Composite Genus One Belyi Maps

Abstract

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.

35 pages, 28 figures

References in corpus (11)

• Gauge Theories from Toric Geometry and Brane Tilings
• D-Brane Gauge Theories from Toric Singularities and Toric Duality
• Belyi functions for hyperbolic hypergeometric-to-Heun transformations
• Gauge Theories and Dessins d'Enfants: Beyond the Torus
• Invariants of Toric Seiberg Duality
• Computation of highly ramified coverings
• Double Handled Brane Tilings
• Brane Geometry and Dimer Models
• Algorithms and differential relations for Belyi functions
• Transformations of algebraic Gauss hypergeometric functions
• Transformations of hypergeometric elliptic integrals

Cited by in corpus (5)

• Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants
• An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups
• Belyi map for the sporadic group J1
• Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry
• Genus One Belyi Maps by Quadratic Correspondences