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

TL;DR

This paper constructs an 11-dimensional metric for AdS/CFT RG flows with SU(3) invariance, revealing geometric structures of compactified manifolds in supergravity and their relation to critical points and domain walls.

Contribution

It provides a detailed geometric construction of the 7-manifold metric in 11D supergravity with SU(3) invariance, including squashing and stretching parameters derived from supergravity vevs.

Findings

Derived the 11D metric as a warped product with squashed/stretch 7-sphere.

Identified differences between SU(3)xU(1) and G_2-invariant sectors.

Shared geometric features across sectors via common CP^2 structure.

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.