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Single-valued integration and superstring amplitudes in genus zero

Francis Brown, Clément Dupont

TL;DR

A canonical regularisation of both open and closed string perturbation amplitudes at tree level is defined, and it is deduced that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values.

Abstract

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper. Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

Single-valued integration and superstring amplitudes in genus zero

TL;DR

A canonical regularisation of both open and closed string perturbation amplitudes at tree level is defined, and it is deduced that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values.

Abstract

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper. Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

Paper Structure

This paper contains 54 sections, 59 theorems, 320 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The beta function
  3. General string amplitudes at tree level
  4. Contents
  5. Acknowledgements
  6. Dihedral coordinates and geometry of M0,S
  7. Dihedral extensions of moduli spaces
  8. Morphisms
  9. Strata
  10. Trivialisation maps
  11. Domains
  12. Logarithmic differential forms
  13. Residues
  14. Summary of structures
  15. Examples
  16. ...and 39 more sections

Key Result

Theorem 1.1

There is a canonical 'renormalisation' indexed by sets $J$ of non-crossing chords in an $N$-gon, where $\Omega_J^{\mathrm{ren}}$ is explicitly defined. The integrals on the right-hand side are convergent around $s_{ij}=0$. They are by definition products of convergent integrals over domains $X^\delta$ of various dimensions.

Figures (2)

  • Figure 1: On the left: three out of the five chords in a pentagon, corresponding to the dihedral coordinates $u_{24}$ (dashed), $u_{35}$, $u_{13}$ (dotted). The figure illustrates the relation $u_{24}=1-u_{13}u_{35}$. On the right: the five divisors on $\mathcal{M}_{0,5}^{\delta}$ defined by $u_{ij}=0$ form a pentagon. Two divisors intersect if and only if the corresponding chords do not cross.
  • Figure 2: An illustration of the trivialisation map $f_c^\gamma$ (the cyclic orientation $\gamma$ is clockwise, induced by the numbering).

Theorems & Definitions (142)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 132 more