Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

TL;DR

This paper analyzes the stable degeneration of K3 surfaces into pairs of rational elliptic surfaces, providing equations and lattice conditions for when such degenerations exist and exploring their implications in F-theory compactifications.

Contribution

It systematically describes the stable degeneration process of K3 surfaces into rational elliptic pairs and investigates lattice conditions for deformation to K3 surfaces with applications in F-theory.

Findings

Stable degeneration exists for pairs of identical rational elliptic surfaces.

An explicit equation describes the stable degeneration of K3 into isomorphic rational elliptic surfaces.

Gauge groups like E6, E7, E8, SU(5), SO(10) are derived from these degenerations in F-theory.

Abstract

F-theory/heterotic duality is formulated in the stable degeneration limit of a K3 fibration on the F-theory side. In this note, we analyze the structure of the stable degeneration limit. We discuss whether stable degeneration exists for pairs of rational elliptic surfaces. We demonstrate that, when two rational elliptic surfaces have an identical complex structure, stable degeneration always exists. We provide an equation that systematically describes the stable degeneration of a K3 surface into a pair of isomorphic rational elliptic surfaces. When two rational elliptic surfaces have different complex structures, whether their sum glued along a smooth fiber admits deformation to a K3 surface can be determined by studying the structure of the K3 lattice. We investigate the lattice theoretic condition to determine whether a deformation to a K3 surface exists for pairs of extremal rational elliptic surfaces. In addition, we discuss the configurations of singular fibers under stable degeneration. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular fibers of which are twice that of the rational elliptic surface. For special situations, singular fibers of the resulting K3 surface collide and they are enhanced to a fiber of another type. Some K3 surfaces become attractive in these situations. We determine the complex structures and the Weierstrass forms of these attractive K3 surfaces. We also deduce the gauge groups in F-theory compactifications on these attractive K3 surfaces times a K3. $E_6$, $E_7$, $E_8$, $SU(5)$, and $SO(10)$ gauge groups arise in these compactifications.