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

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.

arXiv comment71 pages, 6 Figures

References in corpus (30)

• Evidence for F-Theory
• On the Geometry of the String Landscape and the Swampland
• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• The Swampland: Introduction and Review
• Compactifications of F-Theory on Calabi--Yau Threefolds -- I
• Phase Transitions In M-Theory And F-Theory
• F-theory and Orientifolds
• Infinite Distances in Field Space and Massless Towers of States
• Tensionless Strings and the Weak Gravity Conjecture
• A Stringy Test of the Scalar Weak Gravity Conjecture
• Infinite Distance Networks in Field Space and Charge Orbits
• The Swampland Distance Conjecture for Kahler moduli
• Point-like Instantons on K3 Orbifolds
• The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces
• Top Down Approach to 6D SCFTs
• Asymptotic Flux Compactifications and the Swampland
• Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes
• Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions
• Quantum Corrections in 4d N=1 Infinite Distance Limits and the Weak Gravity Conjecture
• Swampland Distance Conjecture for One-Parameter Calabi-Yau Threefolds
• Instanton Corrections and Emergent Strings
• Moduli Space Holography and the Finiteness of Flux Vacua
• Moduli Stabilization in Asymptotic Flux Compactifications
• Physics of Infinite Complex Structure Limits in eight Dimensions
• On TCS $G_2$ manifolds and 4D Emergent Strings
• Bulk Reconstruction in Moduli Space Holography
• Mathematics Underlying the F-Theory/Heterotic String Duality in Eight Dimensions
• Heterotic-Type IIA Duality and Degenerations of K3 Surfaces
• Modular Curves and the Refined Distance Conjecture
• Stability of BPS States and Weak Coupling Limits

Cited by in corpus (3)

• The Asymptotic Weak Gravity Conjecture in M-theory
• Unifying the 6D $\mathcal{N}=(1,1)$ String Landscape
• Affine Algebras at Infinite Distance Limits in the Heterotic String