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 (Brown and Dupont in Single-valued integration and double copy, 2020). 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.
Highlights
I closed(ω, s) = s π m,dR I m(ω, s). It follows that the coefficients in the canonical Laurent expansion of the closed string amplitudes are single-valued multiple zeta values
We prove a folklore result that the Parke–Taylor forms are a basis of cohomology with coefficients
We have a simple normal crossing divisor D ⊂ Mδ0,S whose induced stratification defines an operad structure (15). This is a dihedral operad in the sense of [DV17]
Summary
It follows that the coefficients in the canonical Laurent expansion of the closed string amplitudes are single-valued multiple zeta values This theorem, for periods (i.e., assuming the period conjecture) was conjectured in [Sti14,ST14] and proved independently by a very different method from our own in [SS19]. It uses in an essential way the fact that the zeros of dihedral coordinates are normal crossing. A cyclic structure on S defines an orientation on the cell X δ and fixes the sign of αS,δ if we demand that its integral over X δ be positive In simplicial coordinates it is given by [Bro[09], (7.1)]: n−1. With a sign that is compatible with the single-valued Fubini theorem
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