Associators for AdS string amplitude building blocks

TL;DR

This work identifies a precise algebraic mechanism for AdS string amplitude building blocks: open-string AdS blocks are generated by the Drinfeld associator, while closed-string AdS blocks are generated by the Deligne associator, mirroring the flat-space recursions. By formulating Selberg-type integrals with polylog insertions and solving associated KZ equations, the authors derive all-order Drinfeld and Deligne recursions that relate lower- and higher-point boundary data, producing all-order low-energy expansions with coefficients in the rings of multiple zeta values ($\mathcal{Z}$) for open strings and single-valued multiple zeta values ($\mathcal{Z}^{\text{sv}}$) for closed strings. The framework extends to iterated integrals and higher-point analogs, linking AdS amplitudes to single-valued maps and providing a structured path toward understanding KLT-like relations and cohomological aspects in AdS. The results offer a unifying algebraic viewpoint for AdS amplitude building blocks and open doors to elliptic generalizations and intersection-theory connections in string amplitudes.

Abstract

We show that building blocks for open- and closed-string amplitudes on AdS are generated by the Drinfeld and Deligne associator, respectively. Our formalism lifts the known associator recursions for flat-space string amplitudes to the AdS picture. This delivers another proof that the AdS building blocks admit low-energy expansions with (single-valued) multiple zeta values as coefficients and provides all-order relations for the integral expressions.