Communications in Mathematical Physics

, Volume 264, Issue 1, pp 227–253 | Cite as

Computation of Superpotentials for D-Branes

  • Paul S. Aspinwall
  • Sheldon Katz


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.


Neural Network Statistical Physic Correlation Function Complex System Gauge Theory 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Witten, E.: Bound States Of Strings And p-Branes. Nucl. Phys. B460, 335–350 (1996)Google Scholar
  2. 2.
    Becker, K., Becker, M., Strominger, A.: Five-branes, Membranes and Nonperturbative String Theory. Nucl. Phys. B456, 130–152 (1995)Google Scholar
  3. 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. 1–102Google Scholar
  4. 4.
    Fukaya, K., Seidel, P.: Floer Homology, A -Categories and Topological Field Theory. Lecture Notes in Pure and Appl. Math. 184, 9–32 (1997)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory — Anomaly and Obstruction. 2000, fukaya.html, 2000
  6. 6.
    Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: Proceedings of the International Congress of Mathematicians. Basel-Boston: Birkhäuser, 1995, pp. 120–139Google Scholar
  7. 7.
    Douglas, M.R.: D-Branes, Categories and N=1 Supersymmetry. J. Math. Phys. 42, 2818–2843 (2001)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Aspinwall, P.S., Lawrence, A.E.: Derived Categories and Zero-Brane Stability. JHEP 08, 004 (2001)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Aspinwall, P.S.: D-Branes on Calabi–Yau Manifolds., 2004
  10. 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. 11.
    Morrison, D.R., Plesser, M.R.: Non-Spherical Horizons. I. Adv. Theor. Math. Phys. 3, 1–81 (1999)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Cachazo, F., Katz, S., Vafa, C.: Geometric Transitions and N = 1 Quiver Theories., 2001
  13. 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. 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)CrossRefADSzbMATHGoogle Scholar
  15. 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. 16.
    Herbst, M., Lazaroiu, C.-I., Lerche, W.: D-brane Effective Action and Tachyon Condensation in Topological Minimal Models. JHEP 0503, 078(2005)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Polishchuk, A.: A -Structures on an Elliptic Curve. Commun. Math. Phys. 247, 527–551 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  18. 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. 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. 637–678Google Scholar
  20. 20.
    Gugenheim, V., Stasheff, J.: On Perturbations and A -Structures. Bul. Soc. Math. Belg. A38, 237–246 (1987)Google Scholar
  21. 21.
    Penkava, M., Schwarz, A.: A Algebras and the Cohomology of Moduli Spaces., 1994
  22. 22.
    Lazaroiu, C.I.: String Field Theory and Brane Superpotentials. JHEP 10, 018 (2001)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Tomasiello, A.: A-infinity Structure and Superpotentials. JHEP 09, 030, (2001)CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Herbst, M., Lazaroiu, C., Lerche, W.: Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings. JHEP 0502, 071 (2005)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Keller, B.: Introduction to A-infinity Algebras and Modules. Homology Homotopy Appl. 3, 1–35 (2001)ADSMathSciNetGoogle Scholar
  26. 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 203–263Google Scholar
  27. 27.
    Kadeishvili, T.V.: The Algebraic Structure in the Homology of an A -Algebra. Soobshch. Akad. Nauk. Gruzin. SSR 108, 249–252 (1982)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Merkulov, S.: Strongly Homotopy Algebras of a Kähler Manifold. Internat. Math. Res. Notices (1999) 153–164Google Scholar
  29. 29.
    Brunner, I., Douglas, M.R., Lawrence, A., Römelsberger, C.: D-branes on the Quintic. JHEP 08, 015 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. 30.
    Witten, E.: Mirror Manifolds and Topological Field Theory. In: S.-T. Yau, (ed.), Essays on Mirror Manifolds, Cambridge, MA: International Press, 1992Google Scholar
  31. 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. 120–348Google Scholar
  32. 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. 421–577Google Scholar
  33. 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. 34.
    Aspinwall, P.S., Melnikov, I.V.: D-Branes on Vanishing del Pezzo Surfaces. JHEP 0412, 042 (2004)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Herzog, C.P.: Seiberg Duality is an Exceptional Mutation. JHEP 0408, 064 (2004)CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Kutasov, D.: Geometry on the Space of Conformal Field Theories and Contact Terms. Phys. Lett. B220, 153–158 (1989)Google Scholar
  37. 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. 247–259Google Scholar
  38. 38.
    Polishchuk, A.: Extensions of Homogeneous Coordinate Rings to A -Algebras. Homology Homotopy Appl. 5, 407–421 (2003)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Berlin Heidelberg Newyork: Springer-Verlag, 1977Google Scholar
  40. 40.
    Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Stud. in Adv. Math. 38, Cambridge: Cambridge Univ. Press, 1994Google Scholar
  41. 41.
    Aspinwall, P.S.: A Point's Point of View of Stringy Geometry. JHEP 01, 002 (2003)CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Katz, S.: Versal Deformations and Superpotentials for Rational Curves in Smooth Threefolds. Cont. Math. 312, 129–136 (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of PhysicsSLAC Stanford UniversityStanfordUSA
  2. 2.Center for Geometry and Theoretical PhysicsDuke UniversityDurhamUSA
  3. 3.Departments of Mathematics and PhysicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations