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

TL;DR

The work establishes a representation-theoretic bridge between AdS/CFT dualities and positive-energy unitary representations of noncompact (super)groups, showing how to pass from a compact lowest-weight (Wigner) basis to a covariant coherent-state (Dirac) basis that yields fields with definite conformal dimension. It extends these ideas from conventional AdS/CFT settings to generalized spacetimes defined by Jordan algebras and to their conformal supergroups, using an oscillator-based ULWR construction and coherent-state formalism to obtain covariant fields and supermultiplets. Key contributions include explicit ULWR-to-field correspondences for conformal groups of formally real Jordan algebras (e.g., $SU(2,2)$ for 4D, $SO^{*}(8)$ for 6D), the induction of covariant fields via coherent states, and the extension to Jordan superalgebras with super-twistor realizations, providing a broad framework for organizing AdS/CFT spectra in generalized geometries. The results offer a solid representation-theoretic basis for AdS/CFT and potentially broader dualities, enabling systematic construction of (super)conformal field theories on generalized spacetimes and their twistorial dynamics.

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.