nLab magic supergravity

Idea

Magic Squares (MS’s), arrays of Lie algebras enjoying remarkable symmetry properties under reflection with respect to their main diagonal, were discovered long time ago by Freudenthal, Rozenfeld and Tits, and their structure and fascinating properties have been studied extensively in mathematics and mathematical physics, especially in relation to exceptional Lie algebras.

Following the seminal papers (Gunyadin-Sierra-Townsend 83, Gunyadin-Sierra-Townsend 83) magic squares have been related to the generalized electric-magnetic U-duality symmetries of particular classes of Maxwell-Einstein supergravity theories, called magic supergravities. In particular, non-compact, real forms of Lie algebras, corresponding to non-compact symmetries of (super)gravity theories, have become relevant as symmetries of the corresponding rank-3 simple Jordan algebras, defined over real normed division algebra $(A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O})$ or split $(A = \mathbb{R}, \mathbb{C}_S, \mathbb{H}_S, \mathbb{O}_S)$ composition algebras. Later on, some other MS’s have been constructed in literature through the exploitation of Tits’ formula.

(from Cacciatori-Cerchiai-Marrani 12)

Related concepts

• magic pyramid

References

• Murat Gunaydin, G. Sierra and Paul Townsend, Exceptional Supergravity Theories and the Magic Square, Phys. Phys. Lett. B 133, 72 (1983) (spire)

• Murat Gunaydin, G. Sierra, Paul Townsend, The Geometry of $\mathcal{N}=2$ Maxwell-Einstein Supergravity and Jordan Algebras, Nucl.Phys. B242 (1984) 244-268

• Murat Gunaydin, G. Sierra and Paul Townsend, Nucl. Phys. B 253, 573 (1985).

• Murat Gunaydin, S. McReynolds, M. Zagermann, Unified $\mathcal{N}=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions, JHEP 0509:026,2005 (arXiv:hep-th/0507227)

• Massimo Bianchi, Sergio Ferrara, Enriques and Octonionic Magic Supergravity Models, JHEP 0802:054,2008 (arXiv:0712.2976)

• Sergio L. Cacciatori, Bianca Cerchiai, Alessio Marrani, Squaring the Magic (arXiv:1208.6153)

• Sergio L. Cacciatori, Bianca Cerchiai, Alessio Marrani, Magic coset decompositions, Adv. Theor. Math. Phys. Volume 17, Number 5 (2013), 1077-1128. (euclid:atmp/1408626511^)