2016/02/04 by Yang-Hui He, He, Yang-Hui, Zhi Hu +5
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT) #hep-th #math.NT
paper · pdf · doi:10.48550/arxiv.1602.01780
54+1 pages, 7 figures
arxiv created 2017/09/05 · arxiv updated 2017/09/06
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.