Borcherds symmetries in M-theory

Abstract

It is well known but rather mysterious that root spaces of the $E_k$ Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on $T^k$ corresponds to blow-up of $k$ points in general position with respect to each other. All theories of the Magic triangle that reduce to the $E_n$ sigma model in three dimensions correspond to singular del Pezzo surfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.

30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively arise

References in corpus (1)

• A Mysterious Duality

Cited by in corpus (53)

• The Casimir effect: Recent controversies and progress
• Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB
• The gauge structure of generalised diffeomorphisms
• Very-extended Kac-Moody algebras and their interpretation at low levels
• Spacelike Singularities and Hidden Symmetries of Gravity
• The Symmetry of M-Theories
• A note on spin-s duality
• Tensor Hierarchy and Generalized Cartan Calculus in SL(3)$\times$SL(2) Exceptional Field Theory
• The group theory of oxidation
• The group theory of oxidation II: Cosets of non-split groups
• The tensor hierarchy algebra
• Hyperbolic billiards of pure D=4 supergravities
• Beyond E11
• Hidden Symmetries and Dirac Fermions
• Tensor hierarchies, Borcherds algebras and E11
• A master exceptional field theory
• The tensor hierarchy simplified
• The Action for Twisted Self-Duality
• Real Borcherds Superalgebras and M-theory
• E${}_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB
• Supersymmetric Domain Walls
• Hyperbolic Kac Moody Algebras and Einstein Billiards
• Superalgebras, constraints and partition functions
• The topology of U-duality (sub-)groups
• Maximal supergravity in three dimensions: supergeometry and differential forms
• Heterotic horizons, Monge-Ampere equation and del Pezzo surfaces
• 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
• Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras
• Algebraic Geometry Realization of Quantum Hall Soliton
• Borcherds Algebras and N=4 Topological Amplitudes
• Oxidizing Borcherds symmetries
• Kac-Moody Algebras in M-theory
• Oxidation = group theory
• Kac-Moody algebras in perturbative string theory
• Embedding Integrable Superspin Chain in String Theory
• Non-Supersymmetric Magic Theories and Ehlers Truncations
• Degenerations of K3, Orientifolds and Exotic Branes
• Duality-Symmetric Approach to General Relativity and Supergravity
• Higher spin fields from indefinite Kac-Moody algebras
• 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
• Selfduality of d=2 Reduction of Gravity Coupled to a Sigma-Model
• Aspects of electric-magnetic dualities in maximal supergravity
• Exotic Branes from del Pezzo Surfaces
• Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras
• Kac-Moody and Borcherds Symmetries of Six-Dimensional Chiral Supergravity
• Lattice Vertex Algebras on General Even, Self-dual Lattices
• The many facets of exceptional field theory
• Generators and relations for (generalised) Cartan type superalgebras