On the generality of the LLM geometries in M-theory

In this note we revisit the Lin, Lunin, Maldacena (LLM) class of d = 11 supergravity solutions with symmetry SO(6) × SO(3) × 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- = -γ5ε₊, may be seen as a consequence of identifying two Killing directions identified through the Killing spinor equation corresponding to candidate R-symmetry directions.