Borcherds symmetries in M-theory

It is well known but rather mysterious that root spaces of the $E_k$ Liegroups 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 actuallysubgroups of supergroups of finite dimension over the Grassmann algebra ofdifferential forms on spacetime that have been shown to preserve theself-duality equation obeyed by all bosonic form-fields of the theory. We showhere that the corresponding duality superalgebras are nothing but Borcherdssuperalgebras truncated by the above choice of Grassmann coefficients. The fullBorcherds' root lattices are the second integral cohomology of the del Pezzosurfaces. Our choice of simple roots uses the anti-canonical form and its knownorthogonal complement. Another result is the determination of del Pezzosurfaces associated to other string and field theory models. Dimensionalreduction on $T^k$ corresponds to blow-up of $k$ points in general positionwith respect to each other. All theories of the Magic triangle that reduce tothe $E_n$ sigma model in three dimensions correspond to singular del Pezzosurfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I andheterotic theories if one drops their gauge sector corresponds to non-normal(singular along a curve) del Pezzo's. We comment on previous encounters withBorcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively aris