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

Abstract

Bol operators (Bols for short) 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. Here, we described analogs of Bols: $\mathfrak{pgl}(a+1|b)$-invariant differential operators between spaces of tensor fields on $(a|b)$-dimensional supermanifolds with irreducible, as $\mathfrak{gl}(a|b)$-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 for $(a|b)=(2|0),(0|3)$ and for the case of $1|1$-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 $1|n$-dimensional superstrings with a~contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension $>1$, there are $(a+b-1)$-parameter families of Bols, whereas there are no non-scalar non-zero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.

19 pages; an edited version with typos corrected