Uncovering infinite symmetries on [p,q] 7-branes: Kac-Moody algebras and beyond

Abstract

In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of EN fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of EN and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra Ê9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E8,E9, Ê9 and E10. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E10 ⊕ E10.