Unitary Rigid Supersymmetry for the Chiral Graviton and Chiral Gravitino in de Sitter Spacetime

Abstract

It is commonly believed that a unitary supersymmetric quantum field theory (QFT) involving graviton and gravitino fields on fixed 4-dimensional de Sitter spacetime ($dS_{4}$) cannot exist due to known challenges associated with supersymmetry (SUSY) in $dS_{4}$. In this paper, we contradict this expectation by presenting a new unitary supersymmetric QFT on $dS_{4}$: the free supersymmetric theory of the chiral graviton and chiral gravitino fields. By chiral, we mean that the corresponding field strengths are anti-self-dual, and the gauge potentials are complex, each carrying a single complex propagating degree of freedom. The global SUSY transformations are generated by the standard Dirac Killing spinors of $dS_{4}$. The theory overcomes the known obstacles to unitary global SUSY on $dS_{4}$ by closing the commutator between two SUSY transformations on $so(4,2) \oplus u(1)$ rather than the de Sitter algebra $so(4,1)$. Crucially, the $so(4,2)$ symmetry is realised through unconventional conformal-like transformations. This free theory cannot become interacting while preserving SUSY in a way that makes the spin-2 sector the true graviton sector of General Relativity, as the three-graviton coupling cannot be $u(1)$-invariant. We establish the unitarity of the free supersymmetric theory in two complementary ways. First, by studying the action of the superalgebra generators on the space of physical gravitino and graviton mode solutions. Second, by quantising the fields and explicitly constructing the complex quantum supercharges $Q_{A}$ and $Q^{A\dagger}$, we show that the trace $\sum_{A} \{ Q_{A}, Q^{A \dagger} \}$ is positive-definite.