Yukawa Couplings in F-theory and Non-Commutative Geometry

TL;DR

This paper explores how H-fluxes in F-theory modify Yukawa couplings by inducing non-commutative deformations, leading to hierarchical structures that could explain flavor hierarchies in GUT models.

Contribution

It introduces a non-commutative geometric framework to compute Yukawa couplings in F-theory with fluxes, revealing how fluxes alter Yukawa matrix ranks and structures.

Findings

Yukawa matrices are rank one without fluxes.

H-fluxes induce non-commutative deformations affecting Yukawa couplings.

Fluxes generate hierarchical Yukawa matrices consistent with flavor hierarchies.

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.