We establish a 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.
- String Theory Seminar