# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.11649