Modular Curves and the Refined Distance Conjecture

Abstract

We test the refined distance conjecture in the vector multiplet moduli space of 4D $\mathcal{N}=2$ compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree $d=2N$ of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup ${\Gamma }_0(N{)}^{+}$. In order to probe the large $N$ regime, we initiate the study of Calabi-Yau threefolds fibered by general degree $d>8$ K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

44 pages, 10 figures, comments welcome; v2: references added, improved discussion of log-scaling

References in corpus (20)

• On the Geometry of the String Landscape and the Swampland
• Symmetries and Strings in Field Theory and Gravity
• The Swampland: Introduction and Review
• The String Landscape and the Swampland
• Lectures on the Swampland Program in String Compactifications
• Super-Planckian Spatial Field Variations and Quantum Gravity
• Non-Invertible Global Symmetries and Completeness of the Spectrum
• Quantum Corrections in 4d N=1 Infinite Distance Limits and the Weak Gravity Conjecture
• Fibrations in CICY Threefolds
• Constraining Neutrino Masses, the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture
• Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape
• 8d Supergravity, Reconstruction of Internal Geometry and the Swampland
• The FL bound and its phenomenological implications
• The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics
• Cobordism Classes and the Swampland
• On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups
• Discrete gauge groups in certain F-theory models in six dimensions
• Swampland Conjectures and Infinite Flop Chains
• Analytic Periods via Twisted Symmetric Squares
• On genus one fibered Calabi-Yau threefolds with 5-sections

Cited by in corpus (9)

• Heterotic de Sitter Beyond Modular Symmetry
• Asymptotic Safety: Swampland or Wonderland?
• Membrane Limits in Quantum Gravity
• Elliptic K3 Surfaces at Infinite Complex Structure and their Refined Kulikov models
• 4d strings at strong coupling
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Non-minimal Elliptic Threefolds at Infinite Distance II: Asymptotic Physics
• Analytic Periods via Twisted Symmetric Squares
• Hitchhiker's Guide to the Swampland: The Cosmologist's Handbook to the string-theoretical Swampland Programme