Deformation quantization generates all multiple zeta values

TL;DR

This paper proves that deformation quantization coefficients with the logarithmic propagator generate all multiple zeta values, establishing a deep connection between quantum algebra and number theory.

Contribution

It demonstrates that Kontsevich graph coefficients at each order generate the entire space of multiple zeta values, confirming a conjecture by Banks--Panzer--Pym.

Findings

Coefficients generate the Q-vector space of weight-n MZVs.

All MZVs are generated by coefficients from a specific subset of Kontsevich graphs.

Develops a new integration technique using polylogarithms for Kontsevich graphs.

Abstract

Banks--Panzer--Pym have shown that the volume integrals appearing in Kontsevich's deformation quantization formula always evaluate to integer-linear combinations of multiple zeta values (MZVs). We prove a sort of converse, which they conjectured in their work, namely that with the logarithmic propagator: (1) the coefficients associated to the graphs appearing at order $\hbar^n$ in the quantization formula generate the Q-vector space of weight-$n$ MZVs, and (2) the set of all coefficients generates the Z-module of MZVs. In order to prove this result, we develop a new technique for integrating Kontsevich graphs using polylogarithms and apply it to an infinite subset of Kontsevich graphs. Then, using the binary string representation of MZVs and the Lyndon word decomposition of binary strings, we show that this subset of graphs generates all MZVs.