  We study twists of supergravity theories and supersymmetric field theories,
using a version of the pure spinor superfield formalism. Our results show that,
just as the component fields of supersymmetric multiplets are the vector
bundles associated to the equivariant Koszul homology of the variety of
square-zero elements in the supersymmetry algebra, the component fields of the
holomorphic twists of the corresponding multiplets are the holomorphic vector
bundles associated to the equivariant Koszul homology of square-zero elements
in the twisted supersymmetry algebra. The BRST or BV differentials of the free
multiplet are induced by the brackets of the corresponding super Lie algebra in
each case. We make this precise in a variety of examples; applications include
rigorous computations of the minimal twists of eleven-dimensional and type IIB
supergravity, in the free perturbative limit. The latter result proves a
conjecture by Costello and Li, relating the IIB multiplet directly to a
presymplectic BV version of minimal BCOV theory.
