Orbits of Exceptional Groups, Duality and BPS States in String Theory

TL;DR

This work provides an invariant orbit classification for the fundamental representations of the exceptional groups $E_{7(7)}$ and $E_{6(6)}$ that encode BPS spectra in string and M‑theory compactified to four and five dimensions. It builds on the exceptional Jordan algebra $J_3^{\mathbf O}$ and the exceptional Freudenthal triple system to define cubic and quartic invariants $I_3$ and $I_4$, whose values determine black hole entropy and map BPS states to distinct duality orbits, including light‑like and non‑zero entropy sectors. The authors extend this framework to $N=2$ Maxwell‑Einstein supergravity theories with symmetric scalar manifolds, enumerating possible orbits for both five and four dimensions and highlighting exceptional cases such as $E_{7(-25)}$ and $E_{6(-26)}$. They further discuss nonlinear conformal‑type extensions of duality groups as potential spectrum‑generating symmetries and outline directions for extending the results to less supersymmetric theories and higher dimensions.

Abstract

We give an invariant classification of orbits of the fundamental representations of exceptional groups $E_{7(7)}$ and $E_{6(6)}$ which classify BPS states in string and M theories toroidally compactified to d=4 and d=5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d=5 and d=4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and space-like vectors in Minkowski space. The orbits of BPS black holes in N=2 Maxwell-Einstein supergravity theories in d=4 and d=5 with symmetric space geometries are also classified including the exceptional N=2 theory that has $E_{7(-25)}$ and $E_{6(-26)}$ as its symmety in the respective dimensions.