Differential operators on supercircle: conformally equivariant quantization and symbol calculus

Abstract

We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist.

References in corpus (7)

• Lie superalgebras of string theories
• Superconformally covariant operators and super W algebras
• Invariant operators on supermanifolds and Standard Models
• How to realize Lie algebras by vector fields
• Invariant bilinear differential operators
• Vector fields in the presence of a contact structure
• Symmetries of modules of differential operators

Cited by in corpus (28)

• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols
• Cohomology of $\mathfrak{osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}
• Conformal symbols and the action of contact vector fields over the superline
• Quantizations of modules of differential operators
• Symmetric Spaces in Supergravity
• Deformation of Vect($\mathbb{R})$-Modules of Symbols
• The Binary Invariant Differential Operators on Weighted Densities on the superspace ${\mathbb{R}}^{1|n}$ and Cohomology
• sl(2)-Trivial Deformations of Vect_{Pol}(R)-Modules of Symbols
• Second-Order Conformally Equivariant Quantization in Dimension 1|2
• Projectively equivariant quantizations over the superspace ${\mathbb{R}}^{p|q}$
• On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative
• On osp(p+1,q+1|2r)-equivariant quantizations
• spo(2|2)-Equivariant Quantizations on the Supercircle $S^{1|2}$
• Deformation of sl(2) and osp(1|2)-Modules of Symbols
• Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$
• Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator
• Cohomology of $\mathfrak{osp}(1|2)$ acting on the space of bilinear differential operators on the superspace ${\mathbb{R}}^{1|1}$
• Equivalence classes of subquotients of supersymmetric pseudodifferential operator modules
• Cohomology of $\mathfrak{aff}(1)$ and $\mathfrak{aff}(1|1)$ acting on the space of $n$-ary differential operators on the superspace ${\mathbb{R}}^{1|1}$
• Symetries of modules of differential operators on $S^{1|1}$
• The spaces ${\mathrm{H}}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$
• Conformally equivariant quantization and symbol maps associated with $n$-ary differential operators on weighted densities
• Second Cohomology Space of the Lie Superalgebra of contact vector fields $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on $S^{1|2}$
• On the matrix realization of the Lie superalgebra of contact projective vector fields $\mathfrak{spo}(2l+2|n)$
• The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$
• Heisenberg order of differential operators on the superspaces ${\mathbb{R}}^{2l+1|n}$
• Projectively Equivariant Quantization and Symbol calculus in dimension 1|2
• Cohomology of $\mathfrak{osp}(2|2)$ acting on the spaces of linear differential operators on the superspace ${\mathbb{R}}^{1|2}$