Magic coset decompositions

Abstract

By exploiting a "mixed" non-symmetric Freudenthal-Rozenfeld-Tits magic square, two types of coset decompositions are analyzed for the non-compact special Kähler symmetric rank-3 coset $E_{7(-25)}/ [(E_{6(-78)} \times U(1)) / \mathbb{Z}_3]$, occurring in supergravity as the vector multiplets'scalar manifold in $\mathcal{N} = 2, \mathcal{D} = 4$ exceptional Maxwell-Einstein theory. The first decomposition exhibits maximal manifest covariance, whereas the second (triality-symmetric) one is of Iwasawa type, with maximal $SO(8)$ covariance. Generalizations to conformal non-compact, real forms of nondegenerate, simple groups "of type E7" are presented for both classes of coset parametrizations, and relations to rank-3 simple Euclidean Jordan algebras and normed trialities over division algebras are also discussed.