Hanany, A., He, Y., Jejjala, V., Pasukonis, J. & Ramgoolam, S. (2011). The Beta Ansatz: A Tale of Two Complex Structures. Journal of High Energy Physics, 2011(6), doi: 10.1007/JHEP06(2011)056

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Abstract
Brane tilings, sometimes called dimer models, are a class of bipartite graphs on a torus which encode the gauge theory data of fourdimensional SCFTs dual to D3branes probing toric CalabiYau threefolds. An efficient way of encoding this information exploits the theory of dessin d’enfants, expressing the structure in terms of a permutation triple, which is in turn related to a Belyi pair, namely a holomorphic map from a torus to a P1 with three marked points. The procedure of amaximization, in the context of isoradial embeddings of the dimer, also associates a complex structure to the torus, determined by the Rcharges in the SCFT, which can be compared with the Belyi complex structure. Algorithms for the explicit construction of the Belyi pairs are described in detail. In the case of orbifolds, these algorithms are related to the construction of covers of elliptic curves, which exploits the properties of Weierstraß elliptic functions. We present a counter example to a previous conjecture identifying the complex structure of the Belyi curve to the complex structure associated with Rcharges.
Item Type:  Article 

Uncontrolled Keywords:  Supersymmetry and Duality – Supersymmetric gauge theory 
Subjects:  Q Science > QC Physics 
Divisions:  School of Engineering & Mathematical Sciences > Department of Mathematical Science 
URI:  http://openaccess.city.ac.uk/id/eprint/891 
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