AdS/CFT Dualities and the Unitary Representations of Non-compact Groups and Supergroups: Wigner versus Dirac

TL;DR

This paper explores the connection between AdS/CFT dualities and the unitary representations of non-compact groups and supergroups, highlighting the transition from Wigner to Dirac bases and extending to superspaces.

Contribution

It demonstrates how to relate unitary lowest weight representations to covariant fields and extends these results to higher-dimensional and generalized spacetimes, including superspaces.

Findings

Established the link between Wigner and Dirac bases for conformal groups.

Extended representation theory to higher-dimensional and generalized spacetimes.

Connected oscillator constructions to superconformal and twistor fields.

Abstract

I review the relationship between AdS/CFT (anti-de Sitter / conformal field theory) dualities and the general theory of positive energy unitary representations of non-compact space-time groups and supergroups. I show, in particular, how one can go from the manifestly unitary compact basis of the lowest weight (positive energy) representations of the conformal group (Wigner picture) to the manifestly covariant coherent state basis (Dirac picture). The coherent states labelled by the space-time coordinates correspond to covariant fields with a definite conformal dimension. These results extend to higher dimensional Minkowskian spacetimes as well as generalized spacetimes defined by Jordan algebras and Jordan triple systems. The second part of my talk discusses the extension of the above results to conformal supergroups of Minkowskian superspaces as well as of generalized superspaces defined by Jordan superalgebras. The (super)-oscillator construction of generalized (super)-conformal groups can be given a dynamical realization in terms of generalized (super)-twistor fields.