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.

22 pages, v2 corrected few errors in tables

References in corpus (12)

• E_11 and M Theory
• E10 and a "small tension expansion" of M Theory
• Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities
• Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma
• Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models
• Exotic tensor gauge theory and duality
• Higher-dimensional Origin of D=3 Coset Symmetries
• Borcherds symmetries in M-theory
• The group theory of oxidation
• The group theory of oxidation II: Cosets of non-split groups
• Seiberg Duality in Matrix Model
• Real Algebraic Surfaces

Cited by in corpus (21)

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• The tensor hierarchy simplified
• Hyperbolic Kac Moody Algebras and Einstein Billiards
• The topology of U-duality (sub-)groups
• Exceptional geometry and Borcherds superalgebras
• Half-maximal supergravity in three dimensions: supergeometry, differential forms and algebraic structure
• Cosmological billiards and oxidation
• Forms and algebras in (half-)maximal supergravity theories
• Maximal supergravity in D=10: forms, Borcherds algebras and superspace cohomology
• Algebraic Geometry Realization of Quantum Hall Soliton
• Borcherds Algebras and N=4 Topological Amplitudes
• Non-Supersymmetric Magic Theories and Ehlers Truncations
• Degenerations of K3, Orientifolds and Exotic Branes
• Duality-Symmetric Approach to General Relativity and Supergravity
• Hidden Borcherds symmetries in Z_n orbifolds of M-theory and magnetized D-branes in type 0' orientifolds
• Infinite-Dimensional Gauge Structure of d=2 N=16 Supergravity
• Exotic Branes from del Pezzo Surfaces
• QFT method for indefinite Kac-Moody Theory: A step towards classification