Advances in Theoretical and Mathematical Physics

Volume 19 (2015)

Number 5

Squaring the magic

Pages: 923 – 954



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

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

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


We construct and classify all possible Magic Squares (MS’s) related to Euclidean or Lorentzian rank-$3$ simple Jordan algebras, both on normed division algebras and split composition algebras. Besides the known Freudenthal–Rozenfeld–Tits MS, the single-split Günaydin-Sierra-Townsend MS, and the double-split Barton–Sudbery MS, we obtain other 7 Euclidean and 10 Lorentzian novel MS’s.

We elucidate the role and the meaning of the various noncompact real forms of Lie algebras, entering the MS’s as symmetries of theories of Einstein–Maxwell gravity coupled to nonlinear sigma models of scalar fields, possibly endowed with local supersymmetry, in $D = 3$, $4$ and $5$ space-time dimensions. In particular, such symmetries can be recognized as the U-dualities or the stabilizers of scalar manifolds within space-time with standard Lorentzian signature or with other, more exotic signatures, also relevant to suitable compactifications of the so-called $M^*$- and $M^\prime$- theories. Symmetries pertaining to some attractor $U$-orbits of magic supergravities in Lorentzian space-time also arise in this framework.

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Published 31 March 2016