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.

arXiv comment71 pages; software available at http://bitbucket.org/bpym/starproducts/ and https://bitbucket.org/PanzerErik/kontsevint/

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Cited by in corpus (12)

• Quantizations of local surfaces and rebel instantons
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• Deformations of path algebras of quivers with relations
• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values
• Periods of generalized Tate curves
• On webs, polylogarithms and cluster algebras
• Strict quantization of polynomial Poisson structures
• Kontsevich's deformation quantization: from Dirac to multiple zeta values
• Deformations of categories of coherent sheaves via quivers with relations
• Iterated integrals and unipotent periods on families of marked elliptic curves
• Computation of Kontsevich Weights of Connection and Curvature Graphs for Symplectic Poisson Structures