Projectively equivariant quantization and symbol on supercircle S^1∣3

Abstract

Let ${{\cal D}_{\lambda ,\mu }}$ be the space of linear differential operators on weighted densities from ${{\cal F}_\lambda }$ to ${{\cal F}_\mu }$ as module over the orthosymplectic Lie superalgebra $\mathfrak{osp}(3\left| 2 \right.)$, where ${{\cal F}_\lambda }$, $\lambda \in \mathbb{C}$ is the space of tensor densities of degree λ on the supercircle S^1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.