Reflexions on Mahler: Dessins, Modularity and Gauge Theories

We provide a unified framework of Mahler measure, dessins d'enfants, and gauge theory. With certain physically motivated Newton polynomials from reflexive polygons, the Mahler measure and the dessin are in one-to-one correspondence. From the Mahler measure, one can construct a Hauptmodul for a congruence subgroup of the modular group, which contains the subgroup associated to the dessin. We also discuss their connections to the quantum periods of del Pezzo surfaces, as well as certain elliptic pencils. In brane tilings and quiver gauge theories, the modular Mahler flow might shed light on the inequivalence amongst the three different complex structures $\tau_{R,G,B}$. We also study how, in F-theory, 7-branes and their monodromies arise in the context of dessins.