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

TL;DR

This paper constructs a unique $ ext{spo}(2|2)$-equivariant quantization on the supercircle $S^{1|2}$, providing explicit formulas and extending previous results from second-order operators to a broader class.

Contribution

It establishes the existence and explicit form of a unique $ ext{spo}(2|2)$-equivariant quantization on $S^{1|2}$ for non-critical values of $ ext{delta}$, extending prior work to general differential operators.

Findings

Existence of a unique $ ext{spo}(2|2)$-equivariant quantization for non-critical $ ext{delta}$.

Explicit formula for the $ ext{spo}(2|2)$-equivariant quantization.

Extension of previous second-order results to general differential operators.

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 $\mathbb{R}^{p|q}$.