Ladder Operators and Fermionic Tensor Fields on Maximally Symmetric Spaces

Abstract

We construct first-order ladder operators for spin-$\frac{1}{2}$ Dirac fields and transverse, $\gamma$-traceless spin-$\frac{3}{2}$ Rarita--Schwinger fields on maximally symmetric spaces using non-isometric closed conformal Killing vectors. For both spins, we find three distinct operators: two of them, $\mathcal{D}$ and ${\mathcal{D}}^s$, shift the conformal label as $\Delta \to \Delta \pm 1$, while a third operator, $\tilde{\mathcal{D}}$, reverses the sign of the Dirac eigenvalue at fixed $\Delta$. The latter exists in arbitrary dimensions and reduces to the standard infinitesimal conformal transformation of a primary spinor when acting on massless spin-$\frac{1}{2}$ fields. On $S^N$, the ladder operators relate neighboring fermionic harmonics and generate the spinor tower from Killing-spinor seeds. In Lorentzian signature, we study their action on de Sitter mode spaces. In $d{S}_4$, the spin-$\frac{3}{2}$ ladders connect the zero-Dirac-mass sector with the fermionic gauge points $M=\pm i/\ell$, while $\tilde{\mathcal{D}}$ extends to arbitrary mass the conformal-like transformation previously identified for the gauge field. We explicitly present the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ fermionic harmonics on spheres as well as the de Sitter mode solutions. Finally, we derive the Casimir operators on $S^3$, $d{S}_3$, and $d{S}_4$ corresponding to SO(4), SO(3,1) and SO(4,1), and relate their eigenvalues for UIRs to the allowed masses in the fermionic field equations. These results provide a unified geometric framework relating conformal Killing geometry, fermionic Dirac-type spectra, and the representation theory of maximally symmetric spaces.

70 pages