Coaction and double-copy properties of configuration-space integrals at genus zero

TL;DR

This paper explores the structure of configuration-space integrals on punctured Riemann spheres, revealing new methods for their expansion, coaction, and relations to string theory and conformal field theory, with implications for mathematical physics.

Contribution

It introduces explicit bases for twisted cycles and cocycles, develops efficient expansion techniques, and uncovers a recursion for a generalized KLT kernel linked to intersection numbers.

Findings

Explicit bases simplify coaction calculations.

New generating-function techniques for polylogarithm coaction.

A recursion for the generalized KLT kernel derived from intersection theory.

Abstract

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension $\alpha'$ or the dimensional-regularization parameter $\epsilon$ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in $(\mathfrak{p},2)$ minimal models, which can be normalized to become uniformly transcendental in the $\mathfrak{p} \to \infty$ limit.