Differential operators on supercircle: conformally equivariant quantization and symbol calculus
نویسنده
چکیده
We consider the supercircle S equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S 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.
منابع مشابه
Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.
متن کاملConformally Equivariant Quantization – a Complete Classif ication
Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ + δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, an...
متن کاملConformally equivariant quantization
Let (M, g) be a pseudo-Riemannian manifold and Fλ(M) the space of densities of degree λ on M . We study the space D2 λ,μ(M) of second-order differential operators from Fλ(M) to Fμ(M). If (M, g) is conformally flat with signature p− q, then D2 λ,μ(M) is viewed as a module over the group of conformal transformations of M . We prove that, for almost all values of μ− λ, the O(p+1, q+1)-modules D2 λ...
متن کاملA Geometric Interpretation and Explicit Form for Higher-order Hankel Operators
where x(z) = ∑∞ j=1 xjz j is the symbol of the operator. In terms of the Hardy polarization of odd spinors on S, B1(x) = P+MxP−, where P+, P− are projections and Mx is the multiplication operator corresponding to x. The map x 7−→ B1(x) is conformally equivariant in a sense explained in §2. In this paper we derive explicit expressions for certain well-known higher-order generalizations of this m...
متن کامل0 O ct 1 99 9 Methods of Equivariant Quantization
This article is a survey of recent work [15, 6, 7, 13] developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, we establish a canonical isomorphism between the space of symmetric contravar...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008