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Massless D-Branes on Calabi–Yau Threefolds and Monodromy

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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.

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References

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

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

    Article  Google Scholar 

  3. Lazaroiu, C. I.: Unitarity, D-Brane Dynamics and D-brane Categories. JHEP 12, 031 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Diaconescu, D.-E.: Enhanced D-brane Categories from String Field Theory. JHEP 06, 016 (2001)

    Article  Google Scholar 

  6. Kontsevich, M.: 1996, Rutgers Lecture, unpublished

  7. Horja, R. P.: Hypergeometric Functions and Mirror Symmetry in Toric Varieties. http://arxic.org/list/math.AG/9912109, 1999

  8. Horja, R. P.: Derived Category Automorphisms from Mirror Symmetry. Duke Math. J. 127, 1–34 (2005)

    Article  MathSciNet  Google Scholar 

  9. Douglas, M. R., Fiol, B., Römelsberger, C.: Stability and BPS Branes. http://arxiv.org/list/hep-th/0002037, 2000

  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. Douglas, M. R.: Topics in D-geometry. Class. Quant. Grav. 17, 1057–1070 (2000)

    Article  Google Scholar 

  12. Aspinwall, P. S., Douglas, M. R.: D-Brane Stability and Monodromy. JHEP 05, 031 (2002)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Strominger, A.: Massless Black Holes and Conifolds in String Theory. Nucl. Phys. B451, 96–108 (1995)

  15. Aspinwall, P. S., Karp, R. L.: Solitons in Seiberg–Witten Theory and D-Branes in the Derived Category. JHEP 04, 049 (2003)

    Article  Google Scholar 

  16. Sen, A.: Tachyon Condensation on the Brane Antibrane System. JHEP 08, 012 (1998)

    Article  Google Scholar 

  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. Fukaya, K.: Floer Homology and Mirror Symmetry I. AMS/IP Stud. in Adv. Math. 23, Providence, RI: Amer. Math. Soc., 2001, pp. 15–43

  19. Seidel, P.: Graded Lagrangian Submanifolds. Bull. Soc. Math. France 128, 103–149 (2000)

    Google Scholar 

  20. Seidel, P., Thomas, R. P.: Braid Groups Actions on Derived Categories of Coherent Sheaves. Duke Math. J. 108, 37–108 (2001)

    Article  Google Scholar 

  21. Bridgeland, T., Maciocia, A.: Fourier-Mukai transforms for Quotient Varieties. http://arxiv.org/list/math.AG/9811101, 1998

  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. 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. 899–918

  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. 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. 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. Aspinwall, P. S., Greene, B. R.: On the Geometric Interpretation of N = 2 Superconformal Theories. Nucl. Phys. B437, 205–230 (1995)

    Google Scholar 

  28. Witten, E.: Phases of N=2 Theories in Two Dimensions. Nucl. Phys. B403, 159–222 (1993)

    Google Scholar 

  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)

    Article  Google Scholar 

  30. Aspinwall, P. S., Greene, B. R., Morrison, D. R.: The Monomial-Divisor Mirror Map. Internat. Math. Res. Notices, 1993, pp. 319–338

  31. Gelfand, I. M., Kapranov, M. M., Zelevinski, A. V.: Discriminants, Resultants and Multidimensional Determinants. Basel-Boston: Birkhäuser, 1994

  32. Greene, B. R., Kanter, Y.: Small Volumes in Compactified String Theory. Nucl. Phys. B497, 127–145 (1997)

    Google Scholar 

  33. Candelas, P. et al.: Mirror Symmetry for Two Parameter Models –- I. Nucl. Phys. B416, 481–562 (1994)

    Google Scholar 

  34. Aspinwall, P. S.: Some Navigation Rules for D-brane Monodromy. J. Math. Phys. 42, 5534–5552 (2001)

    Article  Google Scholar 

  35. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Berlin-Heidelberg-New York: Springer-Verlag, 1977

  36. Hartshorne, R.: Residues and Duality. Lecture Notes in Math. 20, Berlin-Heidelberg-New York: Spinger-Verlag, 1966

  37. Diaconescu, D.-E., Gomis, J.: Fractional Branes and Boundary States in Orbifold Theories. JHEP 10, 001 (2000)

    Article  Google Scholar 

  38. Bridgeland, T., Maciocia, A.: Fourier-Mukai Transforms for K3 and Elliptic Fibrations. http://arxiv.org/list/math.AG/9908022, 1999

  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. Beilinson, A. A.: Coherent Sheaves on ℙn and Problems in Linear Algebra. Funct. Anal. Appl. 12, 214–216 (1978)

    Article  Google Scholar 

  41. Beilinson, A. A., Bernstein, J. N., Deligne, P.: Faisceaux pervers. Astérisque 100, (1982)

  42. Aspinwall, P. S.: Enhanced Gauge Symmetries and Calabi–Yau Threefolds. Phys. Lett. B371, 231–237 (1996)

    Google Scholar 

  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. Witten, E.: Phase Transitions in M-Theory and F-Theory. Nucl. Phys. B471, 195–216 (1996)

    Google Scholar 

  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))

  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. 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. Klemm, A. et al.: Self-Dual Strings and N=2 Supersymmetric Field Theory. Nucl. Phys. B477, 746–766 (1996)

    Google Scholar 

  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)

  50. Aspinwall, P. S., Plesser, M. R.: T-Duality Can Fail. J. High Energy Phys. 08, 001 (1999)

    Article  Google Scholar 

  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 

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Authors and Affiliations

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

    Paul S Aspinwall & Robert L. Karp

  2. Department of Mathematics, University of Michigan, East Hall, 525 E University Avenue, Ann Arbor, MI, 48109-1109, USA

    R. Paul Horja

Authors
  1. Paul S Aspinwall
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  2. R. Paul Horja
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  3. Robert L. Karp
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Communicated by N.A. Nekrasov

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Aspinwall, P., Horja, R. & Karp, R. Massless D-Branes on Calabi–Yau Threefolds and Monodromy. Commun. Math. Phys. 259, 45–69 (2005). https://doi.org/10.1007/s00220-005-1378-6

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  • DOI: https://doi.org/10.1007/s00220-005-1378-6

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