ConclusionsThe pure spinor formalism is now well-developed. But the Koszul duality was not widely used, at least not in the string theory literature. We tried to study this duality by applying it to the Type IIB supergravity, more specifically to the classification of the linearized excitations of homogeneous spaces. We had to invent a modification of the duality, which involves the “left” and “right” quadratic algebras glued together in some special way. This gives an elegant description of the ghost number 1 operators, and a puzzle for the ghost number 2 operators.More research is needed in this direction.