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Toric CFTs, Permutation Triples, and Belyi Pairs

Four-dimensional CFTs dual to branes transverse to toric Calabi-Yau threefolds have been described by bipartite graphs on a torus (dimer models). We use the theory of dessins d'enfants to describe these in terms of triples of permutations which multiply to one. These permutations yield an elegant description of zig-zag paths, which have appeared in characterizing the toroidal dimers that lead to consistent SCFTs. The dessins are also related to Belyi pairs, consisting of a curve equipped with a map to P^1, branched over three points on the P^1. We construct explicit examples of Belyi pairs associated to some CFTs, including C^3 and the conifold. Permutation symmetries of the superpotential are related to the geometry of the Belyi pair. The Artin braid group action and a variation thereof play an interesting role. We make a conjecture relating the complex structure of the Belyi curve to R-charges in the conformal field theory.