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.

Tensor hierarchies, Borcherds algebras and E 11

“…This agreement has been considered somewhat mysterious, since neither U n+1 or E 11 appears in the construction of the tensor hierarchy. On the other hand, it is in line with the ideas of gauging as a probe of M-theory degrees of freedom [1,2], and of Borcherds or Kac-Moody algebras as symmetries in M-theory [3][4][5].…”

“…Another way is to embed g into an infinitedimensional Lie (super)algebra, either a Borcherds algebra (which depends on g) or the indefinite Kac-Moody algebra E 11 . In the level decomposition of the Borcherds algebra with respect to g, the representation content on level p coincides with r p , up to level D − 2 [4,12,13]. The same is true for E 11 if the level decomposition is done with respect to g ⊕ sl D , and restricted to tensors that are antisymmetric under sl D [14][15][16][17][18][19].…”

“…General gauge deformations of the lower-dimensional theories have been systematically studied in recent years as a way of exploring the M-theory degrees of freedom beyond supergravity [1,2]. These studies have exhibited features that also appear in other approaches to M-theory, developed during the last decade, where possible symmetries based on Kac-Moody or Borcherds algebras have been investigated [3][4][5]. The present work is an attempt to relate the different approaches to each other via the features that they share.…”

Hyperbolic billiards of pureD= 4 supergravities

“…One of the interests of the billiard analysis is its connection with Udualities [25] and hidden symmetries of the theory, for which various proposals exist [17,26,27,28,29,30]. We reserve for further study a more detailed analysis of the significance of the twist in the symmetry structure of the models where it appears.…”

Kac-Moody and Borcherds symmetries of six-dimensional chiral supergravity

“…This self-duality condition is also properly incorporated in the sigma model formulation [46]. The p-form content is, however, different and this can best be discussed in terms of the underlying Borcherds algebras [47,48].…”