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<p>We study discrete series representations of SL(2, <i>ℝ</i>) 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 <i>imaginary mass parameters</i> on a fixed two-dimensional de Sitter, dS<sub>2</sub>, 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.</p>

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Quite discrete for a fermion

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月1日

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  • Vasileios A. Letsios,
  • Ben Pethybridge,
  • Alan Rios Fukelman

摘要

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, dS2, 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.