Magic coset decompositions. (English) Zbl 1295.81135

Cacciatori, Sergio L.; Cerchiai, Bianca L.; Marrani, Alessio

Adv. Theor. Math. Phys. 17, No. 5, 1077-1128 (2013).

Summary: 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 \(\mathrm{SO}(8)\) covariance. Generalizations to conformal non-compact, real forms of nondegenerate, simple groups “of type \(E_7\)” 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.

MSC:

81R05	Finite-dimensional groups and algebras motivated by physics and their representations
17B25	Exceptional (super)algebras
22E70	Applications of Lie groups to the sciences; explicit representations
83E50	Supergravity