Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2011 (2011), Article ID 357016, 52 pages
http://dx.doi.org/10.1155/2011/357016
Review Article

BPS States, Crystals, and Matrices

1California Institute of Technology, Pasadena, CA 91125, USA
2Faculty of Physics, University of Warsaw, ul. Hoża 69, 00-681 Warsaw, Poland

Received 14 May 2011; Accepted 19 June 2011

Academic Editor: Amihay Hanany

Copyright © 2011 Piotr Sułkowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. Denef and G. Moore, “Split states, entropy enigmas, holes and halos,” . In press, http://arxiv.org/abs/hep-th/0702146.
  2. M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,” . In press, http://arxiv.org/abs/0811.2435.
  3. R. Gopakumar and C. Vafa, “M-theory and topological strings. I,” . In press, http://arxiv.org/abs/hep-th/9809187.
  4. R. Gopakumar and C. Vafa, “M-theory and topological strings. II,” . In press, http://arxiv.org/abs/hep-th/9812127.
  5. H. Ooguri and C. Vafa, “Knot invariants and topological strings,” Nuclear Physics B, vol. 577, no. 3, pp. 419–438, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. Aganagic, A. Klemm, M. Mariño, and C. Vafa, “The topological vertex,” Communications in Mathematical Physics, vol. 254, no. 2, pp. 425–478, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Okounkov, N. Reshetikhin, and C. Vafa, “Quantum Calabi-Yau and classical crystals,” . In press, http://arxiv.org/abs/hep-th/0309208.
  8. A. Iqbal, C. Vafa, N. Nekrasov, and A. Okounkov, “Quantum foam and topological strings,” Journal of High Energy Physics, vol. 2008, no. 4, article 011, 2008. View at Publisher · View at Google Scholar
  9. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, “Gromov-Witten theory and Donaldson-Thomas theory, I,” Compositio Mathematica, vol. 142, no. 5, pp. 1263–1285, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. T. Dimofte and S. Gukov, “Refined, motivic, and quantum,” Letters in Mathematical Physics, vol. 91, no. 1, pp. 1–27, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” Advances in Theoretical and Mathematical Physics, vol. 7, no. 5, pp. 831–864, 2004. View at Google Scholar · View at Zentralblatt MATH
  12. A. Iqbal, C. Kozçaz, and C. Vafa, “The refined topological vertex,” Journal of High Energy Physics, no. 10, p. 69, 2009. View at Publisher · View at Google Scholar
  13. K. Nagao, “Refined open noncommutative Donaldson-Thomas invariants for small crepant resolutions,” . In press, http://arxiv.org/abs/0907.3784.
  14. B. Szendrői, “Non-commutative Donaldson-Thomas theory and the conifold,” Geometry & Topology, vol. 12, no. 2, pp. 1171–1202, 2008. View at Publisher · View at Google Scholar
  15. J. Bryan and B. Young, “Generating functions for colored 3D young diagrams and the Donaldson-Thomas invariants of orbifolds,” Duke Mathematical Journal, vol. 152, no. 1, pp. 115–153, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. P. Sułkowski, “Wall-crossing, free fermions and crystal melting,” Communications in Mathematical Physics, vol. 301, no. 2, pp. 517–562, 2011. View at Publisher · View at Google Scholar
  17. K. Nagao, “Noncommutative Donaldson-Thomas theory and vertex operators,” . In press, http://arxiv.org/abs/0910.5477.
  18. K. Nagao and H. Nakajima, “Counting invariant of perverse coherent sheaves and its wallcrossing,” International Mathematics Research Notices, vol. 2011, no. 13, 2011. View at Publisher · View at Google Scholar
  19. K. Nagao, “Derived categories of small toric Calabi-Yau 3-folds and counting invariants,” . In press, http://arxiv.org/abs/0809.2994.
  20. D. Jafferis and G. Moore, “Wall crossing in local Calabi Yau manifolds,” . In press, http://arxiv.org/abs/0810.4909.
  21. W. Y. Chuang and D. L. Jafferis, “Wall crossing of BPS states on the conifold from Seiberg duality and pyramid partitions,” Communications in Mathematical Physics, vol. 292, no. 1, pp. 285–301, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. H. Ooguri and M. Yamazaki, “Crystal melting and toric Calabi-Yau manifolds,” Communications in Mathematical Physics, vol. 292, no. 1, pp. 179–199, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Aganagic, H. Ooguri, C. Vafa, and M. Yamazaki, “Wall crossing and M-theory,” Nuclear Physics B, vol. 47, no. 2, pp. 569–584, 2011. View at Publisher · View at Google Scholar
  24. M. Aganagic and M. Yamazaki, “Open BPS wall crossing and M-theory,” Nuclear Physics B, vol. 834, no. 1-2, pp. 258–272, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. R. Dijkgraaf, P. Sułkowski, and C. Vafa, unpublished.
  26. K. Nagao and M. Yamazaki, “The non-commutative topological vertex and wall crossing phenomena,” Advances in Theoretical and Mathematical Physics, vol. 14, pp. 1147–1181, 2010. View at Google Scholar
  27. P. Sułkowski, “Wall-crossing, open BPS counting and matrix models,” Journal of High Energy Physics, vol. 2011, no. 3, p. 89, 2011. View at Publisher · View at Google Scholar
  28. B. Eynard, “A matrix model for plane partitions and TASEP,” Journal of Statistical Mechanics, no. 10, Article ID P10011, 2009. View at Google Scholar
  29. H. Ooguri, P. Sułkowski, and M. Yamazaki, “Wall Crossing As Seen By Matrix Models,” Communications in Mathematical Physics, 2011. View at Google Scholar
  30. R. Szabo and M. Tierz, “Matrix models and stochastic growth in Donaldson-Thomas theory,” . In press, http://arxiv.org/abs/1005.5643.
  31. P. Sułkowski, “Refined matrix models from BPS counting,” Physical Review D, vol. 83, no. 8, Article ID 085021, 12 pages, 2011. View at Publisher · View at Google Scholar
  32. N. Saulina and C. Vafa, “D-branes as defects in the Calabi-Yau crystal,” . In press, http://arxiv.org/abs/hep-th/0404246.
  33. N. Halmagyi, A. Sinkovics, and P. Sułkowski, “Knot invariants and Calabi-Yau crystals,” Journal of High Energy Physics, no. 1, article 040, p. 32, 2006. View at Publisher · View at Google Scholar
  34. J. Gomis and T. Okuda, “D-branes as a bubbling Calabi-Yau,” Journal of High Energy Physics, no. 7, article 005, p. 28, 2007. View at Publisher · View at Google Scholar
  35. M. Aganagic and K. Schaeffer, “Wall crossing, quivers and crystals,” . In press, http://arxiv.org/abs/1006.2113.
  36. T. Nishinaka and S. Yamaguchi, “Wall-crossing of D4-D2-D0 and flop of the conifold,” Journal of High Energy Physics, vol. 2010, no. 9, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. P. Sułkowski, “Calabi-Yau crystals in topological string theory,” . In press, http://arxiv.org/abs/0712.2173.
  38. M. Yamazaki, “Crystal melting and wall crossing phenomena,” International Journal of Modern Physics A, vol. 26, no. 7-8, pp. 1097–1228, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. C. Vafa and E. Zaslow, Eds., Mirror Symmetry, CMI/AMS publication.
  40. R. Dijkgraaf, C. Vafa, and E. Verlinde, “M-theory and a topological string duality,” . In press, http://arxiv.org/abs/hep-th/0602087.
  41. J. M. F. Labastida, M. Mariño, and C. Vafa, “Knots, links and branes at large N,” Journal of High Energy Physics, no. 11, p. 7, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. D. Gaiotto, A. Strominger, and X. Yin, “New connections between 4D and 5D black holes,” Journal of High Energy Physics, no. 2, p. 10, article 024, 2006. View at Publisher · View at Google Scholar
  43. M. Jimbo and T. Miwa, “Solitons and infinite-dimensional lie algebras,” Kyoto University. Research Institute for Mathematical Sciences. Publications, vol. 19, no. 3, pp. 943–1001, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, NY, USA, 2nd edition, 1995.
  45. M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño, and C. Vafa, “Topological strings and integrable hierarchies,” Communications in Mathematical Physics, vol. 261, no. 2, pp. 451–516, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  46. P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, “2D gravity and random matrices,” Physics Reports, vol. 254, no. 1-2, pp. 1–133, 1995. View at Publisher · View at Google Scholar
  47. M. Mariño, Chern-Simons Theory, Matrix Models, And Topological Strings, vol. 131 of International Series of Monographs on Physics, The Clarendon Press Oxford University Press, Oxford, UK, 2005. View at Publisher · View at Google Scholar
  48. B. Eynard and N. Orantin, “Invariants of algebraic curves and topological expansion,” . In press, http://arxiv.org/abs/math-ph/0702045.
  49. M. Mariño, “Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants,” Communications in Mathematical Physics, vol. 253, no. 1, pp. 25–49, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  50. M. Aganagic, A. Klemm, M. Mariño, and C. Vafa, “Matrix model as a mirror of Chern-Simons theory,” Journal of High Energy Physics, no. 2, article 010, p. 46, 2004. View at Publisher · View at Google Scholar
  51. V. Bouchard, A. Klemm, M. Mariño, and S. Pasquetti, “Remodeling the B-model,” Communications in Mathematical Physics, vol. 287, no. 1, pp. 117–178, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  52. B. Eynard, “All order asymptotic expansion of large partitions,” Journal of Statistical Mechanics, no. 7, Article ID P07023, 2008. View at Google Scholar
  53. A. Klemm and P. Sułkowski, “Seiberg-Witten theory and matrix models,” Nuclear Physics B, vol. 819, no. 3, pp. 400–430, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  54. P. Sułkowski, “Matrix models for 2* theories,” Physical Review D, vol. 80, no. 8, Article ID 086006, 2009. View at Google Scholar
  55. P. Sułkowski, “Matrix models for β-ensembles from Nekrasov partition functions,” Journal of High Energy Physics, no. 4, article 063, p. 63, 2010. View at Google Scholar
  56. B. Eynard, A. K. Kashani-Poor, and O. Marchal, “A matrix model for the topological string I: deriving the matrix model,” . In press, http://arxiv.org/abs/1003.1737.
  57. M. Taki, “Refined topological vertex and instanton counting,” Journal of High Energy Physics, no. 3, article 048, p. 48, 2008. View at Publisher · View at Google Scholar
  58. H. Awata and H. Kanno, “Instanton counting, Macdonald function and the moduli space of D-branes,” Journal of High Energy Physics, no. 5, article 039, p. 26, 2005. View at Publisher · View at Google Scholar
  59. H. Awata and H. Kanno, “Refined BPS state counting from Nekrasov's formula and Macdonald functions,” International Journal of Modern Physics A, vol. 24, no. 12, pp. 2253–2306, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  60. I. Antoniadis, S. Hohenegger, K. S. Narain, and T. R. Taylor, “Deformed topological partition function and Nekrasov backgrounds,” Nuclear Physics B, vol. 838, no. 3, pp. 253–265, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  61. Y. Nakayama, “Refined topological amplitudes in N=1 flux compactification,” Journal of High Energy Physics, vol. 2010, no. 11, article 117, pp. 1–14, 2010. View at Publisher · View at Google Scholar
  62. L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville correlation functions from four-dimensional Gauge theories,” Letters in Mathematical Physics, vol. 91, no. 2, pp. 167–197, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  63. R. Dijkgraaf and C. Vafa, “Toda theories, matrix models, topological strings, and N=2 Gauge systems,” . In press, http://arxiv.org/abs/0909.2453.
  64. A. Mironov, A. Morozov, and A. Morozov, “Matrix model version of AGT conjecture and generalized Selberg integrals,” Nuclear Physics B, vol. 843, no. 2, pp. 534–557, 2011. View at Publisher · View at Google Scholar
  65. H. Awata and Y. Yamada, “Five-dimensional AGT relation and the deformed beta-ensemble,” Progress of Theoretical Physics, vol. 124, no. 2, pp. 227–262, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  66. M. Huang and A. Klemm, “Direct integration for general Omega backgrounds,” . In press, http://arxiv.org/abs/1009.1126.
  67. A. Brini, M. Mariño, and S. Stevan, “The uses of the refined matrix model recursion,” Journal of Mathematical Physics, vol. 52, no. 5, 2011. View at Publisher · View at Google Scholar
  68. M. Aganagic and S. Shakirov, “Knot homology from refined Chern-Simons theory,” . In press, http://arxiv.org/abs/1105.5117.