Realizations of exceptional U-duality groups as conformal and quasiconformal groups and their minimal unitary representations

TL;DR

This paper explores how exceptional U-duality groups in supergravity can be realized as conformal and quasiconformal groups, leading to their minimal unitary representations and spectrum-generating symmetries in various dimensions.

Contribution

It introduces novel quasiconformal and conformal realizations of exceptional U-duality groups and connects them to minimal unitary representations relevant for supergravity spectra.

Findings

Realization of $E_{8(8)}$ as a quasiconformal group in 57D charge-entropy space.

Identification of $E_{7(7)}$ as a conformal group in 27D charge space.

Correspondence of these groups with spectrum-generating symmetries in supergravity.

Abstract

We review the novel quasiconformal realizations of exceptional U-duality groups whose "quantization" lead directly to their minimal unitary irreducible representations. The group $E_{8(8)}$ can be realized as a quasiconformal group in the 57 dimensional charge-entropy space of BPS black hole solutions of maximal N=8 supergravity in four dimensions and leaves invariant "lightlike separations" with respect to a quartic norm. Similarly $E_{7(7)}$ acts as a conformal group in the 27 dimensional charge space of BPS black hole solutions in five dimensional N=8 supergravity and leaves invariant "lightlike separations" with respect to a cubic norm. For the exceptional N=2 Maxwell-Einstein supergravity theory the corresponding quasiconformal and conformal groups are $E_{8(-24)}$ and $E_{7(-25)}$, respectively. These conformal and quasiconformal groups act as spectrum generating symmetry groups in five and four dimensions and are isomorphic to the U-duality groups of the corresponding supergravity theories in four and three dimensions, respectively. Hence the spectra of these theories are expected to form unitary representations of these groups whose minimal unitary realizations are also reviewed.