Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonicalizing zeta generators: genus zero and genus one

Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Oliver Schlotterer, Leila Schneps, Bram Verbeek

Abstract

Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees $w\geq 2$, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the $f$-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez' elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai's geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.

Canonicalizing zeta generators: genus zero and genus one

Abstract

Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees , through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the -alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez' elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai's geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.
Paper Structure (62 sections, 30 theorems, 282 equations, 4 figures)

This paper contains 62 sections, 30 theorems, 282 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The canonical zeta generators in genus zero
  3. The canonical f-alphabet isomorphism
  4. The canonical zeta generators in genus one
  5. Motivation and outlook
  6. Background on multizeta values
  7. Real and formal multizeta values
  8. Real multizeta values, shuffle and stuffle multiplication
  9. Formal MZVs
  10. The Goncharov--Brown coaction
  11. Motivic MZVs
  12. Motivic versus formal multizetas, coproduct and coaction
  13. Reducible motivic MZVs
  14. Irreducible MZVs
  15. The Z-map associating polynomials to MZVs
  16. ...and 47 more sections

Key Result

Theorem 1.1.1

Write the expansion of $\Phi^{\mathfrak{m}}$ in $x,y$ in any basis of motivic multizetas adapted to the canonical decomposition canondec0, and for each $w\ge 2$, set Then the polynomials $g_w$ lie in ${\mathcal{MZ}}_w^\vee$. Equivalently, $g_w$ can be identified (with no reference to $\Phi^{\mathfrak{m}}$) as the unique polynomial in ${\mathcal{MZ}}_w^\vee$ satisfying the following three properti

Figures (4)

  • Figure 1: Contributions to the coaction formula (\ref{['Gonchco']}) for $\Delta_{GB} I(0;a_{1},a_{2},\ldots,a_n;1)$ from polygons with inner vertices $a_{i_1},a_{i_2}$, i.e. quadrilaterals associated with subsets of $a_1,a_2,\ldots,a_n$ of cardinality $r=2$.
  • Figure 2: Example of a contribution to $\Delta_{GB} \zeta^{\mathfrak{f}}(xxyxy)$ as computed in \ref{['eq:coex']}.
  • Figure 3: The loops ${\cal C}_x$ and ${\cal C}_y$ around $z=0$ and $z=1$ anchored at the origin (upper half) and their homotopy deformation to infinitesimal circles along with straight paths between zero and one in case of ${\cal C}_y$ (lower half) Ihara:1990. Strictly speaking, all the contours start and end at the tangential base point from 0 to 1 as indicated by the arrows at the origin pointing along the positive real axis. The straight line portions of the path in the lower-right panel should be viewed as running along the real axis between 0 and 1; they have been slightly separated for visual convenience.
  • Figure 4: The degeneration $\tau \rightarrow i \infty$ of the torus with coordinate $z$ (left panel) yields a nodal sphere, where the image of the A-cycle connecting $z=0$ with $z=1$ is drawn in two different coordinates $\sigma$ and $\eta$ (right panel). The image of the A-cycle in the $\eta$ coordinate (lower-right panel) matches the deformation of the loop ${\cal C}_y$ around $z=1$ in Figure \ref{['figsphere']}. Similar to Figure \ref{['figsphere']}, the straight line portions of all the paths should be viewed as running along the real axis between 0 and 1; they have been slightly separated for visual convenience.

Theorems & Definitions (73)

  • Theorem 1.1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.2.1
  • Theorem 1.3.1: see Theorem \ref{['thm:522']} (iii)
  • Definition 1
  • Remark 4
  • Definition 2
  • Definition 3
  • ...and 63 more