Unitary Realizations of U-duality Groups as Conformal and Quasiconformal Groups and Extremal Black Holes of Supergravity Theories

TL;DR

This paper reviews the construction of unitary representations of U-duality groups in supergravity theories across five, four, and three dimensions, highlighting their role in black hole entropy and the extension to conformal and quasiconformal groups.

Contribution

It introduces new methods for constructing unitary representations of U-duality groups and explores their extensions to conformal and quasiconformal groups in supergravity contexts.

Findings

U-duality groups relate to black hole entropy invariants.

Extensions to conformal and quasiconformal groups are isomorphic across dimensions.

Oscillator constructions enable positive energy representations.

Abstract

We review the current status of the construction of unitary representations of U-duality groups of supergravity theories in five, four and three dimensions. We focus mainly on the maximal supergravity theories and on the N=2 Maxwell-Einstein supergravity (MESGT) theories defined by Jordan algebras of degree three in five dimensions and their descendants in four and three dimensions. Entropies of the extremal black hole solutions of these theories in five and four dimensions are given by certain invariants of their U-duality groups. The five dimensional U-duality groups admit extensions to spectrum generating generalized conformal groups which are isomorphic to the U-duality groups of corresponding four dimensional theories. Similarly, the U-duality groups of four dimensional theories admit extensions to spectrum generating quasiconformal groups that are isomorphic to the corresponding U-duality groups in three dimensions. We outline the oscillator construction of the unitary representations of generalized conformal groups that admit positive energy representations, which include the U-duality groups of N=2 MESGT's in four dimensions. We conclude with a review of the minimal unitary realizations of U-duality groups that are obtained by quantizations of their quasiconformal actions.