Journal of High Energy Physics

, 2014:40 | Cite as

Dualities and fivebrane instantons

  • Sergei Alexandrov
  • Sibasish Banerjee
Open Access
Regular Article - Theoretical Physics


We derive the fivebrane instanton corrections to the hypermultiplet moduli space \( {\mathrm{\mathcal{M}}}_H \) of Calabi-Yau string vacua using S-duality symmetry of the type IIB formulation. The result is given in terms of a set of holomorphic functions on the twistor space of \( {\mathrm{\mathcal{M}}}_H \). It contains not only all orders of the instanton expansion, but also takes into account the presence of D1-D(-1)-brane instantons. Furthermore, we provide a thorough study of the group of discrete isometries of \( {\mathrm{\mathcal{M}}}_H \) and show that its closure requires a modification of certain symmetry transformations. After this modification, the fivebrane instantons are proven to be consistent with the full duality group.


p-branes Nonperturbative Effects Supersymmetric Effective Theories 


Open Access

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  1. [1]
    S. Alexandrov, Twistor approach to string compactifications: a review, Phys. Rept. 522 (2013) 1 [arXiv:1111.2892] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Alexandrov, J. Manschot, D. Persson and B. Pioline, Quantum hypermultiplet moduli spaces in N =2 string vacua: a review, arXiv:1304.0766 [INSPIRE].
  3. [3]
    B. de Wit, P.G. Lauwers and A. Van Proeyen, Lagrangians of N =2 supergravity-matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    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].ADSCrossRefGoogle Scholar
  5. [5]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    J. Bagger and E. Witten, Matter couplings in N =2 supergravity, Nucl. Phys. B 222 (1983) 1 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    I. Antoniadis, S. Ferrara, R. Minasian and K.S. Narain, R 4 couplings in M and type-II theories on Calabi-Yau spaces, Nucl. Phys. B 507 (1997) 571 [hep-th/9707013] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    H. Gunther, C. Herrmann and J. Louis, Quantum corrections in the hypermultiplet moduli space, Fortsch. Phys. 48 (2000) 119 [hep-th/9901137] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, String loop corrections to the universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079 [hep-th/0307268] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    D. Robles-Llana, F. Saueressig and S. Vandoren, String loop corrected hypermultiplet moduli spaces, JHEP 03 (2006) 081 [hep-th/0602164] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Alexandrov, Quantum covariant c-map, JHEP 05 (2007) 094 [hep-th/0702203] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, D-instantons and twistors, JHEP 03 (2009) 044 [arXiv:0812.4219] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    S. Alexandrov, D-instantons and twistors: some exact results, J. Phys. A 42 (2009) 335402 [arXiv:0902.2761] [INSPIRE].MathSciNetGoogle Scholar
  15. [15]
    S. Alexandrov, D. Persson and B. Pioline, Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces, JHEP 03 (2011) 111 [arXiv:1010.5792] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Alexandrov and S. Banerjee, Fivebrane instantons in Calabi-Yau compactifications, Phys. Rev. D 90 (2014) 041902 [arXiv:1403.1265] [INSPIRE].ADSGoogle Scholar
  17. [17]
    D. Robles-Llana, M. Roček, F. Saueressig, U. Theis and S. Vandoren, Nonperturbative corrections to 4D string theory effective actions from SL(2, Z) duality and supersymmetry, Phys. Rev. Lett. 98 (2007) 211602 [hep-th/0612027] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    S. Alexandrov and S. Banerjee, Modularity, quaternion-Kähler spaces and mirror symmetry, J. Math. Phys. 54 (2013) 102301 [arXiv:1306.1837] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    S. Alexandrov, J. Manschot and B. Pioline, D3-instantons, Mock theta series and twistors, JHEP 04 (2013) 002 [arXiv:1207.1109] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59 [hep-th/9505162] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    R. Bohm, H. Gunther, C. Herrmann and J. Louis, Compactification of type IIB string theory on Calabi-Yau threefolds, Nucl. Phys. B 569 (2000) 229 [hep-th/9908007] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    E.R. Sharpe, D-branes, derived categories and Grothendieck groups, Nucl. Phys. B 561 (1999) 433 [hep-th/9902116] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    M.R. Douglas, B. Fiol and C. Romelsberger, Stability and BPS branes, JHEP 09 (2005) 006 [hep-th/0002037] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Hosono, A. Klemm and S. Theisen, Lectures on mirror symmetry, hep-th/9403096 [INSPIRE].
  26. [26]
    M.-x. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: Modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    C.T.C. Wall, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 1 (1966) 355.ADSCrossRefMATHGoogle Scholar
  28. [28]
    J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Álvarez-Gaumé, G.W. Moore, P.C. Nelson, C. Vafa and J. Bost, Bosonization in arbitrary genus, Phys. Lett. B 178 (1986) 41 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    L. Álvarez-Gaumé, J.B. Bost, G.W. Moore, P.C. Nelson and C. Vafa, Bosonization on higher genus Riemann surfaces, Commun. Math. Phys. 112 (1987) 503 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  32. [32]
    E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    D.S. Freed, Dirac charge quantization and generalized differential cohomology, hep-th/0011220 [INSPIRE].
  34. [34]
    S. Alexandrov, D. Persson and B. Pioline, Wall-crossing, Rogers dilogarithm and the QK/HK correspondence, JHEP 12 (2011) 027 [arXiv:1110.0466] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    D. Belov and G.W. Moore, Holographic action for the self-dual field, hep-th/0605038 [INSPIRE].
  36. [36]
    L. Bao, A. Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, Rigid Calabi-Yau threefolds, Picard eisenstein series and instantons, arXiv:1005.4848 [INSPIRE].
  37. [37]
    S. Alexandrov, D. Persson and B. Pioline, On the topology of the hypermultiplet moduli space in type-II/CY string vacua, Phys. Rev. D 83 (2011) 026001 [arXiv:1009.3026] [INSPIRE].ADSGoogle Scholar
  38. [38]
    S.M. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982) 143.ADSCrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    C. LeBrun, Fano manifolds, contact structures, and quaternionic geometry, Int.. J. Math. 6 (1995) 419 [dg-ga/9409001] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  40. [40]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics, Commun. Math. Phys. 296 (2010) 353 [arXiv:0810.1675] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    K. Galicki, A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Comm. Math. Phys. 108 (1987) 117.ADSCrossRefMATHMathSciNetGoogle Scholar
  42. [42]
    S. Alexandrov and B. Pioline, S-duality in twistor space, JHEP 08 (2012) 112 [arXiv:1206.1341] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    J. Manschot, Wall-crossing of D4-branes using flow trees, Adv. Theor. Math. Phys. 15 (2011) 1 [arXiv:1003.1570] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    J. Manschot, B. Pioline and A. Sen, Wall crossing from Boltzmann black hole halos, JHEP 07 (2011) 059 [arXiv:1011.1258] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE].
  46. [46]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  47. [47]
    S. Alexandrov and F. Saueressig, Quantum mirror symmetry and twistors, JHEP 09 (2009) 108 [arXiv:0906.3743] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    M. de Vroome and S. Vandoren, Supergravity description of spacetime instantons, Class. Quant. Grav. 24 (2007) 509 [hep-th/0607055] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    D.S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math 3 (1999) 819 [hep-th/9907189] [INSPIRE].MATHMathSciNetGoogle Scholar
  50. [50]
    S. Alexandrov, c-map as c =1 string, Nucl. Phys. B 863 (2012) 329 [arXiv:1201.4392] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    B. Pioline and S. Vandoren, Large D-instanton effects in string theory, JHEP 07 (2009) 008 [arXiv:0904.2303] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    B. Pioline and D. Persson, The automorphic N S5-brane, Commun. Num. Theor. Phys. 3 (2009) 697 [arXiv:0902.3274] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  53. [53]
    L. Bao, A. Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, Instanton corrections to the universal hypermultiplet and automorphic forms on SU(2, 1), Commun. Num. Theor. Phys. 4 (2010) 187 [arXiv:0909.4299] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  54. [54]
    D. Persson, Automorphic instanton partition functions on Calabi-Yau threefolds, J. Phys. Conf. Ser. 346 (2012) 012016 [arXiv:1103.1014] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    J. Manschot, Stability and duality in N =2 supergravity, Commun. Math. Phys. 299 (2010) 651 [arXiv:0906.1767] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221MontpellierFrance

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