Ladder Operators and Fermionic Tensor Fields on Maximally Symmetric Spaces
Abstract
We construct first-order ladder operators for spin-1/2 Dirac fields and transverse, γ-traceless spin-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, 𝒟 and 𝒟s, shift the conformal label as Δ→ Δ± 1, while a third operator, 𝒟̃, reverses the sign of the Dirac eigenvalue at fixed Δ. The latter exists in arbitrary dimensions and reduces to the standard infinitesimal conformal transformation of a primary spinor when acting on massless spin-1/2 fields. On SN, 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 dS4, the spin-3/2 ladders connect the zero-Dirac-mass sector with the fermionic gauge points M=± i/ℓ, while 𝒟̃ extends to arbitrary mass the conformal-like transformation previously identified for the gauge field. We explicitly present the spin-1/2 and spin-3/2 fermionic harmonics on spheres as well as the de Sitter mode solutions. Finally, we derive the Casimir operators on S3, dS3, and dS4 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.
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BibTeX
@misc{cruz2026ladder,
title = {{Ladder Operators and Fermionic Tensor Fields on Maximally Symmetric Spaces}},
author = {Facundo L. Cruz and Matias N. Sempé and Guillermo A. Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12088},
eprint = {2609.12088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12088},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ladder Operators and Fermionic Tensor Fields on Maximally Symmetric Spaces
AU - Facundo L. Cruz
AU - Matias N. Sempé
AU - Guillermo A. Silva
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12088
UR - https://arxiv.org/abs/2609.12088
AB - We construct first-order ladder operators for spin-1/2 Dirac fields and transverse, γ-traceless spin-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, 𝒟 and 𝒟^s, shift the conformal label as Δ→ Δ± 1, while a third operator, 𝒟̃, reverses the sign of the Dirac eigenvalue at fixed Δ. The latter exists in arbitrary dimensions and reduces to the standard infinitesimal conformal transformation of a primary spinor when acting on massless spin-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 dS_4, the spin-3/2 ladders connect the zero-Dirac-mass sector with the fermionic gauge points M=± i/ℓ, while 𝒟̃ extends to arbitrary mass the conformal-like transformation previously identified for the gauge field. We explicitly present the spin-1/2 and spin-3/2 fermionic harmonics on spheres as well as the de Sitter mode solutions. Finally, we derive the Casimir operators on S^3, dS_3, and dS_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.
ER -