Wallcrossing, open BPS counting and matrix models

  • Piotr SulkowskiEmail author


We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence.


Matrix Models Topological Strings M-Theory 


  1. [1]
    R. Gopakumar and C. Vafa, M-theory and topological strings. I, hep-th/9809187 [SPIRES].
  2. [2]
    R. Gopakumar and C. Vafa, M-theory and topological strings. II, hep-th/9812127 [SPIRES].
  3. [3]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146 [SPIRES].
  5. [5]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  6. [6]
    D.L. Jafferis and G.W. Moore, Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909 [SPIRES].
  7. [7]
    W.-y. Chuang and D.L. Jafferis, Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions, Commun. Math. Phys. 292 (2009) 285 [arXiv:0810.5072] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. [8]
    T. Dimofte and S. Gukov, Refined, Motivic and Quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [SPIRES].MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    H. Ooguri and M. Yamazaki, Crystal Melting and Toric Calabi-Yau Manifolds, Commun. Math. Phys. 292 (2009) 179 [arXiv:0811.2801] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. [10]
    P. Sulkowski, Wall-crossing, free fermions and crystal melting, Commun. Math. Phys. 301 (2011) 517 [arXiv:0910.5485] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    B. Szendroi, Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol. 12 (2008) 1171 [arXiv:0705.3419] [SPIRES].MathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Young and J . Bryan, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, arXiv:0802.3948 [SPIRES].
  13. [13]
    K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, arXiv:0809.2992 [SPIRES].
  14. [14]
    K. Nagao, Derived categories of small toric Calabi-Yau 3-folds and counting invariants, arXiv:0809.2994.
  15. [15]
    K. Nagao, Non-commutative Donaldson-Thomas theory and vertex operators, arXiv:0910.5477 [SPIRES].
  16. [16]
    M. Aganagic, H. Ooguri, C. Vafa and M. Yamazaki, W all Crossing and M-theory, arXiv:0908.1194 [SPIRES].
  17. [17]
    R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  19. [19]
    B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].
  20. [20]
    V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  21. [21]
    H. Ooguri, P. Sulkowski and M. Yamazaki, Wall Crossing As Seen By Matrix Models, arXiv:1005.1293 [SPIRES].
  22. [22]
    B. Eynard, A Matrix model for plane partitions and TASEP, J. Stat. Mech. (2009) P 10011 [arXiv:0905.0535] [SPIRES].
  23. [23]
    R. Dijkgraaf, P. Sulkowski and C. Vafa, W all-crossing and open topological string theory, (unpublished).Google Scholar
  24. [24]
    M. Aganagic and M. Yamazaki, Open BPS Wall Crossing and M-theory, Nucl. Phys. B 834 (2010) 258 [arXiv:0911.5342] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    P. Sulkowski, Refined matrix models from BPS counting, arXiv:1012.3228 [SPIRES].
  26. [26]
    M. Aganagic and K. Schaeffer, Wall Crossing, Quivers and Crystals, arXiv:1006.2113 [SPIRES].
  27. [27]
    R.J. Szabo and M. Tierz, Matrix models and stochastic growth in Donaldson-Thomas theory, arXiv:1005.5643 [SPIRES].
  28. [28]
    K. Nagao, Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions, arXiv:0907.3784 [SPIRES].
  29. [29]
    K. Nagao and M. Yamazaki, The Non-commutative Topological Vertex and Wall Crossing Phenomena, arXiv:0910.5479 [SPIRES].
  30. [30]
    T. Nishinaka and S. Yamaguchi, Wall-crossing of D4-D2-D0 and flop of the conifold, JHEP 09 (2010) 026 [arXiv:1007.2731] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    S. Cecotti and C. Vafa, BPS Wall Crossing and Topological Strings, arXiv:0910.2615 [SPIRES].
  32. [32]
    N. Saulina and C. Vafa, D-branes as defects in the Calabi-Yau crystal, hep-th/0404246 [SPIRES].
  33. [33]
    N. Halmagyi, A. Sinkovics and P. Sulkowski, Knot invariants and Calabi-Yau crystals, JHEP 01 (2006) 040 [hep-th/0506230] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    P. Sulkowski, Calabi-Yau crystals in topological string theory, arXiv:0712.2173 [SPIRES].
  35. [35]
    J. Gomis and T. Okuda, D-branes as a Bubbling Calabi-Yau, JHEP 07 (2007) 005 [arXiv:0704.3080] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [SPIRES].
  37. [37]
    M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [SPIRES].
  38. [38]
    J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large-N, JHEP 11 (2000) 007 [hep-th/0010102] [SPIRES].ADSCrossRefGoogle Scholar
  39. [39]
    D. Gaiotto, A. Strominger and X. Yin, New Connections Between 4D and 5D Black Holes, JHEP 02 (2006) 024 [hep-th/0503217] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [SPIRES].MathSciNetGoogle Scholar
  41. [41]
    M. Mariño and C. Vafa, Framed knots at large-N, hep-th/0108064 [SPIRES].

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaU.S.A.

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