(Non-)unitarity of strictly and partially massless fermions on de Sitter space

TL;DR

The paper tackles unitarity of strictly and partially massless fermionic fields on $dS_{D}$ by constructing TT tensor-spinor eigenmodes and linking their one-particle Hilbert spaces to spin$(D,1)$ UIRs. A separation-of-variables approach expresses $dS_{D}$ eigenmodes in terms of Dirac-eigenmodes on $S^{D-1}$, enabling precise identification of spin$(D)$ content and Casimir eigenvalues to test against the Ottoson–Schwarz classification. The main finding is that strictly and partially massless fields with spins $s=\tfrac{3}{2},\tfrac{5}{2}$ are non-unitary for $D\neq4$, while in $D=4$ they correspond to Discrete Series UIRs, with a complete dictionary provided for half-odd-integer spins. This work reveals a unique role for four-dimensional de Sitter space in the representation theory of fermionic gauge theories on curved backgrounds and extends Higuchi’s framework to higher-spin fermions on $dS_{D}$.

Abstract

We present the dictionary between the one-particle Hilbert spaces of totally symmetric tensor-spinor fields of spin $s={3}/{2}, {5}/{2}$ with any mass parameter on $D$-dimensional ($D \geq 3$) de Sitter space ($dS_{D}$) and Unitary Irreducible Representations (UIR's) of the de Sitter algebra spin$(D,1)$. Our approach is based on expressing the eigenmodes on global $dS_{D}$ in terms of eigenmodes of the Dirac operator on the ${(D-1)}$-sphere, which provides a natural way to identify the corresponding representations with known UIR's under the decomposition spin$(D,1)$ $\supset$ spin$(D)$. Remarkably, we find that four-dimensional de Sitter space plays a distinguished role in the case of the gauge-invariant theories. In particular, the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on $dS_{D}$, are not unitary unless $D=4$.