Cohomology of 𝔞𝔣𝔣(m|1) acting on the space of superpseudodifferential operators on the supercircle S1|m

Abstract

We investigate the first differential cohomology space associated with the embedding of the affine Lie superalgebra [Formula: see text] on the [Formula: see text]-dimensional supercircle [Formula: see text] in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text]. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles. We study the deformations of the structure of the [Formula: see text]-module [Formula: see text]. We prove that any formal deformation is equivalent to its infinitesimal part.