SIGMA 5 (2009), 111, 11 pages      arXiv:0912.5190      https://doi.org/10.3842/SIGMA.2009.111

Second-Order Conformally Equivariant Quantization in Dimension 1|2

Najla Mellouli
Institut Camille Jordan, UMR 5208 du CNRS, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Received September 22, 2009, in final form December 13, 2009; Published online December 28, 2009

Abstract
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.

Key words: equivariant quantization; conformal superalgebra.

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