City Research Online

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.

Publication Type:	Article
Additional Information:	Electronic version of an article published as He, Y-H. , Huang, Z., Probst, M. and Read, J. (2018). Yang-Mills theory and the ABC conjecture. International Journal of Modern Physics A, 33(13), 1850053. 10.1142/S0217751X18500537 © © 2018 World Scientific Publishing Co https://www.worldscientific.com/worldscinet/ijmpa
Publisher Keywords:	N=4 supersymmetric Yang–Mills theory, ABC conjecture, brane tilings, dessin d’enfant
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics
SWORD Depositor:	Symplectic Administrator

Export

Downloads