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The beta ansatz: a tale of two complex structures

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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 four-dimensional SCFTs dual to D3-branes probing toric Calabi-Yau 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 \( {\mathbb{P}^1} \) with three marked points. The procedure of a-maximization, in the context of isoradial embeddings of the dimer, also associates a complex structure to the torus, determined by the R-charges 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 R-charges.

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Author information

Authors and Affiliations

  1. Theoretical Physics Group, The Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2AZ, U.K.

    Amihay Hanany

  2. Department of Mathematics, City University, London, Northampton Square, London, EC1V 0HB, U.K.

    Yang-Hui He

  3. School of Physics, NanKai University, Tianjin, 300071, China

    Yang-Hui He

  4. Merton College, University of Oxford, Oxford, OX14JD, U.K.

    Yang-Hui He

  5. Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS, U.K.

    Vishnu Jejjala, Jurgis Pasukonis & Sanjaye Ramgoolam

  6. Department of Physics, Technion, Haifa, 3200, Israel

    Diego Rodriguez-Gomez

  7. Department of Mathematics and Physics, University of Haifa at Oranim, Tivon, 36006, Israel

    Diego Rodriguez-Gomez

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  1. Amihay Hanany
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  6. Diego Rodriguez-Gomez
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Correspondence to Vishnu Jejjala.

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ArXiv ePrint: 1104.5490

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Hanany, A., He, YH., Jejjala, V. et al. The beta ansatz: a tale of two complex structures. J. High Energ. Phys. 2011, 56 (2011). https://doi.org/10.1007/JHEP06(2011)056

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