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Derive cubic KK couplings for Dp-brane backgrounds

Derive the cubic effective action and explicit cubic couplings for the Kaluza–Klein supergravity modes around the near-horizon Dp-brane type IIA/IIB supergravity backgrounds given in equation (eq:NHsugra), in order to enable holographic three-point correlation function calculations for maximally supersymmetric Yang–Mills theories via the auxiliary AdS prescription.

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Background

To compute holographic 3-point functions using the auxiliary AdS_{p+2+η} structure, one needs the cubic interactions among the supergravity KK modes around the Dp-brane near-horizon backgrounds. While such cubic couplings are known for AdS5×S5, they have not been derived for the non-conformal Dp-brane geometries described by eq. (eq:NHsugra).

The authors emphasize that obtaining these couplings directly in 10d supergravity is technically challenging and remains unperformed for Dp-branes, preventing a complete holographic 3-point function computation with fully fixed normalization and operator dependence.

References

Deriving this cubic effective action by a direct calculation in 10d supergravity is technically involved even for the AdS$_5 \times S5$ background, see, and has not been performed for the D$p$-brane backgrounds in~eq:NHsugra. This is an important open problem that we leave for the future.

eq:NHsugra:

ds102=(gU)7−p2dsp+12+(gU)p−72(dU2+U2dΩ8−p2) ,eΦ=gs(gU)(p−3)(7−p)4 ,Cp+1=gs−1(gU)7−pvolp+1 ,\begin{split} ds^2_{10} &= (gU)^{\frac{7-p}{2}} ds^2_{p+1} + (gU)^{\frac{p-7}{2}}(dU^2 + U^2 d\Omega^2_{8-p})\,,\\ e^{\Phi}&= g_s (gU)^{\frac{(p-3)(7-p)}{4}}\,, \quad\qquad C_{p+1} = g_s^{-1}(gU)^{7-p} {\rm vol}_{p+1}\,, \end{split}

— Correlation functions for non-conformal D$p$-brane holography (2503.18770 - Bobev et al., 24 Mar 2025) in Subsection 2.2 (Auxiliary AdS)