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Analogs of Bol operators on superstrings

Sofiane Bouarroudj, Dimitry Leites, Irina Shchepochkina

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.

Analogs of Bol operators on superstrings

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 -dimensional supermanifold (superstring) , we classify analogs of Bol operators invariant under the simple maximal subalgebra of the same rank as its simple ambient superalgebra of vector fields on and containing all elements of negative degree of in a -grading. We also consider the Lie superalgebras of vector fields preserving a contact structure on the superstring . We have discovered many new operators.

Paper Structure

This paper contains 25 sections, 134 equations.

Table of Contents

  1. Introduction
  2. Veblen's problem
  3. Feasible versions of Veblen's problem
  4. The problems we consider
  5. Super Bol operators ${\mathcal{F}}_\lambda\longrightarrow {\mathcal{F}}_\mu$ in terms of $\theta$'s and $\xi\eta$'s
  6. Comments
  7. Dualization and permutation of arguments
  8. Weighted densities
  9. Weighted densities on supermanifolds with a contact structure
  10. Case ${\mathfrak{k}}(2n+1|m)$.
  11. Case ${\mathfrak{m}}(n)$.
  12. Generating functions: contact series and their subalgebras
  13. Divergences
  14. Divergences on ${\mathfrak{k}}(2n+1 |m)$.
  15. The case ${\mathfrak{k}}(1 |2)$
  16. ...and 10 more sections

Theorems & Definitions (4)

  • proof
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