On the standard model group in F-theory

Abstract

We analyze the standard model gauge group \(SU(3)\times SU(2) \times U(1)\) constructed in F-theory. The non-Abelian part \(SU(3) \times SU(2)\) is described by a surface singularity of Kodaira type. Blow-up analysis shows that the non-Abelian part is distinguished from the naïve product of \(SU(3)\) and \(SU(2)\), but that it should be a rank three group along the chain of \(E_n\) groups, because it has non-generic gauge symmetry enhancement structure responsible for desirable matter curves. The Abelian part \(U(1)\) is constructed from a globally valid two-form with the desired gauge quantum numbers, using a similar method to the decomposition (factorization) method of the spectral cover. This technique makes use of an extra section in the elliptic fiber of the Calabi–Yau manifold, on which F-theory is compactified. Conventional gauge coupling unification of \(SU(5)\) is achieved, without requiring a threshold correction from the flux along the hypercharge direction.