Quite Discrete for a fermion

TL;DR

This paper explores the realization of discrete series representations of SL(2,R) through fermionic fields on dS2, analyzing their symmetries, zero-modes, and proposing related interacting theories.

Contribution

It provides a classical and quantum analysis of fermionic discrete series representations on dS2, including zero-mode treatment and new interacting theories.

Findings

Fermionic shift symmetry arises at specific mass tuning.

Zero-modes prevent standard two-point function definitions.

Proposed Euclidean procedure yields consistent two-point functions.

Abstract

We study Discrete Series representations of $SL(2,\mathbb{R})$ with half-integer scaling dimension $\Delta$. 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.