Yukawa Couplings in F-theory and Non-Commutative Geometry

TL;DR

This paper investigates Yukawa couplings in F-theory seven-brane configurations, showing that a rank-one structure at a single interaction point is generically lifted by three-form H-flux. The authors formulate a holomorphic, residue-based approach for computing Yukawas in commutative geometry and show that H-flux induces a non-commutative deformation, yielding hierarchical Yukawa textures that align with flavor-hierarchy proposals. They develop a non-commutative extension of the seven-brane theory, derive a quantum residue formula, and demonstrate explicit examples where order-one coefficients generate realistic hierarchies. The results connect flux-induced geometric data to phenomenological flavor structures and provide a framework for analyzing Yukawas in non-perturbative or strongly coupled F-theory GUTs, with implications for masses, mixing angles, and potential extensions to E-type gauge groups.

Abstract

We consider Yukawa couplings generated by a configuration of intersecting seven-branes in F-theory. In configurations with a single interaction point and no fluxes turned on, the Yukawa matrices have rank one. This is no longer true when the three-form H-flux is turned on, which is generically the case for F-theory compactifications on Calabi-Yau fourfolds. In the presence of H-fluxes, the Yukawa coupling is computed using a non-commutative deformation of holomorphic Chern-Simons theory (and its reduction to seven-branes) and subsequently the rank of the Yukawa matrix changes. Such fluxes give rise to a hierarchical structure in the Yukawa matrix in F-theory GUTs of the type which has recently been proposed as a resolution of the flavor hierarchy problem.