Abstract
Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant.
Highlights
Over almost a decade, a fruitful programme of investigating certain extraordinary bipartite structure of supersymmetric gauge theories in four-dimensions has emerged
What began as a convenient method of encoding the matter content and interactions of world-volume gauge theories of D3-branes probing non-compact Calabi-Yau manifolds that admit a toric description [1], has blossomed into a vast field ranging from the field and string theory of configurations of brane tilings [2, 3] to the integrable models of dimers [4, 5], from the geometry of Calabi-Yau algebras and cluster transformations [6] to the systematic outlook of bipartite field theories (BCFTs) [7,8,9] and remarkable relations to scattering amplitudes in N = 4 Super-Yang-Mills theory [10,11,12,13,14]
The key to the Belyi pair is that the parameters therein are algebraic numbers; while the degree of the field extension over Q has been shown to be a Seiberg duality invariant, how these algebraic numbers precisely relate to the R-charges and to volumes in the dual Sasaki-Einstein geometry remains to be understood
Summary
A fruitful programme of investigating certain extraordinary bipartite structure of supersymmetric gauge theories in four-dimensions has emerged. While the archetypal brane tilings and dimer models are bipartite graphs on the torus, i.e., dessins on the elliptic curve, which give us affine Calabi-Yau threefolds, in general, the moduli spaces of gauge theories corresponding to dessins on genus g Riemann surfaces are Calabi-Yau varieties of dimension 2g + 1 [31, 33] We will take this comprehensive viewpoint, calculate where needed and make use of the available datasets from the mathematics literature where possible (extensive use will be made of the excellent interactive website of [39]), to explicitly write down the Belyi pairs, genus by genus, and degree by degree.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
