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Flat connections in open string mirror symmetry

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Abstract

We study a flat connection defined on the open-closed deformation space of open string mirror symmetry for type II compactifications on Calabi-Yau threefolds with D-branes. We use flatness and integrability conditions to define distinguished flat coordinates and the superpotential function at an arbitrary point in the open-closed deformation space. Integrability conditions are given for concrete deformation spaces with several closed and open string deformations. We study explicit examples for expansions around different limit points, including orbifold Gromov-Witten invariants, and brane configurations with several brane moduli. In particular, the latter case covers stacks of parallel branes with non-Abelian symmetry.

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References

  1. C. Vafa, Extending mirror conjecture to Calabi-Yau with bundles, hep-th/9804131 [INSPIRE].

  2. 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] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  3. S. Kachru, S.H. Katz, A.E. Lawrence and J. McGreevy, Mirror symmetry for open strings, Phys. Rev. D 62 (2000) 126005 [hep-th/0006047] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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

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

    Article  MathSciNet  ADS  Google Scholar 

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

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

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. D.R. Morrison and J. Walcher, D-branes and normal functions, arXiv:0709.4028 [INSPIRE].

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. M. Aganagic and C. Beem, The geometry of D-brane superpotentials, JHEP 12 (2011) 060 [arXiv:0909.2245] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. 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 [INSPIRE].

  17. M. Alim et al., Type II/F-theory superpotentials with several deformations and N = 1 mirror symmetry, JHEP 06 (2011) 103 [arXiv:1010.0977] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. A. Tomasiello, A-infinity structure and superpotentials, JHEP 09 (2001) 030 [hep-th/0107195] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. M.R. Douglas, S. Govindarajan, T. Jayaraman and A. Tomasiello, D branes on Calabi-Yau manifolds and superpotentials, Commun. Math. Phys. 248 (2004) 85 [hep-th/0203173] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. M. Herbst, C.-I. Lazaroiu and W. Lerche, Superpotentials, A(∞) relations and WDVV equations for open topological strings, JHEP 02 (2005) 071 [hep-th/0402110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. S.K. Ashok, E. Dell’Aquila, D.-E. Diaconescu and B. Florea, Obstructed D-branes in Landau-Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004) 427 [hep-th/0404167] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  22. K. Hori and J. Walcher, F-term equations near Gepner points, JHEP 01 (2005) 008 [hep-th/0404196] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. P.S. Aspinwall and S.H. Katz, Computation of superpotentials for D-branes, Commun. Math. Phys. 264 (2006) 227 [hep-th/0412209] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. M. Baumgartl, I. Brunner and M.R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP 07 (2007) 061 [arXiv:0704.2666] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. J. Walcher, Extended holomorphic anomaly and loop amplitudes in open topological string, Nucl. Phys. B 817 (2009) 167 [arXiv:0705.4098] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. J. Walcher, Evidence for tadpole cancellation in the topological string, Commun. Num. Theor. Phys. 3 (2009) 111 [arXiv:0712.2775] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Krefl and J. Walcher, Real mirror symmetry for one-parameter hypersurfaces, JHEP 09 (2008) 031 [arXiv:0805.0792] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. J. Knapp and E. Scheidegger, Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces, Adv. Theor. Math. Phys. 13 (2009) 991 [arXiv:0805.1013] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  30. J. Knapp and E. Scheidegger, Matrix factorizations, massey products and F-terms for two-parameter Calabi-Yau hypersurfaces, Adv. Theor. Math. Phys. 14 (2010) 225 [arXiv:0812.2429] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Walcher, Calculations for mirror symmetry with D-branes, JHEP 09 (2009) 129 [arXiv:0904.4905] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. H. Jockers, P. Mayr and J. Walcher, On N = 1 4D effective couplings for F-theory and heterotic vacua, Adv. Theor. Math. Phys. 14 (2010) 1433 [arXiv:0912.3265] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  34. M. Baumgartl, I. Brunner and M. Soroush, D-brane superpotentials: geometric and worldsheet approaches, Nucl. Phys. B 843 (2011) 602 [arXiv:1007.2447] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. H. Fuji, S. Nakayama, M. Shimizu and H. Suzuki, A note on computations of D-brane superpotential, J. Phys. A 44 (2011) 465401 [arXiv:1011.2347] [INSPIRE].

    ADS  Google Scholar 

  36. M. Shimizu and H. Suzuki, Open mirror symmetry for Pfaffian Calabi-Yau 3-folds, JHEP 03 (2011) 083 [arXiv:1011.2350] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  37. T.W. Grimm, A. Klemm and D. Klevers, Five-brane superpotentials, blow-up geometries and SU(3) structure manifolds, JHEP 05 (2011) 113 [arXiv:1011.6375] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. A. Hebecker, S.C. Kraus, D. Lüst, S. Steinfurt and T. Weigand, Fluxbrane inflation, Nucl. Phys. B 854 (2012) 509 [arXiv:1104.5016] [INSPIRE].

    Article  ADS  Google Scholar 

  39. N. Carqueville and M.M. Kay, Bulk deformations of open topological string theory, arXiv:1104.5438 [INSPIRE].

  40. A. Mertens, Reflections of N S5 branes, arXiv:1107.1457 [INSPIRE].

  41. S.H. Katz and C.-C.M. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2002) 1 [math/0103074] [INSPIRE].

    MathSciNet  Google Scholar 

  42. T. Graber and E. Zaslow, Open string Gromov-Witten invariants: calculations and a mirror ’theorem’, hep-th/0109075 [INSPIRE].

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  45. W. Lerche, D. Smit and N. Warner, Differential equations for periods and flat coordinates in two-dimensional topological matter theories, Nucl. Phys. B 372 (1992) 87 [hep-th/9108013] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. S. Hosono and B. Lian, GKZ hypergeometric systems and applications to mirror symmetry, hep-th/9602147 [INSPIRE].

  47. S. Hosono, B. Lian and S.-T. Yau, GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Commun. Math. Phys. 182 (1996) 535 [alg-geom/9511001] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. 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] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [INSPIRE].

    MathSciNet  Google Scholar 

  50. W. Lerche and P. Mayr, On N = 1 mirror symmetry for open type II strings, hep-th/0111113 [INSPIRE].

  51. W. Lerche, Special geometry and mirror symmetry for open string backgrounds with N = 1 supersymmetry, hep-th/0312326 [INSPIRE].

  52. V. Bouchard and R. Cavalieri, On the mathematics and physics of high genus invariants of C 3 /Z 3, Adv. Theor. Math. Phys. 13 (2009) 695 [arXiv:0709.3805] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Brini and R. Cavalieri, Open orbifold Gromov-Witten invariants of C 3 /Z n : Localization and mirror symmetry, arXiv:1007.0934 [INSPIRE].

  54. M. Aganagic, V. Bouchard and A. Klemm, Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  56. V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Topological open strings on orbifolds, Commun. Math. Phys. 296 (2010) 589 [arXiv:0807.0597] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  57. A. Brini and A. Tanzini, Exact results for topological strings on resolved Y**p,q singularities, Commun. Math. Phys. 289 (2009) 205 [arXiv:0804.2598] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. M. Alim, J.D. Lange and P. Mayr, Global properties of topological string amplitudes and orbifold invariants, JHEP 03 (2010) 113 [arXiv:0809.4253] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. A. Brini, Open topological strings and integrable hierarchies: remodeling the A-model, Commun. Math. Phys. 312 (2012) 735 [arXiv:1102.0281] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. D.-E. Diaconescu and J. Gomis, Fractional branes and boundary states in orbifold theories, JHEP 10 (2000) 001 [hep-th/9906242] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. 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 [INSPIRE].

    Article  ADS  Google Scholar 

  62. A. Klemm and S. Theisen, Considerations of one modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps, Nucl. Phys. B 389 (1993) 153 [hep-th/9205041] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  63. R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].

    Article  ADS  Google Scholar 

  64. D. Lüst, P. Mayr, S. Reffert and S. Stieberger, F-theory flux, destabilization of orientifolds and soft terms on D7-branes, Nucl. Phys. B 732 (2006) 243 [hep-th/0501139] [INSPIRE].

    Article  ADS  Google Scholar 

  65. H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B 718 (2005) 203 [hep-th/0502059] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  66. J. Gomis, F. Marchesano and D. Mateos, An open string landscape, JHEP 11 (2005) 021 [hep-th/0506179] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  67. L. Martucci, D-branes on general N = 1 backgrounds: superpotentials and D-terms, JHEP 06 (2006) 033 [hep-th/0602129] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  69. K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].

  70. S.H. Katz, D.R. Morrison and M.R. Plesser, Enhanced gauge symmetry in type-II string theory, Nucl. Phys. B 477 (1996) 105 [hep-th/9601108] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  71. A. Klemm and P. Mayr, Strong coupling singularities and nonAbelian gauge symmetries in N =2 string theory, Nucl. Phys. B 469 (1996) 37 [hep-th/9601014] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A

    Murad Alim

  2. Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, D-80333, Munich, Germany

    Michael Hecht, Peter Mayr & Adrian Mertens

  3. Bethe Center for Theoretical Physics, Physikalisches Institut Universität Bonn, Nussallee 12, 53115, Bonn, Germany

    Hans Jockers & Masoud Soroush

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  1. Murad Alim
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  2. Michael Hecht
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  3. Hans Jockers
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  4. Peter Mayr
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  5. Adrian Mertens
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  6. Masoud Soroush
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Correspondence to Masoud Soroush.

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ArXiv ePrint: 1110.6522

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Alim, M., Hecht, M., Jockers, H. et al. Flat connections in open string mirror symmetry. J. High Energ. Phys. 2012, 138 (2012). https://doi.org/10.1007/JHEP06(2012)138

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