Some cohomology operators in 2-D field theory

Abstract

It is typical for a semi-infinite cohomology complex associated with a graded Lie algebra to occur as a vertex operator (or chiral) superalgebra where all the standard operators of cohomology theory, in particular the differential, are modes of vertex operators (fields). Although vertex operator superalgebras -with the inherent Virasoro action- are regarded as part of Conformal Field Theory (CFT), a VOSA may exhibit a square-zero operator (often, but not always, the semi-infinite cohomology differential) for which the Virasoro algebra acts trivially in the cohomology. Capable of shedding its CFT features, such a VOSA is called a ``topological chiral algebra'' (TCA). We investigate the semi-infinite cohomology of the vertex operator Weil algebra and indicate a number of differentials which give rise to TCA structures.