  We construct a fully interacting holomorphic/topological theory in eleven
dimensions that is defined on products of Calabi-Yau fivefolds with real
one-manifolds. The theory describes a particular deformation of the cotangent
bundle to the moduli space of Calabi-Yau structures on the fivefold. Its field
content matches the holomorphic (or minimal) twist of the eleven-dimensional
supergravity multiplet recently computed by the second two authors, and we
offer numerous consistency checks showing that the interactions correctly
describe interacting twisted eleven-dimensional supergravity at the
perturbative level. We prove that the global symmetry algebra of our model on
flat space is an $L_\infty$ central extension of the infinite-dimensional
simple exceptional super Lie algebra $E(5,10)$, following a recent suggestion
of Cederwall in the context of the relevant pure spinor model. Twists of
superconformal algebras map to the fields of our model on the complement of a
stack of M2 or M5 branes, laying the groundwork for a fully holomorphic version
of twisted holography in this context.
