Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

TL;DR

This paper reviews the algebraic structures and spectrum generating symmetries in supergravity theories, black hole solutions, and their relation to harmonic superspace and minimal unitary representations.

Contribution

It elucidates the spectrum generating symmetry groups in supergravity theories linked to Jordan algebras and explores their connection to harmonic superspace and minimal unitary representations.

Findings

Spectrum generating groups are conformal and quasiconformal groups related to Jordan algebras.

Black hole attractor equations correspond to geodesic motion in reduced supergravity theories.

Harmonic superspace formulation connects to minimal unitary representations of isometry groups.

Abstract

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N=2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.