Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols

Abstract

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this $\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\le 4$. This work is the simplest superization of a result by Bouarroudj [On $\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127]. Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology of the Lie superalgebras of contact vector fields on $1|N$-dimensional superspace.

References in corpus (12)

• Lie superalgebras of string theories
• Differential operators on supercircle: conformally equivariant quantization and symbol calculus
• Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on ${\dot{T}}^{\ast }{S}^1$
• Superconformally covariant operators and super W algebras
• Invariant operators on supermanifolds and Standard Models
• Cohomology of $\mathfrak{osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}
• On the Cohomology of the Lie Superalgebra of Contact Vector Fields on $S^{1|2}$
• Deformations of modules of differential forms
• Three Cocycles on $\Diff(S^1)$ Generalizing the Schwarzian Derivative
• Deformation of Vect($\mathbb{R})$-Modules of Symbols
• Invariant bilinear differential operators
• sl(2)-Trivial Deformations of Vect_{Pol}(R)-Modules of Symbols

Cited by in corpus (9)

• Cohomology of $\mathfrak{osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}
• Conformal symbols and the action of contact vector fields over the superline
• The Binary Invariant Differential Operators on Weighted Densities on the superspace ${\mathbb{R}}^{1|n}$ and Cohomology
• Deformation of sl(2) and osp(1|2)-Modules of Symbols
• Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$
• Cohomology of $\mathfrak{osp}(2|2)$ acting on the spaces of linear differential operators on the superspace ${\mathbb{R}}^{1|2}$
• The Binary $\mathfrak{aff}(n|1)$-Invariant Differential Operators On Weighted Densities On The Superspace ${\mathbb{R}}^{1|n}$ And $\mathfrak{aff}(n|1)$-Relative Cohomology
• Deformation of $\mathfrak{osp}(2|2)$-Modules of Symbols
• Second Cohomology Space of the Lie Superalgebra of contact vector fields $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on $S^{1|2}$