Remarks on quiver gauge theories from open topological string theory

  • Nils Carqueville
  • Alexander Quintero Vélez


We study effective quiver gauge theories arising from a stack of D3-branes on certain Calabi-Yau singularities. Our point of view is a first principle approach via open topological string theory. This means that we construct the natural A -structure of open string amplitudes in the associated D-brane category. Then we show that it precisely reproduces the results of the method of brane tilings, without having to resort to any effective field theory computations. In particular, we prove a general and simple formula for effective superpotentials.


D-branes Topological Strings Supersymmetric Effective Theories 


  1. [1]
    M. Aganagic and C. Beem, The geometry of D-brane superpotentials, arXiv:0909.2245 [SPIRES].
  2. [2]
    M. Alim et al., Hints for off-shell mirror symmetry in type-II/F-theory compactifications, arXiv:0909.1842 [SPIRES].
  3. [3]
    M. Alim, M. Hecht, P. Mayr and A. Mertens, Mirror symmetry for toric branes on compact hypersurfaces, JHEP 09 (2009) 126 [arXiv:0901.2937] [SPIRES].CrossRefADSGoogle Scholar
  4. [4]
    S.K. Ashok, E. Dell’Aquila, D.-E. Diaconescu and B. Florea, Obstructed D-branes in Landau-Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004) 427 [hep-th/0404167] [SPIRES].MathSciNetGoogle Scholar
  5. [5]
    P.S. Aspinwall, Topological D-branes and commutative algebra, hep-th/0703279 [SPIRES].
  6. [6]
    P.S. Aspinwall and L.M. Fidkowski, Superpotentials for quiver gauge theories, JHEP 10 (2006) 047 [hep-th/0506041] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    P.S. Aspinwall and A.E. Lawrence, Derived categories and zero-brane stability, JHEP 08 (2001) 004 [hep-th/0104147] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    M. R. Ballard, Sheaves on local Calabi-Yau varieties, arXiv:0801.3499.
  9. [9]
    M. Baumgartl, I. Brunner and M.R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP 07 (2007) 061 [arXiv:0704.2666] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    M. Baumgartl and S. Wood, Moduli Webs and Superpotentials for Five-Branes, JHEP 06 (2009) 052 [arXiv:0812.3397] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    A.A. Beĭlinson, Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978) 68.MATHGoogle Scholar
  12. [12]
    M. Bender and S. Mozgovoy, Crepant resolutions and brane tilings II: tilting bundles, arXiv:0909.2013.
  13. [13]
    A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003) 1 [math/0204218].MATHMathSciNetGoogle Scholar
  14. [14]
    A.I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 25.Google Scholar
  15. [15]
    T. Bridgeland, T-structures on some local Calabi-Yau varieties, math/0502050.
  16. [16]
    N. Broomhead, Dimer models and Calabi-Yau algebras, arXiv:0901.4662 [SPIRES].
  17. [17]
    I. Brunner, M.R. Douglas, A.E. Lawrence and C. Romelsberger, D-branes on the quintic, JHEP 08 (2000) 015 [hep-th/9906200] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A geometric unification of dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    N. Carqueville, Matrix factorisations and open topological string theory, JHEP 07 (2009) 005 [arXiv:0904.0862] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    K. Costello, Topological conformal field theories and Calabi-Yau categories, math/0412149.
  21. [21]
    A. Craw and G. G. Smith, Projective toric varieties as fine moduli spaces of quiver representations, Amer. J. Math. 130 (2008) 1509 [math/0608183].MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818 [hep-th/0011017] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  23. [23]
    M.R. Douglas, S. Govindarajan, T. Jayaraman and A. Tomasiello, D-branes on Calabi-Yau manifolds and superpotentials, Commun. Math. Phys. 248 (2004) 85 [hep-th/0203173] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  24. [24]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [SPIRES].
  25. [25]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    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
  29. [29]
    S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    M. Futaki and K. Ueda, A-infinity categories associated with dimer models, arXiv:0912.1656.
  31. [31]
    S. Govindarajan and H. Jockers, Effective superpotentials for B-branes in Landau-Ginzburg models, JHEP 10 (2006) 060 [hep-th/0608027] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, The D5-brane effective action and superpotential in N = 1 compactifications, Nucl. Phys. B 816 (2009) 139 [arXiv:0811.2996] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  33. [33]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing brane and flux superpotentials in F-theory compactifications, arXiv:0909.2025 [SPIRES].
  34. [34]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Five-brane superpotentials and heterotic/F-theory duality, arXiv:0912.3250 [SPIRES].
  35. [35]
    D.R. Gulotta, Properly ordered dimers, R-charges and an efficient inverse algorithm, JHEP 10 (2008) 014 [arXiv:0807.3012] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    A. Hanany, C.P. Herzog and D. Vegh, Brane tilings and exceptional collections, JHEP 07 (2006) 001 [hep-th/0602041] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [SPIRES].
  38. [38]
    M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [SPIRES].
  39. [39]
    M. Herbst, C.-I. Lazaroiu and W. Lerche, Superpotentials, A relations and WDVV equations for open topological strings, JHEP 02 (2005) 071 [hep-th/0402110] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    L. Hille and M. van den Bergh, Fourier-Mukai transforms, in Handbook of tilting theory, London Math. Soc. Lecture Note Ser. volume 332, Cambridge University Press, Cambridge U.K. (2007), page 147.Google Scholar
  41. [41]
    C. Hofman and W.-K. Ma, Deformations of topological open strings, JHEP 01 (2001) 035 [hep-th/0006120] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  42. [42]
    A. Ishii and K. Ueda, Dimer models and exceptional collections, arXiv:0911.4529.
  43. [43]
    A.. Ishii and K. Ueda, Dimer models and the special McKay correspondence, arXiv:0905.0059.
  44. [44]
    A. Ishii and K. Ueda, On moduli spaces of quiver representations associated with dimer models, RIMS Kôkyûroku Bessatsu B 9 (2008) 127 [arXiv:0710.1898].MathSciNetGoogle Scholar
  45. [45]
    H. Jockers and M. Soroush, Effective superpotentials for compact D5-brane Calabi-Yau geometries, Commun. Math. Phys. 290 (2009) 249 [arXiv:0808.0761] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  46. [46]
    H. Jockers and M. Soroush, Relative periods and open-string integer invariants for a compact Calabi-Yau hypersurface, Nucl. Phys. B 821 (2009) 535 [arXiv:0904.4674] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  47. [47]
    T. Kadeishvili, On the homology theory of fibre spaces (in Russian), Uspekhi Mat. Nauk 35 (1980) 183 [Math. Surveys 35 (1980) 231] [math.AT/0504437].MathSciNetGoogle Scholar
  48. [48]
    M.M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988) 479.MATHCrossRefMathSciNetADSGoogle Scholar
  49. [49]
    B. Keller, A algebras in representation theory, contribution to the proceedings of International Conference on Representations of Algebras and Related Topics (ICRA X), August 21–September 1, Beijing, China (2000).Google Scholar
  50. [50]
    B. Keller, Introduction to A algebras and modules, Homol. Homotopy Appl. 3 (2001) 1 [math.RA/9910179].MATHADSGoogle Scholar
  51. [51]
    B. Keller, A algebras, modules and functor categories, Trends in representation theory of algebras and related topics, Contemp. Math. volume 406, American Mathematical Society, U.S.A. (2006).Google Scholar
  52. [52]
    A.D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994) 515.MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    A. D. King, Tilting bundles on some rational surfaces, available at∼masadk/papers/.
  54. [54]
    J. Knapp and E. Scheidegger, Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces, arXiv:0805.1013 [SPIRES].
  55. [55]
    M. Kontsevich, Formal (non)commutative symplectic geometry, in The Gelfand Mathematical Seminars, 19901992, Fields Institute Communications, Birkhäuser Boston U.S.A. (1993).Google Scholar
  56. [56]
    M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, math/0011041.
  57. [57]
    C.I. Lazaroiu, Generating the superpotential on a D-brane category. I, hep-th/0610120 [SPIRES].
  58. [58]
    C.I. Lazaroiu, String field theory and brane superpotentials, JHEP 10 (2001) 018 [hep-th/0107162] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  59. [59]
    C.I. Lazaroiu, On the non-commutative geometry of topological D-branes, JHEP 11 (2005) 032 [hep-th/0507222] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  60. [60]
    K. Lefèvre-Hasegawa, Sur les A catégories, math.CT/0310337.
  61. [61]
    D.-M. Lu, J.H. Palmieri, Q.-S. Wu and J.J. Zhang, A structure on Ext-algebras, J. Pure Appl. Algebra 213 (2009) 2017 [math/0606144].MATHCrossRefMathSciNetGoogle Scholar
  62. [62]
    S.A. Merkulov, Strongly homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices 3 (1999) 153 [math.AG/9809172].CrossRefMathSciNetGoogle Scholar
  63. [63]
    D.R. Morrison and J. Walcher, D-branes and normal functions, arXiv:0709.4028 [SPIRES].
  64. [64]
    S. Mozgovoy, Crepant resolutions and brane tilings I: Toric realization, arXiv:0908.3475.
  65. [65]
    S. Mukhopadhyay and K. Ray, Seiberg duality as derived equivalence for some quiver gauge theories, JHEP 02 (2004) 070 [hep-th/0309191] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  66. [66]
    A.N. Rudakov, Exceptional collections, mutations and helices, in Helices and vector bundles, London Math. Soc. Lecture Note Ser. volume 148, Cambridge University Press, Cambridge U.K. (1990), page 1.Google Scholar
  67. [67]
    E. Segal, The A deformation theory of a point and the derived categories of local Calabi-Yaus, J. Alg. 320 (2008) 3232 [math/0702539].MATHCrossRefGoogle Scholar
  68. [68]
    P. Seidel, Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry, arXiv:0907.2063.
  69. [69]
    E.R. Sharpe, D-branes, derived categories and Grothendieck groups, Nucl. Phys. B 561 (1999) 433 [hep-th/9902116] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  70. [70]
    J. Stienstra, Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants, arXiv:0711.0464 [SPIRES].
  71. [71]
    A. Tomasiello, A-infinity structure and superpotentials, JHEP 09 (2001) 030 [hep-th/0107195] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  72. [72]
    J. Walcher, Opening mirror symmetry on the quintic, Commun. Math. Phys. 276 (2007) 671 [hep-th/0605162] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  73. [73]
    J. Walcher, Calculations for mirror symmetry with D-branes, JHEP 09 (2009) 129 [arXiv:0904.4905] [SPIRES].CrossRefADSGoogle Scholar
  74. [74]
    E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [SPIRES].

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Arnold Sommerfeld Center for Theoretical PhysicsLMU MünchenMünchenGermany
  2. 2.Excellence Cluster UniverseGarchingGermany
  3. 3.Department of MathematicsUniversity of GlasgowGlasgowU.K.

Personalised recommendations