Real Borcherds Superalgebras and M-theory

Abstract

The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the Magic triangle that reduce to the pure supergravities in four dimensions correspond to singular real del Pezzo surfaces and the same is true for the Magic square of N=2 SUGRAS. A real del Pezzo surface is the invariant set under an antilinear involution of a complex one. This conjugation induces an involution of the Picard group that preserves the anticanonical class and the intersection form. The known non-split U-duality algebras are embedded into superBorcherds algebras defined by their Cartan matrix (minus the intersection form) and fixed by the anti-involution. These data may be described by Tits-Satake bicoloured diagrams. As in the split case, oxidation results from blowing down disjoint real P^1's of self-intersection -1. The singular del Pezzo surfaces of interest are obtained by degenerating regular surfaces upon contraction of real curves of self-intersection -2. We use the finite classification of real simple singularities to exhibit the relevant normal surfaces. We also give a general construction of more magic triangles like a type I split magic triangle and prove their (approximate) symmetry with respect to their diagonal, this symmetry argument was announced in our previous paper for the split case.