combinations of multiple zeta values.

Developed a concrete algorithm for calculating these integrals.

Created the first software package for symbolic computation of Kontsevich's deformation quantization.

Abstract

Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.