On the generality of the LLM geometries in M-theory

TL;DR

This work tests the claimed generality of the Lin–Lunin–Maldacena (LLM) class in $d=11$ supergravity by keeping all fluxes allowed by the $SO(6) \times SO(3) \times \mathbb{R}$ symmetry and analyzing the Killing spinor equations. The authors prove there are no supersymmetric geometries with nonzero flux density $\mathcal{I}$ within the LLM warped-product framework, extending LLM’s perturbative argument to a full nonperturbative result. They further show that, when $\mathcal{I}=0$, the relation between the two Killing spinors is forced to be $\varepsilon_- = - \gamma_5 \varepsilon_+$; even a broad class of potential generalisations using independent linear combinations of spinors does not yield new geometries. The result clarifies the rigidity of the LLM class and ties the absence of generalisations to the existence of two Killing directions associated with the $R$-symmetry.

Abstract

In this note we revisit the Lin, Lunin, Maldacena (LLM) class of d=11 supergravity solutions with symmetry SO(6) X SO(3) X R, but generalise to allow for all fluxes consistent with the isometries. Using the Killing spinor equation, we prove there are no supersymmetric geometries with additional fluxes beyond the LLM ansatz. In addition, the LLM relationship between Killing spinors, ε_- = - γ_5 ε_+, may be seen as a consequence of identifying two Killing directions identified through the Killing spinor equation corresponding to candidate R-symmetry directions.