Multiple zeta values in 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. Contents

Strict quantization of polynomial Poisson structures

“…Observables Recall that a Poisson manifold (X, η) is given by a smooth manifold X and a smooth bivector field η on X whose associated Poisson bracket {−,−} η satisfies the Jacobi identity. 1 The algebra of smooth functions on a Poisson manifold (X, η) can be viewed as a Poisson algebra of classical observables, where X is the phase space of a classical mechanical system and the Poisson bracket {−,−} η encodes the time evolution via Hamilton's equations of motion. Deformation quantization after Bayen-Flato-Frønsdal-Lichnerowicz-Sternheimer [3] aims to produce an associative algebra of quantum observables from such an algebra of classical observables by deforming the usual pointwise commutative product of smooth functions to an associative star product ⋆.…”

“…for any ∈ Ω. Some natural choices for q = 1 + it + O(t 2 ) would be the power series expansions around t = 0 of the functions 1 + it, e it , or 1 1−it . Remark 2.27.…”

“…). Let R ≥ 1 2 and ∈ C. Then the standard-ordered and the Moyal-Weyl product ⋆ std , * W : P TR (R d ) × P TR (R d ) → P TR (R d ) are continuous and therefore extend uniquely to continuous products ⋆ std , * W : P TR (R d ) × P TR (R d ) → P TR (R d ). For all f, g ∈ P TR (R d ), the maps µ f,g , ∆ f,g : C → P TR (R d ), defined as in (3.6), are holomorphic.…”

“…Although Kontsevich's universal quantization formula [15] can be made very explicit, it is only a finite sum on polynomials for constant or linear Poisson structures [15,8]. For nonlinear Poisson structures, the asymptotics of the multiple zeta values appearing as weights of certain graphs in the formula (see [1]) make the convergence properties of the Kontsevich star product on polynomials difficult to determine and Kontsevich's conjecture on the convergence of the Kontsevich star product [16,Conj. 1] is still widely open (cf.…”

On the Moyal Star Product of Resurgent Series

“…where each G k is a suitable collection of graphs, the c Γ 's are universal coefficients, and the B Γ,π 's are polydifferential operators depending on the graph Γ and the Poisson structure π. Recently, a deep connection between these universal coefficients c Γ and multiple zeta values 1 has been brought to light [BPP20].…”