Single-valued integration and superstring amplitudes in genus zero

TL;DR

This paper develops a mathematical framework using single-valued integration to analyze genus zero string amplitudes, proving they are single-valued projections of open string amplitudes and deriving the KLT formula.

Contribution

It introduces a canonical regularisation of string amplitudes using dihedral coordinates and establishes their Laurent expansions with coefficients as multiple zeta values.

Findings

Closed string amplitudes are single-valued projections of open string amplitudes.

Existence of a motivic Laurent expansion matching string amplitudes.

Derivation of the KLT formula relating closed and open string amplitudes.

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.