First cohomology space of the affine Lie superalgebra 𝔞𝔣𝔣(3|1)

This paper investigates the first cohomology space associated with the embedding of the affine Lie superalgebra [Formula: see text] on the [Formula: see text]-dimensional superspace [Formula: see text] into the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients. Following the methodology of Ovsienko and Roger, we compute the cohomology of the affine Lie superalgebra [Formula: see text] with coefficients in the space of superpseudodifferential symbols [Formula: see text]. Our main result in Theorem 3.1 establishes that this cohomology space is purely even and one-dimensional, spanned by the nontrivial cocycle [Formula: see text]. We then lift this result to the full algebra of superpseudodifferential operators [Formula: see text] using a spectral sequence argument, proving that the first cohomology space of [Formula: see text] in [Formula: see text] is also one-dimensional. This work extends previous results for [Formula: see text] to the more complex case [Formula: see text], with explicit computations of the obstructions preventing the extension of cocycles from subalgebras.