The 11-dimensional Metric for AdS/CFT RG Flows with Common SU(3) Invariance

TL;DR

This work constructs the complete 11-dimensional metric for AdS_4 RG flows with common SU(3) invariance by employing the de Wit–Nicolai–Warner nonlinear metric ansatz in the SU(3)-singlet sector of d=4, ${\cal N}=8$ gauged supergravity. The authors decode the SU(3)-invariant vevs into geometric deformation parameters $(a,b,c,d)$ and derive the 7-manifold metric as a warped, squashed and stretched ${\bf S}^7$, providing explicit local frames and a warp factor, and they compare the SU(3)×U(1) and G_2-invariant sectors, both sharing a ${\bf CP}^2 \cong S^4$ core. Section 5 presents a geometric construction using an ${\mathbb R}^8$ embedding and Hopf fibration on ${\bf CP}^3$ to match the dWNW output with a concrete 7-manifold metric, yielding explicit expressions for the warp factor $\Delta$ and the siebenbeins. The discussion outlines the need to complete the 11D uplift by solving the full Einstein–Maxwell equations and posits a route via an ansatz for the 3-form field, highlighting the framework’s potential to illuminate the M-theory lifts of the entire ${\cal SU}(3)$-invariant sector and its RG flows.

Abstract

The compact 7-manifold arising in the compactification of 11-dimensional supergravity is described by the metric encoded in the vacuum expectation values(vevs) in d=4, N=8 gauged supergravity. Especially, the space of SU(3)-singlet vevs contains various critical points and RG flows(domain walls) developing along AdS_4 radial coordinate. Based on the nonlinear metric ansatz of de Wit-Nicolai-Warner, we show the geometric construction of the compact 7-manifold metric and find the local frames(siebenbeins) by decoding the SU(3)-singlet vevs into squashing and stretching parameters of the 7-manifold. Then the 11-dimensional metric for the whole SU(3)-invariant sector is obtained as a warped product of an asymptotically AdS_4 space with a squashed and stretched 7-sphere. We also discuss the difference in the 7-manifold between two sectors, namely SU(3)xU(1)-invariant sector and G_2-invariant sector. In spite of the difference in base 6-sphere, both sectors share the 4-sphere of CP^2 associated with the common SU(3)-invariance of various 7-manifolds.