Closed-string amplitude recursions from the Deligne associator

TL;DR

This paper derives a recursive relation for closed-string tree-level amplitudes by connecting Selberg integrals, the Knizhnik-Zamolodchikov equation, and the Deligne associator, building on open-string amplitude recursions and single-valued periods.

Contribution

It introduces a novel recursive formula for closed-string amplitudes using the Deligne associator and Lie algebra representations, extending open-string recursion results.

Findings

Closed-form recursion for closed-string amplitudes.

Connection between Selberg integrals and KZ equation.

Identification of amplitude limits via the Deligne associator.

Abstract

Inspired by earlier results on recursions for open-string tree-level amplitudes, and by a result of Brown and Dupont relating open- and closed-string tree-level amplitudes via single-valued periods, we identify a recursive relation for closed-string tree-level amplitudes. We achieve this by showing that closed-string analogues of Selberg integrals satisfy the Knizhnik-Zamolodchikov equation for a suitable matrix representation of the free Lie algebra on two generators, and by identifying the limits at z=1 and z=0, which are related by the Deligne associator, with N-point and (N-1)-point closed-string amplitudes, respectively.