Invariant Differential Operators and Characters of the AdS_4 Algebra

TL;DR

This paper systematically applies modern Lie algebra representation theory tools to AdS_4, deriving invariant differential operators, classifying representations, and exploring implications for string theory and integrable models.

Contribution

It introduces explicit singular vectors, constructs invariant differential operators, and presents a new diagrammatic structure for AdS_4 representations, including massless and singleton cases.

Findings

Explicit singular vectors of AdS_4 Verma modules

New diagram involving reducible representations and finite-dimensional irreps

Classification of so(5,C) irreps and character formulae

Abstract

The aim of this paper is to apply systematically to AdS_4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS_4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS_4 Verma modules. These are used to obtain the AdS_4 invariant differential operators. Using this we display a new structure - a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS_4 algebra. We study in more detail the cases involving UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case we discover the structure of sets of 2s_0-1 conserved currents for each spin s_0 UIR, s_0=1,3/2,... All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s_0. Further we give the classification of the so(5,C) irreps presented in a diagramatic way which makes easy the derivation of all character formulae. The paper concludes with a speculation on the possible applications of the character formulae to integrable models.