The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$

TL;DR

This paper extends the construction of fine equivariant quantizations to super circles $S^{1|n}$ for $n \\geq 3$, using Lie superalgebra techniques and Casimir operators to establish existence and uniqueness.

Contribution

It generalizes known results from $S^{1|1}$ and $S^{1|2}$ to higher super dimensions, introducing a new quantization method for $ ext{spo}(2|n)$-contact vector fields.

Findings

Constructed the fine equivariant quantization on $S^{1|n}$ for $n \\geq 3$.

Proved the uniqueness of the quantization using Casimir operators.

Extended the model from purely even cases to super circles with higher $n$.

Abstract

In this paper, we generalize the known results on the super circles $S^{1|1}$ and $S^{1|2}$. We construct the fine equivariant quantization on the super circle $S^{1|n}$ for $n\geqslant 3$. The equivariant Lie superalgebra is $\spo(2|n)$ which is constituted of the contact projective vector fields on $S^{1|n}$. In order to construct the fine equivariant quantization on $S^{1|n}$, we use the model developed, in the purely even case, by Charles H. Conley and Valentin Ovsienko in \textit{[Linear Differential Operators on Contact manifolds, http://www.arxiv:math-Ph/1205.6562v1,24p, 2012]}. We also use the technical of Casimir operators to prove the uniqueness of the fine quantization on $S^{1|n}$. The technical of Casimir operators used here is the same as the one used by P. Mathonet and F. Radoux in [\textit{Lett. Math. Phys. 98 (2011),311-331}] to prove the existence of a $\pgl(p+1|q)$-equivariant quantization on $\R^{p|q}$.