Matter representations from geometry: under the spell of Dynkin

Abstract

In the traditional Katz-Vafa method, matter representations are determined by decomposing the adjoint representation of a parent simple Lie algebra m as the direct sum of irreducible representations of a semisimple subalgebra g. The Katz-Vafa method becomes ambiguous as soon as m contains several subalgebras isomorphic to g but giving different decompositions of the adjoint representation. We propose a selection rule that characterizes the matter representations observed in generic constructions in F-theory and M-theory: the matter representations in generic F-theory compactifications correspond to linear equivalence classes of subalgebras g⊂m with Dynkin index one along each simple components of g. This simple yet elegant selection rule allows us to apply the Katz-Vafa method to a much large class of models. We illustrate on numerous examples how this proposal streamlines the derivation of matter representations in F-theory and resolves previously ambiguous cases.