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Toric CFTs, permutation triples, and Belyi pairs

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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 \( {\mathbb{P}^1} \), branched over three points on the \( {\mathbb{P}^1} \). We construct explicit examples of Belyi pairs associated to some CFTs, including \( {\mathbb{C}^3} \) 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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References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].

    Article  MathSciNet  MATH  Google Scholar 

  2. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].

    MathSciNet  MATH  Google Scholar 

  4. I.R. Klebanov and E. Witten, Superconformal field theory on threebranes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP 06 (2005) 064 [hep-th/0411264] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. S. Benvenuti and M. Kruczenski, From Sasaki-Einstein spaces to quivers via BPS geodesics: L p,q|r, JHEP 04 (2006) 033 [hep-th/0505206] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. A. Butti, D. Forcella and A. Zaffaroni, The dual superconformal theory for L p,q|r manifolds, JHEP 09 (2005) 018 [hep-th/0505220] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [SPIRES].

  10. S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane Dimers and Quiver Gauge Theories, JHEP 01 (2006) 096 [hep-th/0504110] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. S. Franco and D. Vegh, Moduli spaces of gauge theories from dimer models: Proof of the correspondence, JHEP 11 (2006) 054 [hep-th/0601063] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 3 [hep-th/0511287] [SPIRES].

    MathSciNet  Google Scholar 

  14. K.D. Kennaway, Brane Tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. M. Yamazaki, Brane Tilings and Their Applications, Fortsch. Phys. 56 (2008) 555 [arXiv:0803.4474] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  16. R. Kenyon, An introduction to the dimer model, math/0310326.

  17. J. Stienstra, Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d’Enfants, arXiv:0711.0464 [SPIRES].

  18. S.K. Ashok, F. Cachazo and E. Dell’Aquila, Strebel differentials with integral lengths and Argyres-Douglas singularities, hep-th/0610080 [SPIRES].

  19. S.K. Ashok, F. Cachazo and E. Dell’Aquila, Children’s drawings from Seiberg-Witten curves, hep-th/0611082 [SPIRES].

  20. M. Bauer and C. Itzykson, Triangulations, in [21].

  21. L. Schneps, The Grothendieck theory of dessins d’enfants, LMS Lecture Notes Series 200, Cambridge University Press, Cambridge U.K. (1994).

    Book  MATH  Google Scholar 

  22. E. Looijenga, Intersection theory on Deligne-Mumford compactifications, Seminaire Bourbaki, exp. no. 768, (1992), pg. 187.

  23. R.d.M. Koch and S. Ramgoolam, From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group, arXiv:1002.1634 [SPIRES].

  24. A. Grothendieck, Esquisse d’un programme, in [21].

  25. G.V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 267.

    MathSciNet  MATH  Google Scholar 

  26. R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar graphs with faces of degree 4, math-ph/0305057.

  27. K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. S. Cordes, G.W. Moore and S. Ramgoolam, Large-N 2 – D Yang-Mills theory and topological string theory, Commun. Math. Phys. 185 (1997) 543 [hep-th/9402107] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. S. Lando and A. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences. Volume 141, Springer, New York U.S.A. (2004).

    MATH  Google Scholar 

  30. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York U.S.A (1978).

    MATH  Google Scholar 

  31. L. Khadjavi and V. Scharashkin, Belyi maps and elliptic curves, http://myweb.lmu.edu/lkhadjavi/.

  32. G.A. Jones and M. Streit, Monodromy groups and cartographic groups, in [33].

  33. L. Schneps and P. Lochak, Geometric Galois actions, the inverse Galois problem, moduli spaces and mapping class groups, LMS Lecture Note Series 243, Cambridge University Press, Cambridge U.K. (1997).

    Book  Google Scholar 

  34. SAGE, http://www.sagemath.org/.

  35. D.R. Morrison and M.R. Plesser, Non-spherical horizons. I, Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [SPIRES].

    MathSciNet  MATH  Google Scholar 

  36. A.M. Uranga, Brane Configurations for Branes at Conifolds, JHEP 01 (1999) 022 [hep-th/9811004] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. M.R. Douglas and G.W. Moore, D-branes, Quivers and ALE Instantons, hep-th/9603167 [SPIRES].

  38. A. Hanany, D. Orlando and S. Reffert, Sublattice Counting and Orbifolds, JHEP 06 (2010) 051 [arXiv:1002.2981] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. J. Davey, A. Hanany and R.-K. Seong, Counting Orbifolds, JHEP 06 (2010) 010 [arXiv:1002.3609] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  40. A. Hanany and R.-K. Seong, Symmetries of Abelian Orbifolds, JHEP 01 (2011) 027 [arXiv:1009.3017] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  41. D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  42. I. Garcia-Etxebarria, F. Saad and A.M. Uranga, Quiver gauge theories at resolved and deformed singularities using dimers, JHEP 06 (2006) 055 [hep-th/0603108] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  43. S. Franco et al., Dimers and Orientifolds, JHEP 09 (2007) 075 [arXiv:0707.0298] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  44. P. Joubert, Geometric actions of the absolute Galois group, MSc. Thesis, University of Stellenbosch (2006).

  45. O. Aharony, A. Hanany and B. Kol, Webs of (p,q) 5-branes, five dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [SPIRES].

    Article  ADS  Google Scholar 

  46. K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [SPIRES].

  47. B. Mazur, Algebraic numbers, in [48].

  48. T. Gowers, Princeton companion to mathematics, Princeton University Press, Princeton U.S.A. (2008).

    MATH  Google Scholar 

  49. S. Lang, Introduction to transcendental numbers, Chapter 2.1, Addison-Wesley, Reading U.S.A. (1966).

    Google Scholar 

  50. K. Ramachandra, Contributions to the theory of transcendental numbers I, II, Acta Arith. 14 (1967/68) 65.

    Google Scholar 

  51. M. Waldschmidt, Diophantine approximation on linear algebraic groups, Chapter 1, Springer, New York U.S.A. (2000).

    Google Scholar 

  52. J. Davey, A. Hanany and J. Pasukonis, On the Classiffication of Brane Tilings, JHEP 01 (2010) 078 [arXiv:0909.2868] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  53. T.W. Brown, Complex matrix model duality, arXiv:1009.0674 [SPIRES].

  54. C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, JHEP 12 (2001) 001 [hep-th/0109053] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  55. N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  56. K. Ueda and M. Yamazaki, Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories, JHEP 12 (2008) 045 [arXiv:0808.3768] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  57. Y. Imamura and K. Kimura, Quiver Chern-Simons theories and crystals, JHEP 10 (2008) 114 [arXiv:0808.4155] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  58. A. Hanany and A. Zaffaroni, Tilings, Chern-Simons Theories and M2 Branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  59. A. Hanany, D. Vegh and A. Zaffaroni, Brane Tilings and M2 Branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  60. F. Benini, C. Closset and S. Cremonesi, Chiral avors and M2-branes at toric CY4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  61. D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with avor and their AdS4 duals, arXiv:0911.4324 [SPIRES].

  62. F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

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

    Vishnu Jejjala & Sanjaye Ramgoolam

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

    Diego Rodriguez-Gomez

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

    Diego Rodriguez-Gomez

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  1. Vishnu Jejjala
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  2. Sanjaye Ramgoolam
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  3. Diego Rodriguez-Gomez
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Correspondence to Vishnu Jejjala.

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ArXiv ePrint: 1012.2351

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Jejjala, V., Ramgoolam, S. & Rodriguez-Gomez, D. Toric CFTs, permutation triples, and Belyi pairs. J. High Energ. Phys. 2011, 65 (2011). https://doi.org/10.1007/JHEP03(2011)065

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