Elliptic K3 Surfaces at Infinite Complex Structure and their Refined Kulikov models

TL;DR

This paper classifies infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces, linking them to specific Weierstrass models and their singularities, with implications for quantum gravity and F-theory.

Contribution

It provides a detailed characterization of Weierstrass models for all Kulikov types of elliptic K3 degenerations, especially Type III, and relates these to infinite distance limits in moduli space.

Findings

Type III Kulikov models can be brought into two canonical forms.

All infinite distance limits relate to degenerations with non-minimal singularities.

Type III limits are associated with (partial) decompactification and symmetry enhancement.

Abstract

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in a companion paper. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.