Matter wave functions and Yukawa couplings in F-theory Grand Unification

We study the local structure of zero mode wave functions of chiral matter fields in F-theory unification. We solve the differential equations for the zero modes derived from local Higgsing in the 8-dimensional parent action of F-theory 7-branes. The solutions are found as expansions both in powers and derivatives of the magnetic fluxes. Yukawa couplings are given by an overlap integral of the three wave functions involved in the interaction and can be calculated analytically. We provide explicit expressions for these Yukawas to second order both in the flux and derivative expansions and discuss the effect of higher order terms. We explicitly describe the dependence of the couplings on the U(1) charges of the relevant fields, appropriately taking into account their normalization. A hierarchical Yukawa structure is naturally obtained. The application of our results to the understanding of the observed hierarchies of quarks and leptons is discussed.