Borcherds symmetries in M-theory

TL;DR

This paper reveals that Borcherds superalgebras describe the duality symmetries in M-theory, linking algebraic structures to geometric properties of del Pezzo surfaces and their role in string theory compactifications.

Contribution

It demonstrates that duality superalgebras in M-theory are Borcherds superalgebras truncated by Grassmann coefficients, connecting algebraic, geometric, and physical aspects of string compactifications.

Findings

Borcherds superalgebras correspond to duality symmetries in M-theory.

Del Pezzo surfaces encode the root lattices of these Borcherds algebras.

Classification of del Pezzo surfaces relates to different string and field theory models.

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.