Journal of High Energy Physics

, 2015:135 | Cite as

Gauge theories and dessins d’enfants: beyond the torus

Open Access
Regular Article - Theoretical Physics

Abstract

Dessin denfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic j-invariant.

Keywords

Brane Dynamics in Gauge Theories AdS-CFT Correspondence Differential and Algebraic Geometry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
  3. [3]
    S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    R. Eager, S. Franco and K. Schaeffer, Dimer models and integrable systems, JHEP 06 (2012) 106 [arXiv:1107.1244] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Franco, D. Galloni and Y.-H. He, Towards the continuous limit of cluster integrable systems, JHEP 09 (2012) 020 [arXiv:1203.6067] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    S. Franco, D. Galloni and A. Mariotti, Bipartite field theories, cluster algebras and the grassmannian, J. Phys. A 47 (2014) 474004 [arXiv:1404.3752] [INSPIRE].Google Scholar
  7. [7]
    S. Franco, Bipartite field theories: from D-brane probes to scattering amplitudes, JHEP 11 (2012) 141 [arXiv:1207.0807] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    J.J. Heckman, C. Vafa, D. Xie and M. Yamazaki, String theory origin of bipartite SCFTs, JHEP 05 (2013) 148 [arXiv:1211.4587] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    D. Xie and M. Yamazaki, Network and Seiberg duality, JHEP 09 (2012) 036 [arXiv:1207.0811] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].
  11. [11]
    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    F. Cachazo, L. Mason and D. Skinner, Gravity in twistor space and its grassmannian formulation, SIGMA 10 (2014) 051 [arXiv:1207.4712] [INSPIRE].MathSciNetGoogle Scholar
  13. [13]
    A. Amariti and D. Forcella, Scattering amplitudes and toric geometry, JHEP 09 (2013) 133 [arXiv:1305.5252] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    S. Franco, D. Galloni and A. Mariotti, The geometry of on-shell diagrams, JHEP 08 (2014) 038 [arXiv:1310.3820] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    V. Jejjala, S. Ramgoolam and D. Rodriguez-Gomez, Toric CFTs, permutation triples and Belyi pairs, JHEP 03 (2011) 065 [arXiv:1012.2351] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    A. Hanany et al., The beta ansatz: a tale of two complex structures, JHEP 06 (2011) 056 [arXiv:1104.5490] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    A. Hanany et al., Invariants of toric Seiberg duality, Int. J. Mod. Phys. A 27 (2012) 1250002 [arXiv:1107.4101] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    Y.-H. He, V. Jejjala and D. Rodriguez-Gomez, Brane geometry and dimer models, JHEP 06 (2012) 143 [arXiv:1204.1065] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    L. Schneps, The Grothendieck theory of dessin d’enfants, Cambridge University Press, Cambridge U.K. (1994).CrossRefGoogle Scholar
  20. [20]
    Y.-H. He and J. McKay, N = 2 gauge theories: congruence subgroups, coset graphs and modular surfaces, J. Math. Phys. 54 (2013) 012301 [arXiv:1201.3633] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    Y.-H. He, Bipartita: physics, geometry & number theory, arXiv:1210.4388 [INSPIRE].
  22. [22]
    Y.-H. He, J. McKay and J. Read, Modular subgroups, dessins d’enfants and elliptic K3 surfaces, LMS J. Comp. Math. 16 (2013) 271 [arXiv:1211.1931] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    Y.-H. He, A. Hanany, A. Lukas and B. Ovrut, Computational algebraic geometry in string and gauge theory, Adv. High Energy Phys. (2012) 431898.Google Scholar
  24. [24]
    A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M2 branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    A. Hanany and Y.-H. He, M2-branes and quiver Chern-Simons: a taxonomic study, arXiv:0811.4044 [INSPIRE].
  26. [26]
    J. Davey, A. Hanany and J. Pasukonis, On the classification of brane tilings, JHEP 01 (2010) 078 [arXiv:0909.2868] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  27. [27]
    J. Hewlett and Y.-H. He, Probing the space of toric quiver theories, JHEP 03 (2010) 007 [arXiv:0909.2879] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    A. Hanany, V. Jejjala, S. Ramgoolam and R.-K. Seong, Calabi-Yau orbifolds and torus coverings, JHEP 09 (2011) 116 [arXiv:1105.3471] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    A. Hanany and R.-K. Seong, Brane tilings and reflexive polygons, Fortsch. Phys. 60 (2012) 695 [arXiv:1201.2614] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  30. [30]
    A. Hanany and R.-K. Seong, Brane tilings and specular duality, JHEP 08 (2012) 107 [arXiv:1206.2386] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    S. Cremonesi, A. Hanany and R.-K. Seong, Double handled brane tilings, JHEP 10 (2013) 001 [arXiv:1305.3607] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    Y.-H. He and J. Read, Hecke groups, dessins d’enfants and the archimedean solids, arXiv:1309.2326 [INSPIRE].
  33. [33]
    Y.-H. He and M. van Loon, Gauge theories, tessellations & Riemann surfaces, JHEP 06 (2014) 053 [arXiv:1402.3846] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    N.M. Adrianov et al., Catalog of dessins d’enfants with ≤ 4 edges, J. Math. Sci. 158 (2009) 22 [arXiv:0710.2658].CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    J. Sijsling and J. Voight, On computing Belyi maps, arXiv:1311.2529.
  36. [36]
    M. Klug, M. Musty, S. Schiavone and J. Voight, Numerical calculation of three-point branched covers of the projective line, LMS J. Comput. Math. 17 (2014) 379 [arXiv:1311.2081].CrossRefGoogle Scholar
  37. [37]
    M. van Hoeij and R. Vidunas, Algorithms and differential relations for Belyi functions, arXiv:1305.7218.
  38. [38]
    G.A. Jones, Regular dessins with a given automorphism group, arXiv:1309.5219.
  39. [39]
    A. Elkin, Belyi maps webpage, http://homepages.warwick.ac.uk/~masjaf/belyi/.
  40. [40]
    S.K. Lando and A.K. Zvonkin, Graphs on surfaces and their applications, 1st ed., Springer, Germany (2004).Google Scholar
  41. [41]
    S.K. Ashok, F. Cachazo and E. Dell’Aquila, Children’s drawings from Seiberg-Witten curves, Commun. Num. Theor. Phys. 1 (2007) 237 [hep-th/0611082] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  42. [42]
    G. Megyesi, The j-invariant of an elliptic curve, University of Manchester Lecture Notes, Manchester U.K. (2012), pg. 1.Google Scholar
  43. [43]
    T. Nagell, Sur les propriétés arithmétiques des cubiques du premier genre (in French), Acta Math. 52 (1929) 93.Google Scholar
  44. [44]
    B. Feng, S. Franco, A. Hanany and Y.-H. He, UnHiggsing the del Pezzo, JHEP 08 (2003) 058 [hep-th/0209228] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    J.-I. Igusa, Arithmetic variety of moduli for genus two, Ann. Math. 72 (1960) 612.CrossRefMATHMathSciNetGoogle Scholar
  46. [46]
    J.-F. Mestre, Construction de courbes de genre 2 à partir de leurs modules (in French), Progr. Math. 94 (1991) 313.Google Scholar
  47. [47]
    M. Streng, Computing Igusa class polynomials, arXiv:0903.4766.
  48. [48]
    K. Lauter and T. Yang, Computing genus 2 curves from invariants on the Hilbert moduli space, Lecture Notes for the International Association for Cryptologic Research, (2000), pg. 1.Google Scholar
  49. [49]
    K.E. Lauter and K.A. Ribet, Computational arithmetic geometry, 1st ed., American Mathematical Society, U.S.A. (2008).Google Scholar
  50. [50]
    T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967) 1022.CrossRefMATHMathSciNetGoogle Scholar
  51. [51]
    R. Lercier and C. Ritzenthaler, Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects, J. Alg. 372 (2012) 595 [arXiv:1111.4152].CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.St. Catherine’s CollegeUniversity of OxfordOxfordU.K.
  2. 2.Institute for Computational Cosmology, Department of PhysicsDurham UniversityDurhamU.K.
  3. 3.Somerville CollegeUniversity of OxfordOxfordU.K.
  4. 4.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.
  5. 5.Department of MathematicsCity UniversityLondonU.K.
  6. 6.School of PhysicsNanKai UniversityTianjinP.R. China
  7. 7.Merton CollegeUniversity of OxfordOxfordU.K.

Personalised recommendations