Conformal symbols and the action of contact vector fields over the superline

TL;DR

This paper investigates the action of the Lie superalgebra of contact vector fields on the supersymmetric line, focusing on modules of differential operators, conformal symbols, and related geometric and algebraic structures.

Contribution

It provides explicit computations of the action of contact vector fields on modules, introduces conformal symbols, and explores geometric subsymbols, cohomology, and symmetries of these modules.

Findings

Computed the action of contact vector fields on modules.

Identified geometric subsymbols and 1-cohomology.

Analyzed symmetries and subquotients of modules.

Abstract

Let K be the Lie superalgebra of contact vector fields on the supersymmetric line. We compute the action of K on the modules of differential and pseudodifferential operators between spaces of tensor densities, in terms of their conformal symbols. As applications we deduce the geometric subsymbols, 1-cohomology, and various uniserial subquotients of these modules. We also outline the computation of the K-equivalences and symmetries of their subquotients.