Yang-Mills Theory and the ABC Conjecture

TL;DR

This paper establishes a novel connection between the ABC Conjecture and N=4 super-Yang-Mills theory using elliptic curves, Belyi maps, and string theory, providing a new perspective on the conjecture through gauge theory and string compactification.

Contribution

It introduces a new correspondence linking the ABC Conjecture to gauge theory and string theory via elliptic curves, Belyi maps, and dessins d'enfant, bridging number theory and theoretical physics.

Findings

Mapped high-quality ABC-triples to elliptic curves and Belyi maps.

Connected the ABC Conjecture to the fundamental domain of string compactification.

Demonstrated the correspondence with random ABC-triples.

Abstract

We establish a precise correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.