  We construct a generalization of Poisson-Chern-Simons theory, defined on any
supermanifold equipped with an appropriate filtration of the tangent bundle.
Our construction recovers interacting eleven-dimensional supergravity in
Cederwall's formulation, as well as all possible twists of the theory, and does
so in a uniform and geometric fashion. Among other things, this proves that
Costello's description of the maximal twist is the twist of eleven-dimensional
supergravity in its pure spinor description. It also provides a pure spinor
lift of the interactions in the minimally twisted theory. Our techniques
enhance the BV formulation of the interactions of each theory to a homotopy
Poisson structure by defining a compatible graded-commutative product; this
suggests interpretations in terms of deformations of geometric structures on
superspace, and provides some concrete evidence for a first-quantized origin of
the theories.
