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Gordan-Rankin-Cohen operators on the spaces of weighted densities in superdimension $1\vert 1$

V. Bovdi, D. Leites

Abstract

The modular forms and weighted densities over the 1-dimensional manifold $M$ are transformed ``alike" under the group of linear fractional changes of coordinates, so the classifications of differential operators between spaces of (A) modular forms and (B) weighted densities are sometimes identified, although they are different. Here, we solve problem B for superstrings in superdimension $(1\vert 1)$ -- superizations of the result of arXiv:2404.18222. Open problems are offered.

Gordan-Rankin-Cohen operators on the spaces of weighted densities in superdimension $1\vert 1$

Abstract

The modular forms and weighted densities over the 1-dimensional manifold are transformed ``alike" under the group of linear fractional changes of coordinates, so the classifications of differential operators between spaces of (A) modular forms and (B) weighted densities are sometimes identified, although they are different. Here, we solve problem B for superstrings in superdimension -- superizations of the result of arXiv:2404.18222. Open problems are offered.
Paper Structure (21 sections, 68 equations)

This paper contains 21 sections, 68 equations.

Table of Contents

  1. Introduction
  2. Weighted densities vs. modular forms
  3. Vital difference and a cause of confusions.
  4. On multidimensional generalizations and superizations.
  5. The GRC-operators on the $n=1$ superstring with a contact structure
  6. ${\mathcal{S}}_c^{1|n}$ and $\alpha_{1}$
  7. Classification of ${\mathfrak{osp}}(1|2)$-invariant GRC-operators
  8. The answer.
  9. The GRC-operators and the algebra structure on the space of all weighted densities
  10. Open problem.
  11. The GRC-operators on the general $(1|1)$-dimensional superstring
  12. On ${\mathfrak{vect}}(1|1)$-modules induced from ${\mathfrak{gl}}(1|1)$-irreducibles
  13. A description of irreducible ${\mathfrak{gl}}(1|1)$-modules with highest (or lowest) weight vector
  14. The tensor product $\text{irr}(M^{\lambda;\, \mu})\otimes \text{irr}(M^{\sigma ; \, \rho })$
  15. The 4 cases.
  16. ...and 6 more sections