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

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 $\mathrm{osp}(2|2)$. We study the spaces of linear differential operators on the spaces of weighted densities as modules over $\mathrm{osp}(2|2)$. We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as $\mathrm{osp}(2|2)$-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.