Towards the Standard Model in F-theory

This article explores possible embeddings of the Standard Model gauge group and its matter representations into F-theory. To this end we construct elliptic fibrations with gauge group SU(3)×SU(2)×U(1)×U(1) as suitable restrictions of a Bl 2P2-fibration with rank-two Mordell-Weil group. We analyse the five inequivalent toric enhancements to gauge group SU(3)×SU(2) along two independent divisors W3 and W2 in the base. For each of the resulting smooth fibrations, the representation spectrum generically consists of a bifundamental (3,2), three types of (1,2) representations and five types of (3,1) representations (plus conjugates), in addition to charged singlet states. The precise spectrum of zero-modes in these representations depends on the 3-form background. We analyse the geometrically realised Yukawa couplings among all these states and find complete agreement with field theoretic expectations based on their U(1) charges. We classify possible identifications of the found representations with the Standard Model field content extended by right-handed neutrinos and extra singlets. The linear combination of the two abelian gauge group factors orthogonal to hypercharge acts as a selection rule which, depending on the specific model, can forbid dangerous dimension-four and -five proton decay operators.