Unconventional conformal invariance of maximal depth partially massless fields on $dS_{4}$ and its relation to complex partially massless SUSY

Deser and Waldron have shown that maximal depth partially massless theories of higher integer spin on 4-dimensional de Sitter spacetime ($dS_{4}$) possess infinitesimal symmetries generated by the conformal Killing vectors of $dS_{4}$. However, it was later shown by Barnich, Bekaert, and Grigoriev that these theories are not invariant under the conformal algebra $so(4,2)$. To get some insight into these seemingly contradicting results we write down the full set of infinitesimal transformations generated by the 15 conformal Killing vectors of $dS_{4}$. Although the infinitesimal transformations generated by the 10 dS Killing vectors are known, the transformations generated by the 5 non-Killing conformal Killing vectors were absent from the literature, and we show that they have an `unconventional' form. In the spin-2 case, we show that the field equations and the action are invariant under the unconventional conformal transformations. For spin $s >2$, the invariance is demonstrated only at the level of the field equations. For all spins $s \geq 2$, we reproduce the result that the symmetry algebra does not close on $so(4,2)$. This is due to the appearance of higher-derivative symmetries in the commutator of two unconventional conformal transformations. Our results concerning the closure of the full algebra are inconclusive. Then we shift focus to the question of supersymmetry (SUSY) on $dS_{4}$ and our objective is twofold. First, we uncover a non-interacting supermultiplet that consists of a complex partially massless spin-2 field and a complex spin-3/2 field on $dS_{4}$. Second, we showcase the appearance of the unconventional conformal symmetries in the commutator of two SUSY transformations. Thus, this commutator closes on an algebra that is neither $so(4,1)$ nor $so(4,2)$, while its full structure is an open question. More open questions arising from our findings are also discussed.