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

TL;DR

This paper generalizes Bol operators to supermanifolds, describing new invariant differential operators under super projective Lie superalgebras for specific dimensions, revealing many novel operators and their classifications.

Contribution

It introduces analogs of Bol operators for supermanifolds invariant under rfcpgl(a+1|b), classifies these operators for key dimensions, and discovers many new invariant differential operators.

Findings

Discovered new rfcpgl(a+1|b)-invariant differential operators.

Classified Bol analogs for specific superdimensions such as (2|0), (0|3), and (1|1).

Found that for fibers of dimension >1, families of Bol operators exist, but no non-scalar operators between weighted densities.

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: $\mathfrak{pgl}(a+1\vert b)$-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 for $(a|b)=(2|0), (0|3)$ and for the case of $1\vert 1$-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 $1\vert n$-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.