Skip to main content
SpringerLink
Log in

Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we study the relation between pyramid partitions with a general empty room configuration (ERC) and the BPS states of D-branes on the resolved conifold. We find that the generating function for pyramid partitions with a length n ERC is exactly the same as the D6/D2/D0 BPS partition function on the resolved conifold in particular Kähler chambers. We define a new type of pyramid partition with a finite ERC that counts the BPS degeneracies in certain other chambers.

The D6/D2/D0 partition functions in different chambers were obtained by applying the wall crossing formula. On the other hand, the pyramid partitions describe T 3 fixed points of the moduli space of a quiver quantum mechanics. This quiver arises after we apply Seiberg dualities to the D6/D2/D0 system on the conifold and choose a particular set of FI parameters. The arrow structure of the dual quiver is confirmed by computation of the Ext group between the sheaves. We show that the superpotential and the stability condition of the dual quiver with this choice of the FI parameters give rise to the rules specifying pyramid partitions with length n ERC.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. http://arXiv.org/abs/hep-th/0702146v2, 2007

  2. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arXiv.org/abs/0811.2435v1[math.AG], 2008

  3. Iqbal A., Nekrasov N., Okounkov A., Vafa C.: Quantum foam and topological strings. JHEP 0804, 011 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  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; Gromov-Witten theory and Donaldson-Thomas theory, II. http://arXiv.org/abs/math/0406092v2[math.AG], 2005

  5. Szendrői B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171–1202 (2008)

    Article  MathSciNet  Google Scholar 

  6. Young, B.: Computing a pyramid partition generating function with dimer shuffling. http://arXiv.org/abs/0709.3079v2[math.CO], 2008

  7. Jafferis, D.L., Moore, G.W.: Wall crossing in local Calabi Yau manifolds. http://arXiv.org/abs/0810.4909v1[hep-th], 2008

  8. Klebanov I.R., Witten E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536, 199 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  9. Jafferis, D.L.: Topological Quiver Matrix Models and Quantum Foam. http://arXiv.org/abs/0705.2250v1[hep-th], 2007

  10. King, A.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. 2,45, no. 180, 515–530 (1994)

    Google Scholar 

  11. Denef F.: Supergravity flows and D-brane stability. JHEP 0008, 050 (2000)

    MathSciNet  Google Scholar 

  12. Ginzburg, V.: Calabi-Yau algebras. http://arXiv.org/abs/math/0612139v3[math.AG], 2007

  13. Aspinwall P.S., Katz S.H.: Computation of superpotentials for D-Branes. Commun. Math. Phys. 264, 227 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978)

    MATH  Google Scholar 

  15. Hartshorne R.: Algebraic Geometry. Springer, Berlin-Heidelberg-New York (1997)

    Google Scholar 

  16. Berenstein, D., Douglas, M.R.: Seiberg duality for quiver gauge theories. http://arXiv.org/abs/hep-th/0207027v1, 2002

  17. Young, B., with an appendix by Bryan, J.: Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. http://arXiv.org/abs/0802.3948v2[math.CO], 2008

  18. Nagao, K., Nakajima, H.: Counting invariant of perverse coherent sheaves and its wall-crossing. http://arXiv.org/abs/0809.2992v3[math.AG], 2008

  19. Nagao, K.: Derived categories of small toric Calabi-Yau 3-folds and counting invariants. http://arXiv.org/abs/0809.2994v3[math.AG], 2008

Download references

Author information

Authors and Affiliations

  1. NHETC, Department of Physics, Rutgers University, 126 Frelinghuysen Rd, New Jersey, 08854, USA

    Wu-yen Chuang & Daniel Louis Jafferis

Authors
  1. Wu-yen Chuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Louis Jafferis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wu-yen Chuang.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chuang, Wy., Jafferis, D.L. Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions. Commun. Math. Phys. 292, 285–301 (2009). https://doi.org/10.1007/s00220-009-0832-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0832-2

Keywords

Search

Navigation