Unconventional conformal invariance of maximal depth partially massless fields on dS4 and its relation to complex partially massless SUSY

Abstract

Deser and Waldron have shown that maximal depth partially massless theories of higher (integer) spin on four-dimensional de Sitter spacetime (dS4) possess infinitesimal symmetries generated by the conformal Killing vectors of dS4. However, it was later shown by Barnich, Bekaert, and Grigoriev that these theories are not invariant under the conformal algebra so(2, 4). To get some insight into these seemingly contradicting results we write down the full set of infinitesimal transformations of the fields generated by the fifteen conformal Killing vectors of dS4. In particular, although the infinitesimal transformations generated by the ten dS Killing vectors are well-known (these correspond to the conventional Lie derivatives), the transformations generated by the five non-Killing conformal Killing vectors were absent from the literature, and we show that they have an ‘unconventional’ form. In the spin-2 case (partially massless graviton), 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 ≥ 2, we reproduce the result that the symmetry algebra does not close on the conformal algebra, so(2, 4). This is due to the appearance of new higher-derivative symmetry transformations in the commutator of two unconventional conformal transformations. Our results concerning the closure of the full symmetry algebra are inconclusive. Then we shift focus to the question of supersymmetry (SUSY) on dS4 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 dS4. 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(1, 4) nor so(2, 4), while its full structure is an open question. More open questions arising from our findings are also discussed.