Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories
Reflexions on Mahler: Dessins, modularity and Gauge theories

Reflexions on Mahler

Number theory

Unifying Mahler measure, dessins and gauge theory, Newton polynomials reveal a 1-to-1 correspondence between Mahler measure and dessins.

Reflexions on Mahler: Dessins, modularity and Gauge theories

Submitted

Y. He, J. Bao, A. Zahabi

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 τR,G,B\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.

Submitted

Y. He, J. Bao, A. Zahabi