Conformal and Quasiconformal Realizations of Exceptional Lie Groups

TL;DR

The paper develops a unified geometric framework for exceptional Lie groups by constructing a quasiconformal realization of $E_{8(8)}$ on a 57-dimensional space, anchored by a quartic invariant and a Freudenthal triple-system interpretation. It demonstrates how this structure yields an invariant light-cone and, via truncations, explicit conformal realizations of lower-rank groups such as $E_{7(7)}$ on 27 dimensions. A key contribution is the explicit nonlinear action, including $F$-transformations, and the connection to Kantor/Freudenthal triple systems that underlie these symmetries. The work also outlines a hierarchy of truncations to $E_{7(7)}$, $E_{6(6)}$, $F_{4(4)}$, $G_{2(2)}$, and $SL(3,\mathbb{R})$, with a detailed conformal realization for $E_{7(7)}$ on $\mathbb{R}^{27}$, highlighting potential applications to M-theory, black-hole entropy, and related generalized spacetime constructions.

Abstract

We present a nonlinear realization of E_8 on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined ``light cone'' in 57 dimensions. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E_7 on a 27 dimensional vector space which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned.