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Pré-Publication, Document De Travail Année : 2009

Conformally equivariant second-order differential operators in dimension 1|2: Quantization and symbol calculus

Najla Mellouli
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Résumé

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle $S^{1|2}$ equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on $S^{1|2}$ contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.
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hal-00424791 , version 1 (19-10-2009)

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Najla Mellouli. Conformally equivariant second-order differential operators in dimension 1|2: Quantization and symbol calculus. 2009. ⟨hal-00424791⟩
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