Codimension-3 Singularities and Yukawa Couplings in F-theory

TL;DR

This paper investigates the complex geometry of codimension-3 singularities in F-theory, focusing on their role in generating Yukawa couplings, and introduces new methods to analyze wavefunctions and dualities in these compactifications.

Contribution

It provides a comprehensive analysis of flux configurations, local geometry, and zero-mode behavior at codimension-3 singularities, advancing understanding of Yukawa coupling generation in F-theory.

Findings

Established a relation between local wavefunctions and global zero-modes.

Introduced an intrinsic definition of spectral surfaces in F-theory.

Enhanced the understanding of Heterotic--F theory duality at singularities.

Abstract

F-theory is one of the frameworks where all the Yukawa couplings of grand unified theories are generated and their computation is possible. The Yukawa couplings of charged matter multiplets are supposed to be generated around codimension-3 singularity points of a base complex 3-fold, and that has been confirmed for the simplest type of codimension-3 singularities in recent studies. However, the geometry of F-theory compactifications is much more complicated. For a generic F-theory compactification, such issues as flux configuration around the codimension-3 singularities, field-theory formulation of the local geometry and behavior of zero-mode wavefunctions have virtually never been addressed before. We address all these issues in this article, and further discuss nature of Yukawa couplings generated at such singularities. In order to calculate the Yukawa couplings of low-energy effective theory, however, the local descriptions of wavefunctions on complex surfaces and a global characterization of zero-modes over a complex curve have to be combined together. We found the relation between them by re-examining how chiral charged matters are characterized in F-theory compactification. An intrinsic definition of spectral surfaces in F-theory turns out to be the key concept. As a biproduct, we found a new way to understand the Heterotic--F theory duality, which improves the precision of existing duality map associated with codimension-3 singularities.