An exceptional story: Symmetries and dualities between Maximal supergravity and General relativity

Abstract

We present the historical path from General relativity to the construction of Maximal $\mathcal{N}_4 = 8$ Supergravity with a detour in D=10 and 11 dimensions. The supergravities obtained by toric dimensional reduction and/or by reducing the number of supersymmetry generators have large exceptional duality symmetry groups and exhibit a remarkably uniform pattern across all values of $\mathcal{N}_D$ and D. In particular (bosonic) General relativity fits in as the simplest case and anchors us to the Real world. Dimensional reduction to 2 dimensions brings us to affine Kac-Moody groups and their semi-direct products with a real form of the Witt algebra: there is "integrable Magics". Integrability of 4D Gravity and of its reduction to 2D is considered with their "Twisted self-duality". Hyperbolic Kac-Moody symmetries appear after reduction to 1D: this leads to "chaotic Magics". We then discover "Borcherds"-Kac-Moody symmetries that allow us to rewrite in any dimension all matter equations of motion as Twisted self-duality: "Algebraic geometric Magics". Finally a "BF" metasymmetry $\Sigma$ exchanges negative quartets of Fermionic dimensions with Bosonic ones inside two Magic triangles. A third ubiquitous triangle of symmetries from Invariant theory resists unification despite its strong resemblance to the others. The prospective remarks include seven Challenges.