Reflexions on Mahler: Dessins, Modularity and Gauge Theories

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 $τ_{R,G,B}$. We also study how, in F-theory, 7-branes and their monodromies arise in the context of dessins.

Figures (6)

• Figure 2.1: The 16 inequivalent reflexive polygons (up to SL$(2,\mathbb{Z})$). Figure taken from Hanany:2012hi (with slight modifications). The reflexive polygons are arranged such that the dual pairs are mirror symmetric with respect to the middle line (fourth row), and the four polygons in the middle line are therefore self-dual. In each row, the polygons have the same number of boundary points/(normalized) area. In each column, the polygons have the same number of vertices.
• Figure 2.2: (a) The dimer model. (b) The toric diagram. (c) The quiver diagram.
• Figure 3.1: The dessins for reflexive polygons with maximally tempered coefficients, their passports and the corresponding congruence subgroups. The missing of the two families (No.11, 12 and No.14) will be further discussed at the end of this subsection. Notice that in (a), the ramification is different from the form $\left\{3^V,2^E,n_1^{a_1}\dots n_k^{a_k}\right\}$, and this will be related to the discussions in §\ref{['ellpencils']}.
• Figure 4.1: The dessin associated to $\Gamma_0(3)$ with passport $\{1^13^1,2^2,1^13^1\}$. Here, the numbers are the labels of the edges, and the orange (purple) cycles indicate the permutations around black (white) vertices.
• Figure B.1: The dessins for reflexive polygons with minimally tempered coefficients. As listed, some of them correspond to certain congruence subgroups (as coset graphs).

Theorems & Definitions (45)

• Proposition 2.1
• Example 1
• Example 2
• Example 3
• Example 4
• Remark 1
• Theorem 2.2: Rodriguez-Villegas villegas1999modular
• Example 5
• Theorem 2.3
• Proposition 2.4