Analogs of Bol operators for $\mathfrak{pgl}(a+1\vert b)\subset \mathfrak{vect}(a\vert b)$.

Bol operators are differential operators invariant under the projective action of $\mathfrak{pgl}(2)\simeq\mathfrak{sl}(2)$ between spaces of weighted densities on the 1-dimensional manifold. In arXiv:2110.10504, we classified analogs of Bol operators on the $1\vert n$-dimensional superstrings with a contact structure invariant under the projective action of $\mathfrak{osp}(n\vert 2)$ between spaces of weighted densities (i.e., tensor fields with rank 1 fiber) as well as under $\mathfrak{pgl}(m|2)\subset \mathfrak{k}(1|2m)$, and $\mathfrak{osp}_a(4|2)\subset \mathfrak{k}(1|4)$ for the even fiber, and $\mathfrak{pe}(2)\subset\mathfrak{m}(1)$ for the odd fiber. Here we consider other analogs of Bol operators: $\mathfrak{pgl}(a+1\vert b)$-invariant differential operators between spaces of tensor fields on $(a\vert b)$-dimensional supermanifolds with irreducible, as $\mathfrak{gl}(a\vert b)$-modules, fibers of arbitrary, even infinite, dimension for certain "key" values of $a$ and $b$ - the ones for which the solution is describable. We discovered many new operators; the case of $1\vert 1$-dimensional general superstring looks a most natural superization of Bol's result, additional to the case of a superstring with a contact structure. For $a$ and $b$ generic and fibers of rank $>1$, the solution is indescribable whereas there are no non-scalar non-zero differential operators between spaces with rank $1$ fibers. This justifies the selection of cases in arXiv:2110.10504.