Multiple zeta values in deformation quantization
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Multiple zeta values in deformation quantization Peter Banks1 · Erik Panzer1
· Brent Pym2,3
Received: 30 March 2019 / Accepted: 27 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Mathematics Subject Classification 11M32, 14H10, 53D55, 81Q30
B Brent Pym
[email protected] Peter Banks [email protected] Erik Panzer [email protected]
1
University of Oxford, Oxford, UK
2
University of Edinburgh, Edinburgh, UK
3
McGill University, Montréal, Canada
123
P. Banks et al.
1 Introduction 1.1 Motivation and overview In 1997, Kontsevich solved a long-standing problem in mathematical physics, by showing that every Poisson manifold can be quantized to obtain a noncommutative algebra [38]. He gave an explicit formula that takes a Poisson bracket on Rk as input and produces a “star product”, i.e. a noncommutative deformation of the product on C ∞ (Rk ) as a formal power series in the deformation parameter h¯ . He did so by constructing his “formality morphism”—an explicit homotopy equivalence between the differential graded Lie algebras of polyvector fields and Hochschild cochains on the affine space Rk . The star product (and more generally, the formality morphism) is expressed as a sum over a suitable collection of graphs. Each graph Γ contributes a term involving a polydifferential operator defined by an elementary combinatorial procedure. This operator is then weighted by a constant cΓ :=
Cn,m
ω ∈ R
(1)
defined by integrating an explicit volume form ωΓ over a suitable moduli space Cn,m of marked holomorphic disks. As explained by Cattaneo–Felder [20], Kontsevich’s formula can be interpreted as a perturbative expansion in an appropriate topological string theory, which has the expressions (1) as its Feynman integrals. The coefficients (1) are universal, in the sense that they are independent of the Poisson bracket one seeks to quantize; hence one only needs to compute each integral once and remember its value in perpetuity. However, the integrals are notoriously difficult to evaluate, and their precise values remain unknown. Thus, even with the aid of a computer, it has so far been impossible to calculate the terms in the quantization formula explicitly beyond h¯ 3 for many of the most basic examples of Poisson brackets, such as the simple “log canonical” bracket {x, y} = x y
(2)
on the (x, y)-plane (so named because the logarithms u = log x and v = log y are canonical variables, i.e.
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