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Open mirror symmetry for pfaffian Calabi-Yau 3-folds

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Abstract

We investigate the open mirror symmetry of certain non-complete intersection Calabi-Yau 3-folds, called pfaffian Calabi-Yau. We predict the number of disk invariants of some examples by using the direct integration method proposed recently and the open mirror symmetry. We treat several pfaffian Calabi-Yau 3-folds and branes with two discrete vacua. Some models have two special points in its moduli space, around both of which we can consider different A-model mirror partners. We compute disc invariants for both cases. This study is the first application of the open mirror symmetry to the compact non-complete intersections in toric variety.

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References

  1. J. Walcher, Opening mirror symmetry on the quintic, Commun. Math. Phys. 276 (2007) 671 [hep-th/0605162] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. D.R. Morrison and J. Walcher, D-branes and Normal Functions, arXiv:0709.4028 [SPIRES].

  3. R. Pandharipande, J. Solomon and J. Walcher, Disk enumeration on the quintic 3-fold, J. Amer. Math. Soc. 21 (2008) 1169 math/0610901.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. J . Walcher, Extended Holomorphic Anomaly and Loop Amplitudes in Open Topological String, Nucl. Phys. B 817 (2009) 167 [arXiv:0705.4098] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. J . Walcher, Evidence for Tadpole Cancellation in the Topological String, arXiv:0712.2775 [SPIRES].

  8. H. Jockers and M. Soroush, Effective superpotentials for compact D5-brane Calabi-Yau geometries, Commun. Math. Phys. 290 (2009) 249 [arXiv:0808.0761] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. W. Lerche, P. Mayr and N. Warner, Holomorphic N =1 special geometry of open-closed type-II strings, hep-th/0207259 [SPIRES].

  10. W. Lerche, P. Mayr and N. Warner, N =1 special geometry, mixed Hodge variations and toric geometry, hep-th/0208039 [SPIRES].

  11. M. Alim, M. Hecht, P. Mayr and A. Mertens, Mirror Symmetry for Toric Branes on Compact Hypersurfaces, JHEP 09 (2009) 126 [arXiv:0901.2937] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [SPIRES].

  13. P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [SPIRES].

    Article  ADS  Google Scholar 

  14. H. Jockers and M. Soroush, Relative periods and open-string integer invariants for a compact Calabi-Yau hypersurface, Nucl. Phys. B 821 (2009) 535 [arXiv:0904.4674] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  15. D. Krefl and J . Walcher, Real Mirror Symmetry for One-parameter Hypersurfaces, JHEP 09 (2008) 031 [arXiv:0805.0792] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  16. J. Knapp and E. Scheidegger, Towards Open String Mirror Symmetry for One-Parameter Calabi-Yau Hypersurfaces, arXiv:0805.1013 [SPIRES].

  17. T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, The D5-brane effective action and superpotential in \( \cal{N} \) =1 compactifications, Nucl. Phys. B 816 (2009) 139 [arXiv:0811.2996] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. J . Walcher, Calculations for Mirror Symmetry with D-branes, JHEP 09 (2009) 129 [arXiv:0904.4905] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. M. Aganagic and C. Beem, The Geometry of D-brane Superpotentials, arXiv:0909.2245 [SPIRES].

  20. M. Baumgartl, I. Brunner and M. Soroush, D-brane Superpotentials: Geometric and Worldsheet Approaches, Nucl. Phys. B 843 (2011) 602 [arXiv:1007.2447] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. H. Fuji, S. Nakayama, M. Shimizu and H. Suzuki, A Note on Computations of D-brane Superpotential, to appear.

  22. P. Mayr, N =1 mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys. 5 (2002) 213 [hep-th/0108229] [SPIRES].

    MathSciNet  Google Scholar 

  23. M. Alim et al., Hints for Off-Shell Mirror Symmetry in type-II/F-theory Compactifications, Nucl. Phys. B 841 (2010) 303 [arXiv:0909.1842] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing Brane and Flux Superpotentials in F-theory Compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  25. T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Five-Brane Superpotentials and Heterotic/F-theory Duality, Nucl. Phys. B 838 (2010) 458 [arXiv:0912.3250] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. H. Jockers, P. Mayr and J. Walcher, On N = 1 4d Effective Couplings for F-theory and Heterotic Vacua, arXiv:0912.3265 [SPIRES].

  27. M. Alim et al., Type II/F-theory Superpotentials with Several Deformations and N =1 Mirror Symmetry, arXiv:1010.0977 [SPIRES].

  28. C. van Enckevort and D. van Straten, Electronic data base of Calabi-Yau equations, http://enriques.mathematik.uni-mainz.de/CYequations/.

  29. G. Almkvist, C. van Enckevort, D. van Straten and Wadim Zudilin, Tables of Calabi-Yau equations, math/0507430.

  30. G. Almkvist, Some binomial identities related to Calabi-Yau differential equations, math/0703255.

  31. E. A. Rødland, The Pfaffian Calabi-Yau, its Mirror, and their Link to the Grassmannian Gr(2, 7), Composio Mathematica 122 (2000) 135, math/9801092.

  32. S. Hosono and Y. Konishi, Higher genus Gromov-Witten invariants of the Grassmannian and the Pfaffian Calabi-Yau threefolds, arXiv:0704.2928 [SPIRES].

  33. A. Kanazawa, On Pfaffian Calabi-Yau Varieties and Mirror Symmetry, arXiv:1006.0223.

  34. K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. H. Ooguri, Y. Oz and Z. Yin, D-branes on Calabi-Yau spaces and their mirrors, Nucl. Phys. B 477 (1996) 407 [hep-th/9606112] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. S. Kachru, S.H. Katz, A.E. Lawrence and J. McGreevy, Open string instantons and superpotentials, Phys. Rev. D 62 (2000) 026001 [hep-th/9912151] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B 507 (1997) 658 [hep-th/9706109] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [SPIRES].

    MathSciNet  Google Scholar 

  40. P. Griffiths, Topics in transcendental algebraic geometry, in Proceedings of a seminar held at the Institute for Advanced Study, Princeton U.S.A. (1981–1982), [Annals of Mathematics Studies. Vol. 106, Princeton University Press, Princeton U.S.A. (1984)].

  41. P. Griffiths, On the periods of certain rational integrals: I, Ann. Math. 90 (1969) 460.

    Article  MATH  Google Scholar 

  42. M.L. Green, Infinitesimal methods in Hodge theory, in Algebraic cycles and Hodge theory, Torino Italy (1993) [Lecture Notes in Math. Vol. 1594, Springer, Berlin Germany (1994)].

  43. S. Li, B.H. Lian and S.-T. Yau, Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces, arXiv:0910.4215 [SPIRES].

  44. F. Tonoli, Construction of Calabi-Yau 3-folds in P6 , J. Alg. Geom. 13 (2004) 249.

    Article  MathSciNet  Google Scholar 

  45. J . Bœhm, Mirror symmetry and tropical geometry, arXiv:0708.4402.

  46. E. Tjøtta, Quantum cohomology of a Pfaffian Calabi-Yau variety: verifying mirror symmetry predictions, Composio Mathematica 126 (2001) 78 [math/9906119].

    ADS  Google Scholar 

  47. K. Hori and D. Tong, Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  48. L. Borisov and A. Caldararu, The Pfaffian-Grassmannian derived equivalence, math/0608404.

  49. A. Kuznetsov, Homological projective duality for Grassmannians of lines, math/0610957.

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Correspondence to Masahide Shimizu.

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ArXiv ePrint:1011.2350

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Shimizu, M., Suzuki, H. Open mirror symmetry for pfaffian Calabi-Yau 3-folds. J. High Energ. Phys. 2011, 83 (2011). https://doi.org/10.1007/JHEP03(2011)083

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