Yang-Mills theory and the ABC conjecture - He, Yang-Hui; Hu, Zhi; Probst, Malte; Read, James (International journal of modern physics. A, Particles and fields, gravitation, cosmology, 2018)
We establish a precise correspondence between the ABC Conjecture and N=4𝒩=4𝒩=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=4𝒩=4𝒩=4 SYM.