Communications in Mathematical Physics

, Volume 301, Issue 2, pp 517–562 | Cite as

Wall-Crossing, Free Fermions and Crystal Melting

Open Access
Article

Abstract

We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years.

Keywords

Partition Function Vertex Operator Topological String Free Fermion Toric Diagram 
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

I thank Jim Bryan for discussions which inspired this project. I am also grateful to Robbert Dijkgraaf, Albrecht Klemm, Hirosi Ooguri, Yan Soibelman, Balazs Szendroi, Cumrun Vafa, and Masahito Yamazaki for useful conversations. I appreciate the hospitality and inspiring atmosphere of the Focus Week on New Invariants and Wall Crossing organized at IPMU in Tokyo, International Workshop on Mirror Symmetry organized at the University of Bonn, and 7 th Simons Workshop on Mathematics and Physics, as well as the High Energy Theory Group at Harvard University. This research was supported by the DOE grant DE-FG03-92ER40701FG-02, the Humboldt Fellowship, the Foundation for Polish Science, and the European Commission under the Marie-Curie International Outgoing Fellowship Programme. The contents of this publication reflect only the views of the author and not the views of the European Commission.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and Classical Crystals. http://arxiv.org/abs/hep-th/0309208v2, 2003
  2. 2.
    Iqbal A., Nekrasov N., Okounkov A., Vafa C.: Quantum foam and topological strings. JHEP 0804, 011 (2008)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Sułkowski, P.: Calabi-Yau crystals in topological string theory. PhD thesis, available at http://arxiv.org/abs/0712.2173v1 [hep-th], 2007
  4. 4.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory, I. http://arxiv.org/abs/math/0312059v3 [math.AG], 2004
  5. 5.
    Denef, F., Moore, G.: Split states, entropy enigmas, holes and halos. http://arxiv.org/abs/hep-th/0702146v2, 2007
  6. 6.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arxiv.org/abs/0811.2435v1 [math.AG], 2008
  7. 7.
    Jafferis, D., Moore, G.: Wall crossing in local Calabi Yau manifolds. http://arxiv.org/abs/0810.4909v1 [hep-th], 2008
  8. 8.
    Chunag W., Jafferis D.: Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions. Commun. Math. Phys. 292, 285–301 (2009)CrossRefADSGoogle Scholar
  9. 9.
    Dimofte T., Gukov S.: Refined, Motivic, and Quantum. Lett. Math. Phys. 91, 1 (2010)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Szendroi B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171–1202 (2008)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Young, B.: Computing a pyramid partition generating function with dimer shuffling. http://arxiv.org/abs/0709.3079v2 [math.CO], 2008
  12. 12.
    Pandharipande R., Thomas R.: Stable pairs and BPS invariants. J. Amer. Math. Soc. 23, 267–297 (2010)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Mozgovoy, S., Reineke, M.: On the noncommutative Donaldson-Thomas invariants arising from brane tilings. http://arxiv.org/abs/0809.0117v2 [math.AG], 2008
  15. 15.
    Nagao, K., Nakajima, H.: Counting invariant of perverse coherent sheaves and its wall-crossing. http://arxiv.org/abs/0809.2992v4 [math.AG], 2009
  16. 16.
    Nagao, K.: Derived categories of small toric Calabi-Yau 3-folds and counting invariants. http://arxiv.org/abs/0809.2994 [math.AG]
  17. 17.
    Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. http://arxiv.org/abs/hep-th/0602087v1, 2006
  18. 18.
    Aganagic, M., Ooguri, H., Vafa, C., Yamazaki, M.: Wall Crossing and M-Theory. http://arxiv.org/abs/0908.1194v1 [hep-th], 2009
  19. 19.
    Witten, E.: Quantum Background Independence In String Theory. http://arxiv.org/abs/hep-th/9306122v1, 1993
  20. 20.
    Ooguri H., Vafa C., Verlinde E.: Hartle-Hawking Wave-Function for Flux Compactifications. Lett. Math. Phys. 74, 311 (2005)CrossRefMathSciNetADSMATHGoogle Scholar
  21. 21.
    Aganagic M., Dijkgraaf R., Klemm A., Marino M., Vafa C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006)CrossRefMathSciNetADSMATHGoogle Scholar
  22. 22.
    Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. http://arxiv.org/abs/hep-th/0306238v2, 2003
  23. 23.
    Dijkgraaf R., Hollands L., Sułkowski P., Vafa C.: Supersymmetric gauge theories, intersecting branes and free fermions. JHEP 0802, 106 (2008)CrossRefADSGoogle Scholar
  24. 24.
    Dijkgraaf R., Hollands L., Sułkowski P.: Quantum Curves and \({\mathcal{D}}\)-modules. JHEP 0911, 047 (2009)CrossRefADSGoogle Scholar
  25. 25.
    Jimbo M., Miwa T.: Solitons and Infinite Dimensional Lie Algebras. Kyoto University, RIMS 19, 943–1001 (1983)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, Oxford: Oxford Univ. Press, 1995Google Scholar
  27. 27.
    Iqbal A., Kashani-Poor A.: The vertex on a strip. Adv. Theor. Math. Phys. 10, 317 (2006)MathSciNetMATHGoogle Scholar
  28. 28.
    Ooguri H., Yamazaki M.: Crystal Melting and Toric Calabi-Yau Manifolds. Commun. Math. Phys. 292, 179–199 (2009)CrossRefMathSciNetADSMATHGoogle Scholar
  29. 29.
    Nagao, K.: Noncommutative Donaldson-Thomas theory and vertex operators. http://arxiv.org/abs/0910.5477v3 [math.AG], 2009
  30. 30.
    Nagao, K.: Refined open noncommutative Donaldson-Thomas invariants for small crepant resolutions. http://arxiv.org/abs/0907.3784v2 [math.AG], 2009
  31. 31.
    Nagao, K., Yamazaki, M.: The Non-commutative Topological Vertex and Wall Crossing Phenomena. http://arxiv.org/abs/0910.5479v1 [hep-th], 2009
  32. 32.
    Young, B.: To appearGoogle Scholar
  33. 33.
    Bryan, J., Cadman, C., Young, B.: The orbifold topological vertex. http://arxiv.org/abs/1008.4205v1 [math.AG], 2010
  34. 34.
    Sułkowski P.: Crystal Model for the Closed Topological Vertex Geometry. JHEP 0612, 030 (2006)CrossRefADSGoogle Scholar
  35. 35.
    Sułkowski P.: Deformed boson-fermion correspondence, Q-bosons, and topological strings on the conifold. JHEP 0810, 104 (2008)CrossRefADSGoogle Scholar
  36. 36.
    Iqbal A., Kozcaz C., Vafa C.: The Refined Topological Vertex. JHEP 0910, 069 (2009)CrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Ooguri H., Vafa C.: Knot Invariants and Topological Strings. Nucl. Phys. B577, 419–438 (2000)CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Saulina, N., Vafa, C.: D-branes as defects in the Calabi-Yau crystal. http://arxiv.org/abs/hep-th/0404246v2, 2004
  39. 39.
    Halmagyi N., Sinkovics A., Sułkowski P.: Knot invariants and Calabi-Yau crystals. JHEP 0601, 040 (2006)CrossRefADSGoogle Scholar
  40. 40.
    Cecotti, S., Vafa, C.: BPS Wall Crossing and Topological Strings. http://arxiv.org/abs/0910.2615v2 [hep-th], 2009
  41. 41.
    Hollowood T., Iqbal A., Vafa C.: Matrix Models, Geometric Engineering and Elliptic Genera. JHEP 0803, 069 (2008)CrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Sułkowski P.: Matrix models for 2* theories. Phys. Rev. D 80, 086006 (2009)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

Personalised recommendations