spo(2|2)-Equivariant Quantizations on the Supercircle $S^{1|2}$

Abstract

We consider the space of differential operators ${\mathcal{D}}_{\lambda \mu }$ acting between $\lambda$- and $\mu$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on ${\mathcal{D}}_{\lambda \mu }$ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space ${\mathcal{D}}_{\lambda \mu }$ and the associated graded space of symbols ${\mathcal{S}}_{\delta }$ ($\delta =\mu -\lambda$) can be considered as $\mathfrak{spo}(2|2)$-modules, where $\mathfrak{spo}(2|2)$ is the Lie superalgebra of contact projective vector fields on $S^{1|2}$. We show in this paper that there is a unique isomorphism of $\mathfrak{spo}(2|2)$-modules between ${\mathcal{S}}_{\delta }$ and ${\mathcal{D}}_{\lambda \mu }$ that preserves the principal symbol (i.e. an $\mathfrak{spo}(2|2)$-equivariant quantization) for some values of $\delta$ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages, arXiv:0912.5190] which were established for second-order differential operators. The method used here to build the $\mathfrak{spo}(2|2)$-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331, arXiv:1003.3320] to prove the existence of a $\mathfrak{pgl}(p+1|q)$-equivariant quantization on ${\mathbb{R}}^{p|q}$.

References in corpus (2)

• Natural and Projectively Invariant Quantizations on Supermanifolds
• On osp(p+1,q+1|2r)-equivariant quantizations

Cited by in corpus (1)

• Conformally equivariant quantization and symbol maps associated with $n$-ary differential operators on weighted densities