Skip to main content
SpringerLink
Log in

Computation of Superpotentials for D-Branes

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

Abstract

We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A ∞-structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern–Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.

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. Witten, E.: Bound States Of Strings And p-Branes. Nucl. Phys. B460, 335–350 (1996)

  2. Becker, K., Becker, M., Strominger, A.: Five-branes, Membranes and Nonperturbative String Theory. Nucl. Phys. B456, 130–152 (1995)

    Google Scholar 

  3. Fukaya, K.: Morse Homotopy, A ∞-Category, and Floer Homologies. In: proc. of the 1993 Garc workshop on Geomentry and Topology. Lecture Notes Ser. 18, Seoul: Seoul Nat.Univ., 1993, pp. 1–102

  4. Fukaya, K., Seidel, P.: Floer Homology, A ∞-Categories and Topological Field Theory. Lecture Notes in Pure and Appl. Math. 184, 9–32 (1997)

    MATH  MathSciNet  Google Scholar 

  5. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory — Anomaly and Obstruction. 2000, http://www.kusm.kyoto-u.ac.jp/~fukaya/ fukaya.html, 2000

  6. Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: Proceedings of the International Congress of Mathematicians. Basel-Boston: Birkhäuser, 1995, pp. 120–139

  7. Douglas, M.R.: D-Branes, Categories and N=1 Supersymmetry. J. Math. Phys. 42, 2818–2843 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Aspinwall, P.S., Lawrence, A.E.: Derived Categories and Zero-Brane Stability. JHEP 08, 004 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  9. Aspinwall, P.S.: D-Branes on Calabi–Yau Manifolds. http://arxiv.org/list/hep-th/0403166, 2004

  10. Klebanov, I.R., Witten, E.: Superconformal Field Theory on Threebranes at a Calabi–Yau Singularity. Nucl. Phys. B536, 199–218 (1998)

    Google Scholar 

  11. Morrison, D.R., Plesser, M.R.: Non-Spherical Horizons. I. Adv. Theor. Math. Phys. 3, 1–81 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Cachazo, F., Katz, S., Vafa, C.: Geometric Transitions and N = 1 Quiver Theories. http://arxiv.org/list/hep-th/0108120, 2001

  13. Cachazo, F., Fiol, B., Intriligator, K.A., Katz, S., Vafa, C.: A Geometric Unification of Dualities. Nucl. Phys. B628, 3–78, (2002)

    Google Scholar 

  14. Douglas, M.R., Govindarajan, S., Jayaraman, T., Tomasiello, A.: D-branes on Calabi-Yau Manifolds and Superpotentials. Commun. Math. phys. 248, 85–118 (2004)

    Article  ADS  MATH  Google Scholar 

  15. Feng, A., Hanany, A., He, Y.-H.: D-brane Gauge Theories from Toric Singularities and Toric Duality. Nucl. Phys. B595, 165–200 (2001)

    Google Scholar 

  16. Herbst, M., Lazaroiu, C.-I., Lerche, W.: D-brane Effective Action and Tachyon Condensation in Topological Minimal Models. JHEP 0503, 078(2005)

    Article  ADS  MathSciNet  Google Scholar 

  17. Polishchuk, A.: A ∞-Structures on an Elliptic Curve. Commun. Math. Phys. 247, 527–551 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Ashok, S.K., Dell'Aquila, E., Diaconescu, D.-E., Florea, B.: Obstructed D-branes in Landau– Ginzburg orbifolds. Adv. Theor. Math. Phys. 8, 427–472 (2004)

    Google Scholar 

  19. Witten, E.: Chern-Simons Gauge Theory as a String Theory. In: H. Hofer et al., (ed.), The Floer Memorial Volume, Basel-Boston: Birkhäuser, 1995, pp. 637–678

  20. Gugenheim, V., Stasheff, J.: On Perturbations and A ∞-Structures. Bul. Soc. Math. Belg. A38, 237–246 (1987)

  21. Penkava, M., Schwarz, A.: A ∞ Algebras and the Cohomology of Moduli Spaces. http://arxiv.org/list/hep-th/9408064, 1994

  22. Lazaroiu, C.I.: String Field Theory and Brane Superpotentials. JHEP 10, 018 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  23. Tomasiello, A.: A-infinity Structure and Superpotentials. JHEP 09, 030, (2001)

    Article  ADS  MathSciNet  Google Scholar 

  24. Herbst, M., Lazaroiu, C., Lerche, W.: Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings. JHEP 0502, 071 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  25. Keller, B.: Introduction to A-infinity Algebras and Modules. Homology Homotopy Appl. 3, 1–35 (2001)

    ADS  MathSciNet  Google Scholar 

  26. Kontsevich, M., Soibelman, Y.: Homological Mirror Symmetry and Torus Fibrations. In: K. Fukaya et al., (ed.), Symplectic Geometry and Mirror Symmetry. River Edge, NJ: World Scientific, 2001, pp 203–263

  27. Kadeishvili, T.V.: The Algebraic Structure in the Homology of an A ∞-Algebra. Soobshch. Akad. Nauk. Gruzin. SSR 108, 249–252 (1982)

    MATH  MathSciNet  Google Scholar 

  28. Merkulov, S.: Strongly Homotopy Algebras of a Kähler Manifold. Internat. Math. Res. Notices (1999) 153–164

  29. Brunner, I., Douglas, M.R., Lawrence, A., Römelsberger, C.: D-branes on the Quintic. JHEP 08, 015 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Witten, E.: Mirror Manifolds and Topological Field Theory. In: S.-T. Yau, (ed.), Essays on Mirror Manifolds, Cambridge, MA: International Press, 1992

  31. Dubrovin, B.: Geometry of 2D Topological Field Theories. In: Integrable Systems and Quantum Groups. Lecture Notes in Math. 1620, Berlin Heidelberg Newyork: Springer, 1996, pp. 120–348

  32. Segal, G.: The Definition of Conformal Field Theory. In: Topology, Geometry and Quantum Field Theory. London Math. Soc. Lecture Note Ser. 308, Cambridge: London Math. Soc., 2004, pp. 421–577

  33. Lazaroiu, C.I.: On the Structure of Open-Closed Topological Field Theory in Two Dimensions. Nucl. Phys. B603, 497–530 (2001)

    Google Scholar 

  34. Aspinwall, P.S., Melnikov, I.V.: D-Branes on Vanishing del Pezzo Surfaces. JHEP 0412, 042 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  35. Herzog, C.P.: Seiberg Duality is an Exceptional Mutation. JHEP 0408, 064 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  36. Kutasov, D.: Geometry on the Space of Conformal Field Theories and Contact Terms. Phys. Lett. B220, 153–158 (1989)

    Google Scholar 

  37. Polishchuk, A.: Homological Mirror Symmetry with Higher Products. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., Proxidence, RI: AMS, 2001, pp. 247–259

  38. Polishchuk, A.: Extensions of Homogeneous Coordinate Rings to A ∞-Algebras. Homology Homotopy Appl. 5, 407–421 (2003)

    MATH  MathSciNet  Google Scholar 

  39. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Berlin Heidelberg Newyork: Springer-Verlag, 1977

  40. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Stud. in Adv. Math. 38, Cambridge: Cambridge Univ. Press, 1994

  41. Aspinwall, P.S.: A Point's Point of View of Stringy Geometry. JHEP 01, 002 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  42. Katz, S.: Versal Deformations and Superpotentials for Rational Curves in Smooth Threefolds. Cont. Math. 312, 129–136 (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, SLAC Stanford University, Stanford, CA, 94305/94309, USA

    Paul S. Aspinwall

  2. Center for Geometry and Theoretical Physics, Duke University, Box 90318, Durham, NC, 27708-0318, USA

    Paul S. Aspinwall

  3. Departments of Mathematics and Physics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Sheldon Katz

Authors
  1. Paul S. Aspinwall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sheldon Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul S. Aspinwall.

Additional information

Communicated by N.A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aspinwall, P., Katz, S. Computation of Superpotentials for D-Branes. Commun. Math. Phys. 264, 227–253 (2006). https://doi.org/10.1007/s00220-006-1527-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-1527-6

Keywords

Search

Navigation