2010/12/31 by Vishnu Jejjala, Sanjaye Ramgoolam, Diego Rodriguez-Gomez · 29 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Conformal map #Conjecture #Homogeneous space #Permutation (music) #Quiver #Superpotential #Toroid #Torus #Triality #Vertex (graph theory) #hep-th #math-ph #math.AG #math.MP #math.NT
paper · pdf · doi:10.1007/jhep03(2011)065
published in Journal of High Energy Physics 2011(3) (Springer Nature) · 64 pages, 16 figures, LaTeX; version 2: minor typo corrections, slight editing of text; version 3: minor typo corrections, version published in JHEP
openalex publication_date 2011/03/01 · arxiv created 2011/03/25 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 P1, branched over three points on the P1. We construct explicit examples of Belyi pairs associated to some CFTs, including C3 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.