Real borcherds superalgebras and M-theory -: art. no. 060

The correspondence between del Pezzo surfaces and field theory models, discussed in [1] and in [2] 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 [3] 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 [4]. 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 Borcherds superalgebras defined by their Cartan matrix (minus the intersection form) and fixed by the anti-involution. These data may be described by Tits-Satake bicoloured superdiagrams. As in the split case, oxidation results from blowing down disjoint real RP1'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.