The Langlands Program and String Modular K3 Surfaces

TL;DR

This paper explores the deep connections between number theory, string theory, and algebraic geometry by linking K3 surface modular forms to string compactification via the Langlands program.

Contribution

It introduces a novel motivic approach to relate Galois orbits of K3 surfaces to modular forms derived from string worldsheet theories, bridging geometry and physics.

Findings

Hecke eigenforms from K3 surfaces relate to string-derived modular forms

String geometry can be reconstructed from spacetime partition functions

K3 modularity follows from mirror symmetry and the Shimura-Taniyama conjecture

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.