Wall Crossing as Seen by Matrix Models

  • Hirosi Ooguri
  • Piotr Sułkowski
  • Masahito Yamazaki
Open Access
Article

Abstract

The number of BPS bound states of D-branes on a Calabi-Yau manifold depends on two sets of data, the BPS charges and the stability conditions. For D0 and D2-branes bound to a single D6-brane wrapping a Calabi-Yau 3-fold X, both are naturally related to the Kähler moduli space \({{\mathcal M}(X)}\) . We construct unitary one-matrix models which count such BPS states for a class of toric Calabi-Yau manifolds at infinite ’t Hooft coupling. The matrix model for the BPS counting on X turns out to give the topological string partition function for another Calabi-Yau manifold Y, whose Kähler moduli space \({{\mathcal M}(Y)}\) contains two copies of \({{\mathcal M}(X)}\) , one related to the BPS charges and another to the stability conditions. The two sets of data are unified in \({{\mathcal M}(Y)}\) . The matrix models have a number of other interesting features. They compute spectral curves and mirror maps relevant to the remodeling conjecture. For finite ’t Hooft coupling they give rise to yet more general geometry \({\widetilde{Y}}\) containing Y.

Keywords

Matrix Model Topological String Hooft Coupling Toric Diagram Wall Crossing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank Mina Aganagic, Vincento Bouchard, Kentaro Hori, and Yan Soibelman for discussions. H. O. and P. S. thank Hermann Nicolai and the Max-Planck-Institut für Gravitationsphysik for hospitality. Our work is supported in part by the DOE grant DE-FG03-92-ER40701. H. O. and M. Y. are also supported in part by the World Premier International Research Center Initiative of MEXT. H. O. is supported in part by JSPS Grant-in-Aid for Scientific Research (C) 20540256 and by the Humboldt Research Award. P. S. acknowledges the support of the European Commission under the Marie-Curie International Outgoing Fellowship Programme and the Foundation for Polish Science. M. Y. is supported in part by the JSPS Research Fellowship for Young Scientists and the Global COE Program for Physical Science Frontier at the University of Tokyo.

Open Access

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References

  1. 1.
    Gopakumar, R., Vafa, C.: M-theory and topological strings. I, http://arXiv.org.abs/hep-th/9809187v1, 1998; M-theory and topological strings. II, http://arXiv.org.abs/hep-th/9812127vl, 1998
  2. 2.
    Ooguri H., Strominger A., Vafa C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Aganagic M., Ooguri H., Vafa C., Yamazaki M.: Wall crossing and M-theory. Pub. RIMS 47, 569 (2011)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Okounkov A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals. http://arXiv.org/abs/hep-th/0309208v2, 2003
  5. 5.
    Ooguri H., Yamazaki M.: Crystal Melting and Toric Calabi-Yau Manifolds. Commun. Math. Phys. 292, 179 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  6. 6.
    Sułkowski P.: Wall-crossing, free fermions and crystal melting. Commun. Math. Phys. 301, 517 (2011)ADSCrossRefMATHGoogle Scholar
  7. 7.
    Nagao, K.: Non-commutative Donaldson-Thomas theory and vertex operators. http://arXiv.org/abs/0910.5477v4 [math.AG], 2010
  8. 8.
    Aganagic M., Dijkgraaf R., Klemm A., Marino M., Vafa C.: Topological strings and integrable hierarchies. Commun. Math. Phys 261, 451 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  9. 9.
    Dijkgraaf R., Hollands L., Sułkowski P., Vafa C.: Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions. JHEP 0802, 106 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Dijkgraaf R., Hollands L., Sułkowski P.: Quantum Curves and D-Modules. JHEP 0911, 047 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    Eynard B.: A Matrix model for plane partitions and TASEP. J. Stat. Mech. 0910, P10011 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Marino M.: Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants. Commun. Math. Phys. 253, 25 (2004)ADSGoogle Scholar
  13. 13.
    Aganagic M., Klemm A., Marino M., Vafa C.: Matrix model as a mirror of Chern-Simons theory. JHEP 0402, 010 (2004)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Bouchard V., Klemm A., Marino M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  15. 15.
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. http://arXiv.org/abs/math-ph/0702045v4, 2007
  16. 16.
    Okuda T.: Derivation of Calabi-Yau crystals from Chern-Simons gauge theory. JHEP 0503, 047 (2005)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Eynard B.: All orders asymptotic expansion of large partitions. J. Stat. Mech. 0807, P07023 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Klemm A., Sułkowski P.: Seiberg-Witten theory and matrix models. Nucl. Phys. B 819, 400 (2009)ADSCrossRefMATHGoogle Scholar
  19. 19.
    Sułkowski P.: Matrix models for 2* theories. Phys. Rev. D 80, 086006 (2009)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Sułkowski P.: Matrix models for β-ensembles from Nekrasov partition functions. JHEP 1004, 063 (2010)ADSCrossRefGoogle Scholar
  21. 21.
    Eynard, B., Kashani-Poor, A. K., Marchal, O.: A matrix model for the topological string I: Deriving the matrix model. http://arXiv.org/abs/1003.1737v2 [hep-th], 2010
  22. 22.
    Dijkgraaf, R., Sułkowski, P., Vafa, C.: In progress.Google Scholar
  23. 23.
    Aganagic, M.: In progressGoogle Scholar
  24. 24.
    Aganagic M., Klemm A., Marino M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005)MathSciNetADSCrossRefMATHGoogle Scholar
  25. 25.
    Iqbal A., Kashani-Poor A. K.: The vertex on a strip. Adv. Theor. Math. Phys. 10, 317 (2006)MathSciNetMATHGoogle Scholar
  26. 26.
    Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3 (2002); On geometry and matrix models. Nucl. Phys. B 644, 21 (2002)Google Scholar
  27. 27.
    Marino M.: Chern-Simons Theory, Matrix Models, And Topological Strings. Oxford University Press, Oxford (2005)CrossRefMATHGoogle Scholar
  28. 28.
    Sułkowski P.: Crystal model for the closed topological vertex geometry. JHEP 0612, 030 (2006)ADSCrossRefGoogle Scholar
  29. 29.
    Imamura Y., Isono H., Kimura K., Yamazaki M.: Exactly marginal deformations of quiver gauge theories as seen from brane tilings. Prog. Theor. Phys. 117, 923 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  30. 30.
    Ooguri H., Yamazaki M.: Emergent Calabi-Yau Geometry. Phys. Rev. Lett. 102, 161601 (2009)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Yamazaki M.: Crystal Melting and Wall Crossing Phenomena. Int. J. Mod. Phts A 26, 1097–1228 (2011)ADSCrossRefMATHGoogle Scholar
  32. 32.
    Hori, K., Vafa, C.: Mirror symmetry. http://arXiv.org/abs/hep-th/0002222v3, 2000
  33. 33.
    Witten E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993)MathSciNetADSCrossRefMATHGoogle Scholar
  34. 34.
    Ooguri H., Vafa C.: Worldsheet Derivation of a Large N Duality. Nucl. Phys. B 641, 3 (2002)MathSciNetADSCrossRefMATHGoogle Scholar
  35. 35.
    Bryan, J., Young, B.: Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. http://arXiv.org/abs/0802.3948v2 [math.CO], 2008
  36. 36.
    Nagao, K., Yamazaki, M.: The Non-commutative Topological Vertex and Wall Crossing Phenomena. http://arXiv.org/abs/0910.5479vL [hep-th], 2009
  37. 37.
    Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  38. 38.
    Harish-Chandra : Differential operators on a semisimple Lie algebra. Amer. J. Math. 79, 87 (1957)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Itzykson C., Zuber J. B.: The Planar Approximation. 2. J Math. Phys. 21, 411 (1980)MathSciNetADSCrossRefMATHGoogle Scholar
  40. 40.
    Mozgovoy, S., Reineke, M., On the noncommutative Donaldson-Thomas invariants arising from brane tilings. http://arXiv.org/abs/0809.0117v2 [math.AG], 2008
  41. 41.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. http://arXiv.org/abs/math-ph/0311005v1, 2003
  42. 42.
    Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994)MathSciNetADSCrossRefMATHGoogle Scholar
  43. 43.
    Witten, E.: Quantum background independence in string theory. http://arXiv.org/abs/hep-th/9306122v1, 1993
  44. 44.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arXiv.org/abs/0811.2435v1 [math.AG], 2008
  45. 45.
    Caporaso N., Griguolo L., Marino M., Pasquetti S., Seminara D.: Phase transitions, double-scaling limit, and topological strings. Phys. Rev. D 75, 046004 (2007)MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    Jimbo M., Miwa T.: Solitons and Infinite Dimensional Lie Algebras. Kyoto University, RIMS 19, 943 (1983)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Lindström B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85 (1973)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Gessel I., Viennot G.: Binomial determinants, paths, and hook length formulae. Adv. in Math. 58, 300 (1985)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Aganagic M., Yamazaki M.: Open BPS Wall Crossing and M-theory. Nucl. Phys. B 834, 258 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  50. 50.
    Sułkowski P.: Wall-crossing, open BPS counting and matrix models. JHEP 1103, 089 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Sułkowski P.: Refined matrix models from BPS counting. Phys. Rev. D 83, 085021 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    Mandal G.: Phase Structure Of Unitary Matrix Models. Mod. Phys. Lett. A 5, 1147–1158 (1990)MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Hirosi Ooguri
    • 1
    • 2
    • 3
  • Piotr Sułkowski
    • 1
  • Masahito Yamazaki
    • 2
  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan
  3. 3.Max-Planck-Institut für GravitationsphysikPotsdamGermany

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