Twisted de Rham theory for string double copy in AdS

This work is motivated by the recent evidence for a double-copy relationship between open- and closed-string amplitudes in Anti-de Sitter (AdS) space. At present, the evidence has the form of a double-copy relation for string-amplitude building blocks, which are combined using the multiple-polylogarithm (MPL) generating functions. These generate MPLs relevant for all-order AdS curvature corrections of four-point string amplitudes. In this paper, we prove this building-block double copy using a new, noncommutative version of twisted de Rham theory. In flat space, the usual twisted de Rham theory is already known to be a natural framework to describe the Kawai-Lewellen-Tye (KLT) double-copy map from open- to closed-string amplitudes, in which the KLT kernel can be computed from the intersections of the open-string amplitude integration contours. We formulate twisted de Rham theory for noncommutative-ring-valued differential forms on complex manifolds and use it to derive the intersection number of two open-string contours, which are closed in the noncommutative twisted homology sense. The inverse of this intersection number is precisely the AdS double-copy kernel for the four-point open- and closed-string generating functions.

💡 Research Summary

The paper establishes a rigorous geometric foundation for the double‑copy relation between open‑ and closed‑string amplitudes in Anti‑de Sitter (AdS) space by extending twisted de Rham theory to non‑commutative, ring‑valued differential forms. In flat space, the Kawai‑Lewellen‑Tye (KLT) kernel that maps color‑ordered open‑string partial amplitudes to closed‑string amplitudes can be interpreted as the inverse of an intersection pairing between twisted cycles. The authors show that the same geometric picture survives in AdS once the curvature‑induced multiple‑polylogarithm (MPL) corrections are incorporated via a non‑commutative generating function.

The key technical obstacle is that individual MPLs are multivalued, preventing the usual definition of a single‑valued twist (\tau = d\log I). By packaging all MPLs into a non‑commutative series (L(e_0,e_1;z)=\sum_{\mathbf w} L_{\mathbf w}(z) , \mathbf e_{\mathbf w}) with formal letters (e_0, e_1) that do not commute, the authors obtain a twist (\tau = d\log L) that is single‑valued. Consequently, the flat connection (\nabla_\tau = d + \tau\wedge) can be used to define twisted homology and cohomology groups on the punctured complex plane (M = \mathbb C\setminus{0,1}).

Two families of twisted cycles are introduced:

• Open‑string cycles (\mathcal C_{\text{open}}) representing the generating function \