Second-Order Conformally Equivariant Quantization in Dimension 1|2

Details Second-Order Conformally Equivariant Quantization in Dimension 1|2 Second-Order Conformally Equivariant Quantization in Dimension 1|2

2009-12-28

arxiv.org/abs/0912.5190v1

SIGMA 5 (2009), 111, 11 pages

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2009.111

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 $\mathcal{K}(2)$ 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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