Quite discrete for a fermion

A bstract We study discrete series representations of SL(2, ℝ ) with half-integer scaling dimension ∆. At the classical level, we show that these UIRs are realised in the space of mode solutions of spinor fields with imaginary mass parameters on a fixed two-dimensional de Sitter, dS 2 , background. Upon such tuning of the mass, the field develops a fermionic shift symmetry that we characterise. We show that in the Euclidean section this manifests itself in the presence of zero-modes which preclude the definition of a Hadamard two-point function for these UIRs. We propose a Euclidean procedure to deal with the zero-modes, define a two-point function with the right singularity structure, and analyse its late-time behaviour. We end this note by proposing two interacting theories containing the fermionic discrete series in their spectrum.