Abstract
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.
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