The Noether-Lefschetz Problem and Gauge-Group-Resolved Landscapes: F-Theory on K3 x K3 as a Test Case

Abstract

Four-form flux in F-theory compactifications not only stabilizes moduli, but gives rise to ensembles of string vacua, providing a scientific basis for a stringy notion of naturalness. Of particular interest in this context is the ability to keep track of algebraic information (such as the gauge group) associated with individual vacua while dealing with statistics. In the present work, we aim to clarify conceptual issues and sharpen methods for this purpose, using compactification on $K3\times K3$ as a test case. Our first approach exploits the connection between the stabilization of complex structure moduli and the Noether-Lefschetz problem. Compactification data for F-theory, however, involve not only a four-fold (with a given complex structure) $Y_4$ and a flux on it, but also an elliptic fibration morphism $Y_4⟶B_3$, which makes this problem complicated. The heterotic-F-theory duality indicates that elliptic fibration morphisms should be identified modulo isomorphism. Based on this principle, we explain how to count F-theory vacua on $K3\times K3$ while keeping the gauge group information. Mathematical results reviewed/developed in our companion paper are exploited heavily. With applications to more general four-folds in mind, we also clarify how to use Ashok-Denef-Douglas' theory of the distribution of flux vacua in order to deal with statistics of sub-ensembles tagged by a given set of algebraic/topological information. As a side remark, we extend the heterotic/F-theory duality dictionary on flux quanta and elaborate on its connection to the semistable degeneration of a K3 surface.

81 pages, 5 figures

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• Enhancements in F-theory models on moduli spaces of K3 surfaces with $ADE$ rank 17
• Distribution of the Number of Generations in Flux Compactifications
• Statistics of Flux Vacua for Particle Physics
• Nongeometric heterotic strings and dual F-theory with enhanced gauge groups
• Gauge Groups and Matter Fields on Some Models of F-theory without Section
• Issues in Complex Structure Moduli Inflation
• F-theory models on K3 surfaces with various Mordell-Weil ranks -constructions that use quadratic base change of rational elliptic surfaces
• K3 surfaces without section as double covers of Halphen surfaces, and F-theory compactifications
• Revisiting arithmetic solutions to the $W=0$ condition
• F-theory models with 3 to 8 U(1) factors on K3 surfaces
• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
• Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces