Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 5

Magic coset decompositions

Pages: 1077 – 1128



Sergio L. Cacciatori (Dipartimento di Scienze ed Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy; INFN, Sezione di Milano, Italy )

Bianca L. Cerchiai (Dipartimento di Matematica, Università degli Studi di Milano, Italy; INFN, Sezione di Milano, Italy )

Alessio Marrani (Physics Department, Theory Unit, CERN, Geneva, Switzerland)


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.

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