Screenings and a universal Lie-de Rham cocycle

Abstract

Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology classes of a configuration space. 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 Kac-Moody) 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 groups of the above local systems to appropriate higher Ext-groups between the Fock space representations. Our construction is motivated by, and in a special case reduces to the construction of Bowknegt et al, see [BMP1], [BMP2].