A Walk Through $Spin(1,d+1)$

TL;DR

This paper constructs and analyzes unitary irreducible representations of the de Sitter group, focusing on higher-spin fields, exceptional series, and the existence of unitary gravitino representations across different dimensions.

Contribution

It provides a detailed construction of de Sitter group representations, clarifies the role of exceptional series, and determines the conditions for unitary higher-spin and gravitino fields.

Findings

No exceptional fermionic representations for d>3.

Existence of unitary gravitino in d=3 and d=2.

Clarification of non-invariance in certain correlators.

Abstract

We construct unitary irreducible representation of the de Sitter group, that forms the basis for the study of $dS_{d+1}$ QFT. Using the intertwining kernel analysis, we discuss bosonic symmetric tensor, and fermionic higher-spinors. Particular care is given to the structure and construction of exceptional series and fermionic operators. We discuss the discrete series, and explain how and why the exceptional series give rise to seemingly non-invariant correlators in de Sitter. Using tools from Clifford analysis, we show that for $d>3$, there are no exceptional fermionic representations, and so no unitary (higher)-gravitino fields. We also consider the structure of representations for $d=3$ and $d=2$, as relevant for the study of $dS_4$, the only space allowing for unitary gravitino and its generalisation, and of $dS_3$.