Analogs of Bol operators on superstrings

TL;DR

This paper classifies and discovers new analogs of Bol operators on superstrings, invariant under specific superalgebra substructures, extending classical differential operator theory into supergeometry.

Contribution

It provides a classification of Bol operator analogs on superstrings invariant under certain superalgebras, including new operators not previously known.

Findings

Classified analogs of Bol operators on superstrings.

Discovered many new invariant differential operators.

Extended classical invariant operator theory to supergeometry.

Abstract

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the $1|n$-dimensional supermanifold (superstring) $\mathcal{M}$, we classify analogs of Bol operators invariant under the simple maximal subalgebra $\mathfrak{h}$ of the same rank as its simple ambient superalgebra $\mathfrak{g}$ of vector fields on $\mathcal{M}$ and containing all elements of negative degree of $\mathfrak{g}$ in a $\mathbb{Z}$-grading. We also consider the Lie superalgebras of vector fields $\mathfrak{g}$ preserving a contact structure on the superstring $\mathcal{M}$. We have discovered many new operators.