Holographic Correlators Beyond Maximal Supersymmetry

TL;DR

This work provides explicit holographic calculations of 3pt-functions in a 4d N=1 SCFT (the Leigh-Strassler theory) using a 10-scalar consistent truncation of 5d gauged supergravity, offering rare analytic results for non-BPS scalar operators in a strongly coupled CFT. The authors carefully navigate derivative-couplings, extremal configurations, and finite boundary terms, establishing a robust framework that passes multiple checks against 4d N=1 superconformal Ward identities and blocks. They demonstrate that a frame where all derivative cubic couplings vanish can reproduce the same physical correlators as the original frame, clarifying ambiguities in extremal correlators via the single-particle operator construction. The work also analyzes an alternative truncation including gauge fields to probe higher-point functions and discusses future directions toward 4pt-functions and broader classes of holographic CFTs, highlighting the broader utility of top-down AdS/CFT in less-than-maximally supersymmetric settings.

Abstract

We use the AdS/CFT correspondence to explicitly calculate some of the three-point functions in the planar limit of the 4d $\mathcal{N}=1$ Leigh-Strassler SCFT. This strongly interacting CFT can be obtained as a mass deformation of the 4d $\mathcal{N}=4$ SYM theory and admits a dual description in terms of an AdS$_5$ background of type IIB supergravity. Our analysis is based on the existence of a consistent truncation of the 10d supergravity to a tractable 5d gravitational theory with 10 scalar fields dual to some of the low-lying operators in the spectrum of the LS SCFT. We apply standard holographic techniques to this 10-scalar model to analytically calculate the correlators of interest and thus provide a rare example of explicit three-point functions of scalar non-BPS operators in strongly coupled 4d CFTs. Using superconformal Ward identities we perform several consistency checks of these holographic correlators. As a byproduct of our analysis we discuss some subtleties related to the calculation of extremal correlators in AdS/CFT and the contribution of scalar derivative couplings to the evaluation of Witten diagrams. Our work provides a blueprint for the holographic calculation of other correlators in the LS SCFT and similar top-down holographic models.

Figures (3)

• Figure 1: A schematic representation of the UV and IR conformal manifolds and the RG flow between them. The orange lines represent the 1d complex submanifolds of $\mathcal{M}_{\mathbb C}^{\rm UV}$ and $\mathcal{M}_{\mathbb C}^{\rm IR}$ along which the global symmetry is not broken, $\rm SO(6)$ in the UV and ${\rm U(1)_R}\times{\rm SU(2)_F}$ in the IR, respectively. The blue surface is the 2d complex submanifold along which the global symmetry is broken to its maximal Cartan subgroup, i.e. $\rm U(1)^3$ in the UV and ${\rm U(1)_R}\times{\rm U(1)_F}$ in the IR. Finally, on a generic point of the conformal manifolds belonging to the grey region there is no continuous flavor symmetry and only the $\rm U(1)_R$ symmetry of the SCFT is preserved. We hasten to add that this is only a cartoon since not much is known about the global properties of both conformal manifolds.
• Figure 2: 3pt-Witten diagrams. On the left: non-derivative cubic vertex $c_{ijk}$. On the right: cubic vertex $d_{ijk}$ with derivatives on $\Phi_j$ and $\Phi_k$.
• Figure 3: Witten diagrams relevant for the computation of 4pt-correlators of $\mathcal{O}_{\beta_1}$ and $\mathcal{O}_{\beta_2}$. From left to right: contact diagram, $s$-channel scalar and graviton exchange diagrams.