Symmetries of abelian orbifolds

  • Amihay Hanany
  • Rak-Kyeong Seong


Using the Polya Enumeration Theorem, we count with particular attention to \( {{{{\mathbb{C}^3}}} \left/ {\Gamma } \right.} \) up to \( {{{{\mathbb{C}^6}}} \left/ {\Gamma } \right.} \), abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S D . This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form \( {{{{\mathbb{C}^D}}} \left/ {\Gamma } \right.} \) where the order of Γ is p α. We propose a generalization of these sequences for any D and any p.


Differential and Algebraic Geometry Superstring Vacua D-branes Conformal Field Models in String Theory 


  1. [1]
    A. Hanany, D. Orlando and S. Reffert, Sublattice counting and orbifolds, JHEP 06 (2010) 051 [arXiv:1002.2981] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    J. Davey, A. Hanany and R.-K. Seong, Counting orbifolds, JHEP 06 (2010) 010 [arXiv:1002.3609] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [SPIRES].
  4. [4]
    M.R. Douglas and B.R. Greene, Metrics on D-brane orbifolds, Adv. Theor. Math. Phys. 1 (1998) 184 [hep-th/9707214] [SPIRES].MathSciNetGoogle Scholar
  5. [5]
    M.R. Douglas, B.R. Greene and D.R. Morrison, Orbifold resolution by D-branes, Nucl. Phys. B 506 (1997) 84 [hep-th/9704151] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    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].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    B.S. Acharya, J.M. Figueroa-O’Farrill, C.M. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [SPIRES].MathSciNetGoogle Scholar
  8. [8]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [SPIRES].
  9. [9]
    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].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    M. Yamazaki, Brane tilings and their applications, Fortsch. Phys. 56 (2008) 555 [arXiv:0803.4474] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  11. [11]
    J. Davey, A. Hanany and J. Pasukonis, On the classification of brane tilings, arXiv:0909.2868 [SPIRES].
  12. [12]
    A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    K.D. Kennaway, Brane tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    T. Muto, D-branes on orbifolds and topology change, Nucl. Phys. B 521 (1998) 183 [hep-th/9711090] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    S. Kachru and E. Silverstein, 4d conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. [17]
    A.E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories in four dimensions, Nucl. Phys. B 533 (1998) 199 [hep-th/9803015] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    M. Bershadsky, Z. Kakushadze and C. Vafa, String expansion as large-N expansion of gauge theories, Nucl. Phys. B 523 (1998) 59 [hep-th/9803076] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    A. Hanany and Y.-H. He, Non-Abelian finite gauge theories, JHEP 02 (1999) 013 [hep-th/9811183] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    C. Beasley, B.R. Greene, C.I. Lazaroiu and M.R. Plesser, D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds, Nucl. Phys. B 566 (2000) 599 [hep-th/9907186] [SPIRES].CrossRefMathSciNetGoogle Scholar
  21. [21]
    A.M. Uranga, From quiver diagrams to particle physics, hep-th/0007173 [SPIRES].
  22. [22]
    B. Feng, A. Hanany, Y.H. He and A. Iqbal, Quiver theories, soliton spectra and Picard-Lefschetz transformations, JHEP 02 (2003) 056 [hep-th/0206152] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  23. [23]
    J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [SPIRES].MathSciNetADSGoogle Scholar
  29. [29]
    A. Hanany and A. Zaffaroni, Tilings, Chern-Simons theories and M2 branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    A. Hanany and Y.-H. He, M2-branes and quiver Chern-Simons: A taxonomic study, arXiv:0811.4044 [SPIRES].
  31. [31]
    A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M2 branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    J. Davey, A. Hanany, N. Mekareeya and G. Torri, Phases of M2-brane theories, JHEP 06 (2009) 025 [arXiv:0903.3234] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  33. [33]
    J. Davey, A. Hanany, N. Mekareeya and G. Torri, Higgsing M2-brane theories, JHEP 11 (2009) 028 [arXiv:0908.4033] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    J. Davey, A. Hanany, N. Mekareeya and G. Torri, Brane tilings, M2-branes and Chern-Simons theories, arXiv:0910.4962 [SPIRES].
  35. [35]
    S. Lee, Superconformal field theories from crystal lattices, Phys. Rev. D 75 (2007) 101901 [hep-th/0610204] [SPIRES].ADSGoogle Scholar
  36. [36]
    S. Lee, S. Lee and J. Park, Toric AdS 4 /CFT 3 duals and M-theory crystals, JHEP 05 (2007) 004 [hep-th/0702120] [SPIRES].CrossRefADSGoogle Scholar
  37. [37]
    M. Taki, M2-branes theories without 3+1 dimensional parents via un-higgsing, arXiv:0910.0370 [SPIRES].
  38. [38]
    C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, JHEP 12 (2001) 001 [hep-th/0109053] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  39. [39]
    B. Feng, S. Franco, A. Hanany and Y.-H. He, Symmetries of toric duality, JHEP 12 (2002) 076 [hep-th/0205144] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    Y.-H. He, On fields over fields, arXiv:1003.2986 [SPIRES].
  41. [41]
    G. Polya and R.C. Reed, Combinatorial enumeration of groups, graphs, and chemical compounds, Springer-Verlag, Heidelberg Germany (1987).CrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

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

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