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We give a formulation of linearized 11D supergravity in 4D, $N=1$ superspacekeeping all eleven bosonic coordinates. The fields are fluctuations around$\mathbf M=\mathbf R^{4|4}\times Y$, where $Y$ is a background Riemannian7-manifold admitting a $G_2$ structure. We embed the 11D fields into superfieldrepresentations of the 4D, $N=1$ superconformal algebra. These consist of theconformal graviton superfield, seven conformal gravitino superfields, a tensorhierarchy of superfields describing the 11D 3-form, and a non-abelianKaluza-Klein vector multiplet gauging the tensor hierarchy by diffeomorphismson $Y$. The quadratic action consists of the linearization of a superspacevolume term and a Chern-Simons action for the gauged hierarchy coupled to thesupergravity and gravitino superfields, and the full structure is fixed bysuperconformal and gauge invariance. When this action is projected tocomponents, we recover the full linearized action of 11D supergravity.

Eleven-Dimensional Supergravity in 4D, $N=1$ Superspace