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Multiple zeta values in deformation quantization. (English) Zbl 1454.11161

Invent. Math. 222, No. 1, 79-159 (2020); correction ibid. 229, No. 1, 449 (2022).
In 1997 M. Kontsevich [Lett. Math. Phys. 48, No. 1, 35–72 (1999; Zbl 0945.18008)] showed that every Poisson manifold can be quantized to obtain a noncommutative algebra. In his quantization formula a formal power series appears in the deformation parameter \(\hbar\). It is conjectured that the coefficients in this power series are expressible as \(\mathbb{Q}[1/(2\pi i)]\)-linear combinations of multiple zeta values (MZVs) which are infinite sums of the form \[\zeta\left(n_{1}, \ldots,n_{d}\right)=\sum_{0<k_{1}<\cdots<k_{d}}\frac{1}{k_{1}^{n_{1}} \cdots k_{d}^{n_{d}}} \in \mathbb{R}\] with positive integers \(n_1,\dots,n_d\) such that \(\sum_jn_j=n\), and \(n_d\ge2\).
The conjecture is not only proven in the paper, but the authors give bounds on the weights of the MZVs that appear at a given order of \(\hbar\). Moreover, it is also shown that the coefficients of the linear combinations are, in fact, integers.
The analysis of the paper is based on a systematic theory of integration on the moduli spaces of marked disks via suitable algebras of polylogarithms. This theory is developed in the paper, based on earlier works of Brown and Goncharov.
The authors’ results are constructive in the sense that they present an algorithm to determine the power series coefficients in the quantization formula.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
14H10 Families, moduli of curves (algebraic)
53D55 Deformation quantization, star products
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry

Citations:

Zbl 0945.18008

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Traces; nauty; HyperInt
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References:

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