English

Borcherds symmetries in M-theory

High Energy Physics - Theory 2009-11-07 v2 Algebraic Geometry

Abstract

It is well known but rather mysterious that root spaces of the EkE_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 TkT^k corresponds to blow-up of kk points in general position with respect to each other. All theories of the Magic triangle that reduce to the EnE_n sigma model in three dimensions correspond to singular del Pezzo surfaces with A8nA_{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.

Keywords

Cite

@article{arxiv.hep-th/0203070,
  title  = {Borcherds symmetries in M-theory},
  author = {Pierre Henry-Labordere and Bernard Julia and Louis Paulot},
  journal= {arXiv preprint arXiv:hep-th/0203070},
  year   = {2009}
}

Comments

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

R2 v1 2026-07-22T15:09:32.315Z