Determine the integration cycles for four-point Grassmannian integrals

Determine the integration cycles and contour prescriptions required to evaluate the four non-trivial Grassmannian integrals for the AdS5 boundary four-point correlators in spinor-helicity variables, and clarify their relation to the discontinuities represented by the Grassmannian construction.

Background

The paper constructs AdS5 exchange correlators by matching their Grassmannian residues to products of three-point functions. For four-point functions, the Grassmannian representation contains four non-trivial integrations, whereas the discontinuity used in the factorization construction localizes two minors. The authors note that evaluating the remaining integrals requires choices of contours and prescriptions, and that understanding which choices reproduce the relevant momentum-space discontinuities remains unresolved.

References

Given the fact that there are four non-trivial integrals to perform to obtain the spinor-helicity result, we have many prescriptions and possible contour choices to evaluate the integral. A detailed study of the integration procedure would be very interesting and is left to future work.

— Four-point functions, Twistors and Supersymmetry in the Symplectic Bi-Grassmannian for CFT$_4$ and AdS$_5$  (2608.14451 - S, 14 Aug 2026) in Section 2, subsection “The singularity structure” (discussion following the Euclidean AdS5 four-point function)

It consistently factorizes at \mathcal{S}=0 and \mathcal{T}=0 and thus represents the correct correlator up to possible contact diagram contributions. We leave a more detailed study of supergravity to the future.

— Four-point functions, Twistors and Supersymmetry in the Symplectic Bi-Grassmannian for CFT$_4$ and AdS$_5$  (2608.14451 - S, 14 Aug 2026) in Section 6, “Extension to Supergravity”