Reflexions on Mahler: Dessins, Modularity and Gauge Theories

TL;DR

This paper explores the deep connections between Mahler measures, dessins d'enfants, and gauge theories, revealing how they relate through modular functions, quantum periods, and string theory constructs.

Contribution

It introduces a unified framework linking Mahler measures, dessins, and gauge theories, and constructs Hauptmodul functions from Mahler measures associated with reflexive polygons.

Findings

Mahler measure and dessins d'enfants are in one-to-one correspondence for certain polynomials.

Hauptmodul functions can be constructed from Mahler measures, relating to modular groups.

Connections to quantum periods of del Pezzo surfaces and F-theory monodromies are established.

Abstract

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.