Twisted de Rham theory for string double copy in AdS

TL;DR

This work develops a noncommutative twisted de Rham framework to realize a string-theoretic AdS double copy, extending flat-space KLT concepts to curved backgrounds. By encoding AdS curvature corrections in a noncommutative MPL generating function and pairing open/closed string building blocks via twisted (co)homology, the authors show the AdS double-copy kernel arises as the inverse of a twisted intersection pairing. The approach unifies open- and closed-string AdS amplitudes through a geometry of twisted cycles and cocycles, with Poincaré duality and intersection numbers acting as organizing principles. The four-point analysis demonstrates a concrete, kernel-based relation $I = J K J^{ m R}$, suggesting a robust, background-spanning framework with potential extensions to higher points, loops, and other curved spacetimes.

Abstract

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.

Figures (4)

• Figure 1: All relevant pairings of twisted de Rham theory as applied to string amplitudes.
• Figure 2: Regularized twisted cycle ${\rm reg}(0,1)$. It consists of two circles $S(i,\epsilon)$ centered at $i \in \{0,1\}$ of radius $\epsilon$ oriented anticlockwise. These are connected via a closed interval $[\epsilon, 1-\epsilon]$ along the real axis. The wiggly lines denote the branch cuts, both oriented along the negative imaginary axis.
• Figure 3: Sinusoid-shaped twisted cycle ${\rm sin}(0,1)$. The black contour is the regularized cycle ${\rm reg}(0,1)$, whereas the blue contour is the sinusoidal deformation of the real-axis interval $(0,1)$. ${\rm sin}(0,1)$ differs from $(0,1)$ by two semi-circular regions enclosed between them and is therefore in the same equivalence class.
• Figure 4: Arc-shaped twisted cycle ${\rm arc}(0,1)$. The black contour is the regularized cycle ${\rm reg}(0,1)$, whereas the blue contour is an arched deformation of the real-axis interval $(0,1)$. ${\rm arc}(0,1)$ differs from $(0,1)$ by the big semi-circular region enclosed between them and is therefore in the same equivalence class.