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Communications in Mathematical Physics

, Volume 259, Issue 1, pp 45–69 | Cite as

Massless D-Branes on Calabi–Yau Threefolds and Monodromy

  • Paul S Aspinwall
  • R. Paul Horja
  • Robert L. Karp
Article

Abstract

We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi–Yau’s. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various possibilities within the classification.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Modulus Space 
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.

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References

  1. 1.
    Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: “Proceedings of the International Congress of Mathematicians”, Basel-Boston: Birkhäuser, 1995, pp. 120–139Google Scholar
  2. 2.
    Douglas, M. R.: D-Branes, Categories and N=1 Supersymmetry. J. Math. Phys. 42, 2818–2843 (2001)CrossRefGoogle Scholar
  3. 3.
    Lazaroiu, C. I.: Unitarity, D-Brane Dynamics and D-brane Categories. JHEP 12, 031 (2001)CrossRefGoogle Scholar
  4. 4.
    Aspinwall, P. S., Lawrence, A. E.: Derived Categories and Zero-Brane Stability. JHEP 08, 004 (2001)CrossRefGoogle Scholar
  5. 5.
    Diaconescu, D.-E.: Enhanced D-brane Categories from String Field Theory. JHEP 06, 016 (2001)CrossRefGoogle Scholar
  6. 6.
    Kontsevich, M.: 1996, Rutgers Lecture, unpublishedGoogle Scholar
  7. 7.
    Horja, R. P.: Hypergeometric Functions and Mirror Symmetry in Toric Varieties. http://arxic.org/list/math.AG/9912109, 1999
  8. 8.
    Horja, R. P.: Derived Category Automorphisms from Mirror Symmetry. Duke Math. J. 127, 1–34 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Douglas, M. R., Fiol, B., Römelsberger, C.: Stability and BPS Branes. http://arxiv.org/list/hep-th/0002037, 2000
  10. 10.
    Douglas, M. R., Fiol, B., Romelsberger, C.: The Spectrum of BPS Branes on a Noncompact Calabi-Yau. http://arxiv.org/list/hep-th/0003263, 2000
  11. 11.
    Douglas, M. R.: Topics in D-geometry. Class. Quant. Grav. 17, 1057–1070 (2000)CrossRefGoogle Scholar
  12. 12.
    Aspinwall, P. S., Douglas, M. R.: D-Brane Stability and Monodromy. JHEP 05, 031 (2002)CrossRefGoogle Scholar
  13. 13.
    Aspinwall, P. S.: A Point’s Point of View of Stringy Geometry. JHEP 01, 002 (2003)CrossRefGoogle Scholar
  14. 14.
    Strominger, A.: Massless Black Holes and Conifolds in String Theory. Nucl. Phys. B451, 96–108 (1995)Google Scholar
  15. 15.
    Aspinwall, P. S., Karp, R. L.: Solitons in Seiberg–Witten Theory and D-Branes in the Derived Category. JHEP 04, 049 (2003)CrossRefGoogle Scholar
  16. 16.
    Sen, A.: Tachyon Condensation on the Brane Antibrane System. JHEP 08, 012 (1998)CrossRefGoogle Scholar
  17. 17.
    Distler, J., Jockers, H., Park, H.: D-Brane Monodromies, Derived Categories and Boundary Linear Sigma Models. http://arxiv.org/list/hep-th/0206242, 2002
  18. 18.
    Fukaya, K.: Floer Homology and Mirror Symmetry I. AMS/IP Stud. in Adv. Math. 23, Providence, RI: Amer. Math. Soc., 2001, pp. 15–43Google Scholar
  19. 19.
    Seidel, P.: Graded Lagrangian Submanifolds. Bull. Soc. Math. France 128, 103–149 (2000)Google Scholar
  20. 20.
    Seidel, P., Thomas, R. P.: Braid Groups Actions on Derived Categories of Coherent Sheaves. Duke Math. J. 108, 37–108 (2001)CrossRefGoogle Scholar
  21. 21.
    Bridgeland, T., Maciocia, A.: Fourier-Mukai transforms for Quotient Varieties. http://arxiv.org/list/math.AG/9811101, 1998
  22. 22.
    Candelas, P., de la Ossa, X. C., Green, P. S., Parkes, L.: A Pair of Calabi–Yau Manifolds as an Exactly Soluble Superconformal Theory. Nucl. Phys. B359, 21–74 (1991)Google Scholar
  23. 23.
    Morrison, D. R.: Geometric Aspects of Mirror Symmetry. In: Enquist, B., Schmid, W. (eds.), “Mathematics Unlimited – 2001 and Beyond”, Berlin-Heidelberg-New York: Springer-Verlag, 2001, pp. 899–918Google Scholar
  24. 24.
    Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Measuring Small Distances in N=2 Sigma Models. Nucl. Phys. B420, 184–242 (1994)Google Scholar
  25. 25.
    Morrison, D. R., Plesser, M. R.: Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties. Nucl. Phys. B440, 279–354 (1995)Google Scholar
  26. 26.
    Batyrev, V. V.: Dual Polyhedra and Mirror Symmetry for Calabi–Yau Hypersurfaces in Toric Varieties. J. Alg. Geom. 3, 493–535 (1994)Google Scholar
  27. 27.
    Aspinwall, P. S., Greene, B. R.: On the Geometric Interpretation of N = 2 Superconformal Theories. Nucl. Phys. B437, 205–230 (1995)Google Scholar
  28. 28.
    Witten, E.: Phases of N=2 Theories in Two Dimensions. Nucl. Phys. B403, 159–222 (1993)Google Scholar
  29. 29.
    Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Multiple Mirror Manifolds and Topology Change in String Theory. Phys. Lett. 303B, 249–259 (1993)CrossRefGoogle Scholar
  30. 30.
    Aspinwall, P. S., Greene, B. R., Morrison, D. R.: The Monomial-Divisor Mirror Map. Internat. Math. Res. Notices, 1993, pp. 319–338Google Scholar
  31. 31.
    Gelfand, I. M., Kapranov, M. M., Zelevinski, A. V.: Discriminants, Resultants and Multidimensional Determinants. Basel-Boston: Birkhäuser, 1994Google Scholar
  32. 32.
    Greene, B. R., Kanter, Y.: Small Volumes in Compactified String Theory. Nucl. Phys. B497, 127–145 (1997)Google Scholar
  33. 33.
    Candelas, P. et al.: Mirror Symmetry for Two Parameter Models –- I. Nucl. Phys. B416, 481–562 (1994)Google Scholar
  34. 34.
    Aspinwall, P. S.: Some Navigation Rules for D-brane Monodromy. J. Math. Phys. 42, 5534–5552 (2001)CrossRefGoogle Scholar
  35. 35.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Berlin-Heidelberg-New York: Springer-Verlag, 1977Google Scholar
  36. 36.
    Hartshorne, R.: Residues and Duality. Lecture Notes in Math. 20, Berlin-Heidelberg-New York: Spinger-Verlag, 1966Google Scholar
  37. 37.
    Diaconescu, D.-E., Gomis, J.: Fractional Branes and Boundary States in Orbifold Theories. JHEP 10, 001 (2000)CrossRefGoogle Scholar
  38. 38.
    Bridgeland, T., Maciocia, A.: Fourier-Mukai Transforms for K3 and Elliptic Fibrations. http://arxiv.org/list/math.AG/9908022, 1999
  39. 39.
    Andreas, B., Curio, G., Hernandez Ruiperez, D., Yau, S.-T.: Fourier–Mukai Transforms and Mirror Symmetry for D-Branes on Elliptic Calabi–Yau. http://arxiv.org/list/math.AG/0012196, 2000
  40. 40.
    Beilinson, A. A.: Coherent Sheaves on ℙn and Problems in Linear Algebra. Funct. Anal. Appl. 12, 214–216 (1978)CrossRefGoogle Scholar
  41. 41.
    Beilinson, A. A., Bernstein, J. N., Deligne, P.: Faisceaux pervers. Astérisque 100, (1982)Google Scholar
  42. 42.
    Aspinwall, P. S.: Enhanced Gauge Symmetries and Calabi–Yau Threefolds. Phys. Lett. B371, 231–237 (1996)Google Scholar
  43. 43.
    Katz, S., Morrison, D. R., Plesser, M. R.: Enhanced Gauge Symmetry in Type II String Theory. Nucl. Phys. B477, 105–140 (1996)Google Scholar
  44. 44.
    Witten, E.: Phase Transitions in M-Theory and F-Theory. Nucl. Phys. B471, 195–216 (1996)Google Scholar
  45. 45.
    Seiberg, N., Witten, E.: Electric - Magnetic Duality, Monopole Condensation, and Confinement in N=2 Supersymmetric Yang-Mills Theory. Nucl. Phys. B426, 19–52 (1994) (erratum-ibid. B430, 485–486 (1994))Google Scholar
  46. 46.
    Kachru, S. et al.: Nonperturbative Results on the Point Particle Limit of N=2 Heterotic String Compactifications. Nucl. Phys. B459, 537–558 (1996)Google Scholar
  47. 47.
    Katz, S., Mayr, P., Vafa, C.: Mirror Symmetry and Exact Solution of 4D N = 2 Gauge Theories. I. Adv. Theor. Math. Phys. 1, 53–114 (1998)Google Scholar
  48. 48.
    Klemm, A. et al.: Self-Dual Strings and N=2 Supersymmetric Field Theory. Nucl. Phys. B477, 746–766 (1996)Google Scholar
  49. 49.
    Vafa, C., Witten, E.: Dual String Pairs With N=1 and N=2 Supersymmetry in Four Dimensions. In: “S-Duality and Mirror Symmetry”, Nucl. Phys. (Proc. Suppl.) B46, 225–247 (1996)Google Scholar
  50. 50.
    Aspinwall, P. S., Plesser, M. R.: T-Duality Can Fail. J. High Energy Phys. 08, 001 (1999)CrossRefGoogle Scholar
  51. 51.
    Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Calabi–Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory. Nucl. Phys. B416, 414–480 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Paul S Aspinwall
    • 1
  • R. Paul Horja
    • 2
  • Robert L. Karp
    • 1
  1. 1.Center for Geometry and Theoretical PhysicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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