The binary invariant differential operators on weighted densities on the superspace R1∣n and cohomology

Abstract

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.