Cohomology and Deformation of $\mathfrak{aff}(1|1)$ Acting on Differential Operators

We consider the 𝔞𝔣𝔣(1|1)-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the second differential cohomology of the Lie superalgebra 𝔞𝔣𝔣(1|1) with coefficients in differential operators acting on the spaces of weighted densities. We classify formal deformations of the 𝔞𝔣𝔣(1|1)-module structure on the superspaces of symbols of differential operators. We prove that any formal deformation of a given infinitesimal deformation of this structure is equivalent to its infinitesimal part. This work is the simplest superization of a result by Basdouri [Deformation of 𝔞𝔣𝔣(1)-modules of pseudo-differential operators and symbols, J. Pseudo-differ. Oper. Appl. 7(2) (2016) 157–179] and application of work by Basdouri et al. [First cohomology of 𝔞𝔣𝔣(1) and 𝔞𝔣𝔣(1|1) acting on linear differential operators, Int. J. Geom. Methods Mod. Phys. 13(1) (2016)].