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. E6, E7, E8, SU(5), and SO(10) gauge groups arise in these compactifications.
Highlights
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
K3 surface stably degenerates into two rational elliptic surfaces in two distinct ways: i) K3 surface splits into two rational elliptic surfaces with an identical complex structure ii) K3 surface splits into two rational elliptic surfaces with different complex structures
Singular fibers of the same type of the K3 surface, that is obtained as the quadratic base change of a rational elliptic surface, collide and they are enhanced to a fiber of another type
Summary
2.1 Equation for the degeneration of K3 to a pair of isomorphic rational elliptic surfaces. We consider the case wherein two rational elliptic surfaces that are glued together are isomorphic For this particular case, we explicitly provide an equation that describes the stable degeneration of a K3 surface into the pair of isomorphic. Splits into two isomorphic rational elliptic surfaces, each given by the cubic pencil f This is the stable degeneration limit of the K3 surface (2.2) splitting into two copies of rational elliptic surfaces X. We observe from equation (2.6) that the K3 surface (2.2) is the quadratic base change of the rational elliptic surface (2.5) into which the K3 surface splits in the stable degeneration limit, when g is the cubic pencil of the same type as the pencil f. The aforementioned argument applies to the deformation of every pair of two isomorphic rational elliptic surfaces with a global section glued along smooth fiber to a K3 surface
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