Screenings and a Universal Lie-de Rham Cocycle

In the pioneering paper [FF], Feigin and Fuchs have constructed intertwining operators between ”Fock-type” modules over the Virasoro algebra via contour integrals of certain operator-valued one dimensional local systems over top homology classes of a configuration space. Similar constructions exist for affine Lie algebras. Key ingredients in such a construction are the so called ”screening operators”. The main observation of the present paper is that the screening operators contain more information. Specifically, at the chain level, the screening operators provide a certain canonical cocycle of the Virasoro (resp. affine) Lie algebra with coefficients in the de Rham complex of an operator-valued local system on the configuration space. This way we obtain canonical morphisms from higher homology spaces of the above local systems to appropriate higher Ext -groups between the Fock space representations. The screening operators that we are interested in this paper are linear maps S : M1 → M2[[z, z]] , where M1 and M2 are certain modules over the Lie algebra g in question, e.g., an affine Lie algebra. We think of such an operator as