Scalar Quartic Couplings in Type IIB Supergravity on $AdS_5\times S^5$

TL;DR

This work derives the complete quartic action for scalars s^I in type IIB supergravity on AdS_5×S^5, which are dual to extended chiral primary operators in N=4 SYM_4, by leveraging covariant equations of motion and careful field redefinitions. It reveals that the quartic action contains both 2- and 4-derivative vertices, and demonstrates that six-derivative terms can be removed, enabling a coherent Lagrangian formulation. The authors prove a consistent KK truncation to the massless N=8, d=5 supergravity multiplet at the quartic level and show the 4-derivative quartic couplings vanish when all fields are massless, reinforcing the KK consistency and supporting non-renormalization implications for certain n-point functions. Extremal quartic couplings are shown to vanish after an additional shift, implying non-renormalization of extremal correlators of single-trace CPOs; together with the KK-consistency results, this work provides a solid framework for computing 4-point functions of CPOs and exploring their coupling dependence at strong coupling.

Abstract

All quartic couplings of scalar fields $s^I$ that are dual to extended chiral primary operators in ${\cal N}=4$ SYM$_4$ are derived by using the covariant equations of motion for type IIB supergravity on $AdS_5\times S^5$. It is shown that despite some expectations if one keeps the structure of the cubic terms untouched, the quartic action obtained contains terms with two and four derivatives. It is shown that the quartic action vanishes on shell in the extremal case, e.g. k_1=k_2+k_3+k_4. Consistency of the truncation of the quartic couplings to the massless multiplet of the ${\cal N}=8$, d=5 supergravity is proven and the explicit values of the couplings are found. It is argued that the consistency of the KK reduction implies non-renormalization of $n$-point functions of $n-1$ operators dual to the fields from the massless multiplet and one operator dual to a field from a massive multiplet.