Double copy relation and KLT relation in AdS space

Double copy relation and KLT relation reveal that gravity and closed strings are, in a precise sense, built from two copies of gauge theory and open strings. Beyond their computational power, they point to a deeper structural connection between gauge theory, gravity, and the geometry of the string worldsheet. It is therefore natural to ask how much of this structure survives away from flat spacetime, in particular in Anti-de Sitter space.

Substantial progress has already been made in AdS. At the supergravity level, Xinan Zhou showed that tree-level four-point reduced Mellin amplitudes in AdS$_5$ obey an AdS analogue of color-kinematics duality and double copy: reduced amplitudes of bi-adjoint scalars, super Yang-Mills, and supergravity can be organized into a common cubic structure, with the gravity result obtained by a kinematic replacement analogous to the BCJ double copy in flat space. This demonstrated explicitly that double-copy structures are not restricted to translationally invariant flat backgrounds. More recently, higher-point AdS supergluon amplitudes have become increasingly tractable through factorization, flat-space limits, recursion, and suitable R-symmetry bases.

At the string level, four-point AdS Virasoro-Shapiro and AdS Veneziano amplitudes have now been constructed perturbatively in the inverse AdS radius, most naturally using reduced amplitudes and emergent worldsheet representations. These results suggest a striking open/closed-string relation also in curved spacetime. Indeed, recent work has established an AdS analogue of the KLT relation at the level of the worldsheet building blocks: the open- and closed-string integral bases are related by an exactly computable AdS KLT kernel, and this structure admits an interpretation in terms of a noncommutative version of twisted de Rham theory.

What is still missing, however, is a comparably direct and universal KLT relation formulated at the level of the physical AdS amplitudes themselves. In particular, it remains unclear how the KLT structure of the underlying worldsheet basis combines with the dynamical coefficients, Kaluza-Klein dependence, crossing structure, and curvature corrections that assemble the full AdS Veneziano and Virasoro-Shapiro amplitudes.

This leads to two closely related questions:

1. Can the AdS double copy be extended systematically to higher-point amplitudes?

In flat space, color-kinematics duality and double copy are intrinsically (n)-point structures. In AdS, the best-understood explicit double-copy construction is still at four points, while higher-point Mellin amplitudes are now accessible by independent bootstrap and factorization methods. Can one identify an appropriate higher-point kinematic algebra or representation in which AdS gauge-theory amplitudes satisfy generalized Jacobi relations, and from which higher-point gravitational amplitudes follow by a genuine double-copy operation?

2. Can one derive an amplitude-level AdS KLT relation?

The existing AdS KLT construction relates the worldsheet integral building blocks underlying four-point open- and closed-string amplitudes. Can this be lifted to a relation directly between the complete AdS Veneziano and AdS Virasoro-Shapiro amplitudes, with an AdS deformation of the flat-space KLT kernel? More ambitiously, can such a formulation be generalized to arbitrary external Kaluza-Klein modes and eventually to higher multiplicity?

A successful answer would provide a conceptual bridge between the emerging AdS worldsheet description and the algebraic double-copy structure familiar from flat-space scattering amplitudes. It could also clarify which ingredients of color-kinematics duality and open/closed-string factorization are genuinely background independent, and which are special consequences of flat-space momentum conservation.

Note. At the level of unreduced amplitudes, Yichao Tang and Bo Wang have already developed a construction of the AdS Veneziano amplitude with general Kaluza-Klein dependence; see the accompanying problem. An analogous strategy should in principle be applicable to constructing the unreduced AdS Virasoro-Shapiro amplitude. Once both unreduced open- and closed-string amplitudes are available in a common language, they provide a particularly clean arena in which to search for an amplitude-level AdS KLT relation.