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 surface 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 is given by combining mirror symmetry with the proof of the Shimura–Taniyama conjecture.