Analogs of Bol operators for 𝔭𝔤𝔩(a + 1|b) ⊂ 𝔳𝔢𝔠𝔱(a|b)

Abstract

Bol operators (Bols for short) are differential operators invariant under the projective action of [Formula: see text] between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: [Formula: see text]-invariant differential operators between spaces of tensor fields on [Formula: see text]-dimensional supermanifolds with irreducible, as [Formula: see text]-modules, fibers of arbitrary, even infinite, dimension for certain “key” values of [Formula: see text] and [Formula: see text] — the ones for which the solution is describable. We discovered many new operators for [Formula: see text] and for the case of [Formula: see text]-dimensional general superstring, which looks like a most natural superization of Bol’s result, additional to the cases of super analogs of Bols between spaces of weighted densities on the [Formula: see text]-dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension [Formula: see text], there are [Formula: see text]-parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.