The Langlands Program and String Modular K3 Surfaces

Abstract

A number theoretic approach to string compactification is developed for Calabi-Yau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic two-form of a particular type of K3 surfaces can be expressed in terms of modular forms constructed from the worldsheet theory. The process of deriving string physics from spacetime geometry can be reversed, allowing the construction of K3 surface geometry from the string characters of the partition function. A general argument for K3 modularity follows from mirror symmetry, in combination with the proof of the Shimura-Taniyama conjecture.

33 pages

References in corpus (12)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Mirror Symmetry is T-Duality
• Arithmetic and Attractors
• The Arithmetic of Calabi--Yau Manifolds and Mirror Symmetry
• The Shimura-Taniyama Conjecture and Conformal Field Theory
• Heterotic Gauge Structure of Type II K3 Fibrations
• Geometric Kac-Moody Modularity
• Arithmetic Spacetime Geometry from String Theory
• Arithmetic of Calabi-Yau Varieties and Rational Conformal Field Theory
• Complex Multiplication Symmetry of Black Hole Attractors
• Complex Multiplication of Exactly Solvable Calabi-Yau Varieties
• On the modularity of three Calabi-Yau threefolds with bad reduction at 11

Cited by in corpus (11)

• Emergent spacetime from modular motives
• String Modular Phases in Calabi-Yau Families
• Flux Vacua and Modularity for ${\mathbb{Z}}_2$ Symmetric Calabi-Yau Manifolds
• A General Framework of Automorphic Inflation
• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
• A modularity test for elliptic mirror symmetry
• String Automorphic Motives of nondiagonal Varieties
• Emergent spacetime and black hole probes from automorphic forms
• K-Rational D-Brane Crystals
• Applied String Theory
• Motivic L-Function Identities from CFT and Arithmetic Mirror Symmetry