The authors discuss the necessary and sufficient conditions (Theorems 2 and 3) for warped products of three-folds (\(M_3\)) and eight-folds (\(M_8\)) supporting fluxes to be solutions to the \(D=11\) supergravity equations of motion including the Duff-Stelle multi-membrane solution as well as its five-parameter non-supersymmetric deformations.
On their way to Theorems 2 and 3, in Lemma 1 the authors show that the existence of supersymmetric solutions (i.e., non-trivial sections of an 11D Spin bundle – Killing spinor) imply \(M_3\) is Einsteinian. Theorems 1–3 show that \(M_3\) is Ricci flat, and the \(M_8\) metric conformally scaled by the warp factor admits a covariantly constant spinor (or is Ricci flat) with the Levi-Civita connection constructed from the warped metric. The scalar appearing in the four-form flux ansatz is determined in terms of the warp factor and the scalar function in the warp factor is a harmonic function with respect to the original eight-dimensional metric. For simply connected eight-folds and irreducible conformal metrics therein, Theorem 4 is on a holonomy classification of the conformal eight-folds tied up to the relationship between the scalars figuring in the four-form flux and the warp factor. Given a finite set of points on the conformal eight fold and a finite set (of the same order) of positive numbers, Theorem 5 is about constructing a function in terms of Green’s function on the conformal eight-fold related to the warp factor, which would satisfy the \(D=11\) EOMs.
Examples of compact and non-compact Ricci-flat eight-folds are then discussed. The examples of eight-folds as warped products of six-folds with Riemann surfaces is discussed in detail. EOMs required the six-fold to be Ricci flat, and a warp-factor-like scalar function \(K\) constructed from a linear combination of the 11D and 8D warp-factor scalars, turns out to be harmonic on the Riemann surface. If \(K\) is a constant, the authors show that one obtains a special case of Theorem 2 – the conformal eight-fold metric is Ricci-flat – supersymmetry would further require the six-fold to be Calabi-Yau. For non-constant \(K\), some special cases are considered.
The authors then discuss the Duff-Stelle solution for multi-membrane \(D=11\) supergravity solutions (involving a Ricci-flat three-fold, a conformally flat eight-fold and the 11D and 8D warp factors and the scalar in the four-form flux, all dependent only on the radial coordinate) in their framework, by first rederiving the same. Next assuming only a conformally flat eight-fold and radial-coordinate-dependent warp factors and four-form flux components, the \(D=11\) EOMs yield a system of coupled second-order differential equations in the same. In Theorem 6, the authors show that the system is equivalent to a single third-order differential equation for a function constructed from first-order derivatives of the two scalars that figure in the 11D and 8D warp factors and the radial coordinate. (This, to the reviewer, is reminiscent of a single ordinary differential equation satisfied by a linear combination of metric perturbations invariant under infinitesimal diffeomorphisms, preserving a certain gauge, which is equivalent to a system of second-order differential equations in the individual metric perturbations.) This yields a five-parameter family of solutions to \(D=11\) supergravity. Some solutions corresponding to special cases, are explicitly obtained. The Duff-Stelle and Freund-Rubin solutions are re-obtained as special cases.