Twisting pure spinor superfields, with applications to supergravity

We study a functor from two-step nilpotent super Lie algebras to sheaves of commutative differential graded algebras on the site of smooth $d$-manifolds, where $d$ is the dimension of the even subalgebra. The functor generalizes the pure spinor superfield formalism as studied in the physics literature. We prove that the functor commutes with deformations of the super Lie algebra by a Maurer–Cartan element, and apply the result to compute twists of various free supergravity theories and supersymmetric field theories of physical interest. 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.