Advertisement

Brane tilings and specular duality

  • Amihay Hanany
  • Rak-Kyeong Seong
Article

Abstract

We study a new duality which pairs 4d \( \mathcal{N} = 1 \) supersymmetric quiver gauge theories. They are represented by brane tilings and are worldvolume theories of D3 branes at Calabi-Yau 3-fold singularities. The new duality identifies theories which have the same combined mesonic and baryonic moduli space, otherwise called the master space. We obtain the associated Hilbert series which encodes both the generators and defining relations of the moduli space. We illustrate our findings with a set of brane tilings that have reflexive toric diagrams.

Keywords

Supersymmetric gauge theory D-branes Differential and Algebraic Geometry Superstring Vacua 

References

  1. [1]
    W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau Manifolds in Weighted P(4), Nucl. Phys. B 341 (1990) 383 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    B.R. Greene and M. Plesser, Duality in Calabi-Yau moduli space, Nucl. Phys. B 338 (1990) 15 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    D.R. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993) 223 [alg-geom/9202004].MathSciNetCrossRefGoogle Scholar
  5. [5]
    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  6. [6]
    V. Batyrev and D. Dais, Strong McKay correspondence, string theoretic Hodge numbers and mirror symmetry, alg-geom/9410001 [INSPIRE].
  7. [7]
    V.V. Batyrev and L.A. Borisov, Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, in Mirror symmetry II, B. Greene and S.T. Yau eds., American Mathematical Society Press, Providence U.S.A. (2000), pg. 71.Google Scholar
  8. [8]
    D. Cox and S. Katz, Mathematical surveys and monographs. Vol. 68: Mirror symmetry and algebraic geometry, American Mathematical Society Press, Providence U.S.A. (1999).Google Scholar
  9. [9]
    K. Hori et al., Clay mathematics monographs. Vol. 1: Mirror symmetry, American Mathematical Society, Providence U.S.A. (2003).Google Scholar
  10. [10]
    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].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    B. Feng, A. Hanany and Y.-H. He, Phase structure of D-brane gauge theories and toric duality, JHEP 08 (2001) 040 [hep-th/0104259] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    B. Feng, S. Franco, A. Hanany and Y.-H. He, Symmetries of toric duality, JHEP 12 (2002) 076 [hep-th/0205144] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    B. Feng, A. Hanany, Y.-H. He and A.M. Uranga, Toric duality as Seiberg duality and brane diamonds, JHEP 12 (2001) 035 [hep-th/0109063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, JHEP 12 (2001) 001 [hep-th/0109053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    S. Franco, A. Hanany and Y.-H. He, A trio of dualities: walls, trees and cascades, Fortsch. Phys. 52 (2004) 540 [hep-th/0312222] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. [17]
    A. Hanany and R.-K. Seong, Brane Tilings and Reflexive Polygons, Fortsch. Phys. 60 (2012) 695 [arXiv:1201.2614] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. [18]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
  19. [19]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    A. Hanany, C.P. Herzog and D. Vegh, Brane tilings and exceptional collections, JHEP 07 (2006) 001 [hep-th/0602041] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    K.D. Kennaway, Brane tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    M. Yamazaki, Brane tilings and their applications, Fortsch. Phys. 56 (2008) 555 [arXiv:0803.4474] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  25. [25]
    M. Kreuzer and H. Skarke, On the classification of reflexive polyhedra, Commun. Math. Phys. 185 (1997) 495 [hep-th/9512204] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. [26]
    M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three-dimensions, Adv. Theor. Math. Phys. 2 (1998) 847 [hep-th/9805190] [INSPIRE].MathSciNetGoogle Scholar
  27. [27]
    M. Kreuzer and H. Skarke, Reflexive polyhedra, weights and toric Calabi-Yau fibrations, Rev. Math. Phys. 14 (2002) 343 [math/0001106] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetGoogle Scholar
  29. [29]
    V. Batyrev and M. Kreuzer, Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions, Adv. Theor. Math. Phys. 14 (2010) 879 [arXiv:0802.3376] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  30. [30]
    P. Candelas and R. Davies, New Calabi-Yau Manifolds with Small Hodge Numbers, Fortsch. Phys. 58 (2010) 383 [arXiv:0809.4681] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. [31]
    S. Benvenuti and A. Hanany, New results on superconformal quivers, JHEP 04 (2006) 032 [hep-th/0411262] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    S. Benvenuti and A. Hanany, Conformal manifolds for the conifold and other toric field theories, JHEP 08 (2005) 024 [hep-th/0502043] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    A. Hanany and R.-K. Seong, work in progress.Google Scholar
  34. [34]
    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) 489 [hep-th/0511287] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  35. [35]
    S. Franco, Dimer models, integrable systems and quantum Teichmüller space, JHEP 09 (2011) 057 [arXiv:1105.1777] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    J. Stienstra, Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins dEnfants, arXiv:0711.0464 [INSPIRE].
  37. [37]
    A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni, Counting chiral operators in quiver gauge theories, JHEP 11 (2007) 092 [arXiv:0705.2771] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    S. Franco et al., Dimers and orientifolds, JHEP 09 (2007) 075 [arXiv:0707.0298] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    A. Hanany, D. Vegh and A. Zaffaroni, Brane Tilings and M2 Branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The Master Space of N = 1 Gauge Theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, Mastering the Master Space, Lett. Math. Phys. 85 (2008) 163 [arXiv:0801.3477] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. [42]
    A. Ishii and K. Ueda, On moduli spaces of quiver representations associated with dimer models, arXiv:0710.1898.
  43. [43]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    A. Hanany and A. Zaffaroni, The master space of supersymmetric gauge theories, Adv. High Energy Phys. 2010 (2010) 427891.MathSciNetGoogle Scholar
  45. [45]
    S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    A. Hanany, Counting BPS operators in the chiral ring: the plethystic story, AIP Conf. Proc. 939 (2007) 165 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    D. Forcella, A. Hanany and A. Zaffaroni, Master space, Hilbert series and Seiberg duality, JHEP 07 (2009) 018 [arXiv:0810.4519] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
  50. [50]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
  51. [51]
    R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar graphs with faces of degree 4, Trans. Amer. Math. Soc. 357 (2005) 3443 [math-ph/0305057].MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Hanany, D. Orlando and S. Reffert, Sublattice counting and orbifolds, JHEP 06 (2010) 051 [arXiv:1002.2981] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    J. Davey, A. Hanany and R.-K. Seong, An Introduction to Counting Orbifolds, Fortsch. Phys. 59 (2011) 677 [arXiv:1102.0015] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  55. [55]
    A. Hanany and R.-K. Seong, Symmetries of Abelian orbifolds, JHEP 01 (2011) 027 [arXiv:1009.3017] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    J. Davey, A. Hanany and R.-K. Seong, Counting orbifolds, JHEP 06 (2010) 010 [arXiv:1002.3609] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    A. Hanany, V. Jejjala, S. Ramgoolam and R.-K. Seong, Calabi-Yau orbifolds and torus coverings, JHEP 09 (2011) 116 [arXiv:1105.3471] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Theoretical Physics Group, The Blackett LaboratoryImperial College LondonLondonU.K.

Personalised recommendations