Differential operators on supercircle: conformally equivariant quantization and symbol calculus

TL;DR

This paper explores the structure of differential operators on the supercircle, establishing a conformally equivariant symbol calculus and identifying special resonant cases where canonical isomorphisms fail.

Contribution

It introduces a canonical isomorphism between differential operators and symbols on the supercircle and analyzes the conditions under which this isomorphism exists or breaks down.

Findings

Established a conformally equivariant symbol calculus on the supercircle

Identified resonant cases where the isomorphism does not exist

Analyzed the module structure over the M"obius superalgebra osp(1|2)

Abstract

We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist.