Y. He

J. Bao

A. Zahabi

Using Newton polynomials from reflexive polygons, we find that the Mahler measure and dessin d’enfants are in one-to-one correspondence.

In string theory, three complex structures, one from dessins d’enfants in number theory, another from QFT and the third from algebraic geometry, describe the torus of a brane tiling. For brane tilings described by reflexive polygons, lattice polygons with a single interior point, we show these three different structures are actually different limits of the Mahler measure, an indicator of a polynomial’s complexity.

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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. In brane tilings and quiver gauge theories, the modular Mahler flow gives a natural resolution of 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. Moreover, we give a dictionary on how Mahler measure generates Gromov-Witten invariants.

Reflexions on Mahler: Dessins, modularity and Gauge theories