The beta ansatz: a tale of two complex structures

  • Amihay Hanany
  • Yang-Hui He
  • Vishnu Jejjala
  • Jurgis Pasukonis
  • Sanjaye Ramgoolam
  • Diego Rodriguez-Gomez
Article

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.

Keywords

Supersymmetry and Duality Supersymmetric gauge theory 

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Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Amihay Hanany
    • 1
  • Yang-Hui He
    • 2
    • 3
    • 4
  • Vishnu Jejjala
    • 5
  • Jurgis Pasukonis
    • 5
  • Sanjaye Ramgoolam
    • 5
  • Diego Rodriguez-Gomez
    • 6
    • 7
  1. 1.Theoretical Physics Group, The Blackett LaboratoryImperial CollegeLondonU.K.
  2. 2.Department of MathematicsCity University, LondonLondonU.K.
  3. 3.School of PhysicsNanKai UniversityTianjinChina
  4. 4.Merton CollegeUniversity of OxfordOxfordU.K.
  5. 5.Department of Physics, Queen MaryUniversity of LondonLondonU.K.
  6. 6.Department of PhysicsTechnion, HaifaIsrael
  7. 7.Department of Mathematics and PhysicsUniversity of Haifa at OranimTivonIsrael

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