Anti-de Sitter Supersymmetry

TL;DR

This work presents a cohesive framework for understanding supersymmetry in anti-de Sitter space by examining masslike terms, Casimir operators, and the structure of unitary representations. It develops the connection between AdS wave equations and the quadratic Casimir, demonstrates multiplet shortening and the existence of singleton representations, and employs the oscillator method and osp(1|4) superalgebra to systematically construct all unitary irreducible representations in four dimensions. The analysis clarifies how AdS curvature modifies mass relations within supermultiplets and how gauge symmetries lead to massless shortening, providing essential tools for studying AdS/CFT and gauged supergravity. Overall, the paper integrates representation theory with field equations to elucidate the unique features of AdS supersymmetry and its multiplet structure.

Abstract

We give a pedagogical introduction to certain aspects of supersymmetric field theories in anti-de Sitter space. Among them are the presence of masslike terms in massless wave equations, irreducible unitary representations and the phenomenon of multiplet shortening.

Figures (5)

• Figure 1: States of the $s=0$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Each point has a $(2j+1)$-fold degeneracy.
• Figure 2: States of the $s={\frac{1}{2}}$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Each point has a $(2j+1)$-fold degeneracy. The small circles denote the original $s=0$ multiplet from which the spin-${\frac{1}{2}}$ multiplet has been constructed by taking a direct product.
• Figure 3: States of the $s=1$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Observe that there are now points with double occupancy, indicated by the circle superimposed on the dots. These points could combine into an $s=0$ multiplet with ground state $\vert E_0+1,s=0\rangle$. This $s=0$ multiplet becomes reducible and can be dropped when $E_0=2$, as is explained in the text. The remaining points then constitute a massless $s=1$ multiplet, shown in Fig. 4.
• Figure 4: States of the massless $s=1$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Now $E_0$ is no longer arbitrary but it is fixed to $E_0=2$.
• Figure 5: The spin-0 and spin-${\frac{1}{2}}$ singleton representations. The solid dots indicate the states of the $s=0$ singleton, the circles the states of the $s={\frac{1}{2}}$ singleton. It is obvious that singletons contain much less degrees of freedom than a generic local field. The value of $E_0$, which denotes the spin-0 ground state energy, is equal to $E_0={\frac{1}{2}}$. The $s={\frac{1}{2}}$ singleton ground state has an energy equal to unity, as is explained in the text.