魔法广场
Squaring the Magic
摘要：
我们构造和分类所有可能的幻方（MS的）相关的欧氏或洛伦兹秩-3简单的约旦代数，无论是赋范除代数和分裂复合代数。除了已知的Freudenthal-Rozenfeld-Tits MS，单分裂Günaydin-Sierra-Townsend MS和双分裂Barton-Sudbery MS外，我们还得到了7个新的欧氏MS和10个新的洛伦兹MS.我们阐明的作用和意义的各种非紧实形式的李代数，进入MS的对称性理论的爱因斯坦-麦克斯韦引力耦合到非线性西格玛模型的标量场，可能赋予局部超对称性，在D = 3，4和5的时空维度。特别地，这种对称性可以被认为是U对偶或时空中标量流形的稳定器，具有标准洛伦兹签名或其他更奇异的签名，也与所谓的M* 和M '理论的适当紧化有关。在洛伦兹时空中，一些关于魔超引力吸引子U轨道的对称性也出现在这个框架中。
英文摘要：
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 non-compact real forms of Lie algebras, entering the MS's as symmetries of theories of Einstein-Maxwell gravity coupled to non-linear 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'- theories. Symmetries pertaining to some attractor U-orbits of magic supergravities in Lorentzian space-time also arise in this framework.
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