Journal of High Energy Physics

, 2018:77 | Cite as

Infinitely many M2-instanton corrections to M-theory on G2-manifolds

  • Andreas P. BraunEmail author
  • Michele Del Zotto
  • James Halverson
  • Magdalena Larfors
  • David R. Morrison
  • Sakura Schäfer-Nameki
Open Access
Regular Article - Theoretical Physics


We consider the non-perturbative superpotential for a class of four-dimensional \( \mathcal{N}=1 \) vacua obtained from M-theory on seven-manifolds with holonomy G2. The class of G2-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over S3. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of [1], which relates M-theory on TCS G2-manifolds to E8 × E8 heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the G2-compactification.


Differential and Algebraic Geometry String Duality F-Theory M-Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    A.P. Braun and S. Schäfer-Nameki, Compact, singular G 2 -holonomy manifolds and M/Heterotic/F-theory duality, JHEP 04 (2018) 126 [arXiv:1708.07215] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    D.R. Morrison, What is F-theory?, to appear.Google Scholar
  3. [3]
    A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003) 125.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A. Corti, M. Haskins, J. Nordström and T. Pacini, G2 -manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J. 164 (2015) 1971 [arXiv:1207.4470] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Corti, M. Haskins, J. Nordström and T. Pacini, Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds, Geom. Topol. 17 (2013) 1955.MathSciNetCrossRefGoogle Scholar
  6. [6]
    R. Donagi, A. Grassi and E. Witten, A nonperturbative superpotential with E 8 symmetry, Mod. Phys. Lett. A 11 (1996) 2199 [hep-th/9607091] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Sethi, C. Vafa and E. Witten, Constraints on low dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
  9. [9]
    J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G 2 holonomy, JHEP 04 (2015) 047 [arXiv:1412.4123] [INSPIRE].
  10. [10]
    J. Halverson and D.R. Morrison, On gauge enhancement and singular limits in G 2 compactifications of M-theory, JHEP 04 (2016) 100 [arXiv:1507.05965] [INSPIRE].
  11. [11]
    T.C. da C. Guio, H. Jockers, A. Klemm and H.-Y. Yeh, Effective action from M-theory on twisted connected sum G 2 -manifolds, Commun. Math. Phys. 359 (2018) 535 [arXiv:1702.05435] [INSPIRE].
  12. [12]
    A.P. Braun and M. Del Zotto, Mirror symmetry for G 2 -manifolds: twisted connected sums and dual tops, JHEP 05 (2017) 080 [arXiv:1701.05202] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A.P. Braun and M. Del Zotto, Towards generalized mirror symmetry for twisted connected sum G 2 manifolds, JHEP 03 (2018) 082 [arXiv:1712.06571] [INSPIRE].
  14. [14]
    M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nonperturbative effects on the string world sheet, Nucl. Phys. B 278 (1986) 769 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nonperturbative effects on the string world sheet. 2., Nucl. Phys. B 289 (1987) 319 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    J. Distler, Resurrecting (2, 0) compactifications, Phys. Lett. B 188 (1987) 431 [INSPIRE].
  17. [17]
    J. Distler and B.R. Greene, Aspects of (2, 0) string compactifications, Nucl. Phys. B 304 (1988) 1 [INSPIRE].
  18. [18]
    E. Silverstein and E. Witten, Criteria for conformal invariance of (0, 2) models, Nucl. Phys. B 444 (1995) 161 [hep-th/9503212] [INSPIRE].
  19. [19]
    C. Beasley and E. Witten, Residues and world sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    E.I. Buchbinder and B.A. Ovrut, Non-vanishing superpotentials in heterotic string theory and discrete torsion, JHEP 01 (2017) 038 [arXiv:1611.01922] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    E. Buchbinder, A. Lukas, B. Ovrut and F. Ruehle, Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds, JHEP 10 (2017) 032 [arXiv:1707.07214] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    G. Curio and D. Lüst, A class of N = 1 dual string pairs and its modular superpotential, Int. J. Mod. Phys. A 12 (1997) 5847 [hep-th/9703007] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [INSPIRE].
  24. [24]
    R.C. Mclean, Deformations of calibrated submanifolds, Commun. Analy. Geom. 6 (1996) 705.MathSciNetCrossRefGoogle Scholar
  25. [25]
    O.J. Ganor, A note on zeros of superpotentials in F-theory, Nucl. Phys. B 499 (1997) 55 [hep-th/9612077] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    S. Gukov, S.-T. Yau and E. Zaslow, Duality and fibrations on G 2 manifolds, hep-th/0203217 [INSPIRE].
  28. [28]
    D.R. Morrison, Half K3 surfaces, talk given at Strings 2002, July 15-20, Cambridge, U.K. (2002).Google Scholar
  29. [29]
    S. Donaldson, Adiabatic limits of co-associative Kovalev Lefschetz fibrations, in Algebra, geometry, and physics in the 21st century, D. Auroux et al. eds., Progress in Mathematics volume 324, Birkhauser, Germany (2017).Google Scholar
  30. [30]
    C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988) 177.MathSciNetCrossRefGoogle Scholar
  31. [31]
    P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [INSPIRE].
  34. [34]
    L. Martucci, Warping the Kähler potential of F-theory/IIB flux compactifications, JHEP 03 (2015) 067 [arXiv:1411.2623] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    L. Martucci, Warped Kähler potentials and fluxes, JHEP 01 (2017) 056 [arXiv:1610.02403] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    L.B. Anderson et al., Instanton superpotentials, Calabi-Yau geometry and fibrations, Phys. Rev. D 93 (2016) 086001 [arXiv:1511.05188] [INSPIRE].
  37. [37]
    E. Witten, World sheet corrections via D instantons, JHEP 02 (2000) 030 [hep-th/9907041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    M.-H. Saito, Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry, in The arithmetic and geometry of algebraic cycles, B.B. Gordon et al. eds., NATO Sci. Ser. C Math. Phys. Sci. volume 548, Kluwer Academic Publishers, Dordrecht, The Netherland 2000Google Scholar
  39. [39]
    M. Cvetič, R. Donagi, J. Halverson and J. Marsano, On seven-brane dependent instanton prefactors in F-theory, JHEP 11 (2012) 004 [arXiv:1209.4906] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    B. Andreas and G. Curio, Three-branes and five-branes in N = 1 dual string pairs, Phys. Lett. B 417 (1998) 41 [hep-th/9706093] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    O. DeWolfe and B. Zwiebach, String junctions for arbitrary Lie algebra representations, Nucl. Phys. B 541 (1999) 509 [hep-th/9804210] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Grassi, J. Halverson and J.L. Shaneson, Matter from geometry without resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    A. Grassi, J. Halverson and J.L. Shaneson, Non-abelian gauge symmetry and the Higgs mechanism in F-theory, Commun. Math. Phys. 336 (2015) 1231 [arXiv:1402.5962] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    A. Grassi, J. Halverson and J.L. Shaneson, Geometry and topology of string junctions, arXiv:1410.6817 [INSPIRE].
  46. [46]
    O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering infinite symmetries on [p, q] 7-branes: Kac-Moody algebras and beyond, Adv. Theor. Math. Phys. 3 (1999) 1835 [hep-th/9812209] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  47. [47]
    G. Lopes Cardoso, G. Curio, D. Lüst and T. Mohaupt, On the duality between the heterotic string and F-theory in eight-dimensions, Phys. Lett. B 389 (1996) 479 [hep-th/9609111] [INSPIRE].
  48. [48]
    J. McOrist, D.R. Morrison and S. Sethi, Geometries, Non-Geometries and Fluxes, Adv. Theor. Math. Phys. 14 (2010) 1515 [arXiv:1004.5447] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  49. [49]
    A.P. Braun, F. Fucito and J.F. Morales, U-folds as K3 fibrations, JHEP 10 (2013) 154 [arXiv:1308.0553] [INSPIRE].
  50. [50]
    A. Malmendier and D.R. Morrison, K3 surfaces, modular forms and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015) 1085 [arXiv:1406.4873] [INSPIRE].
  51. [51]
    I. García-Etxebarria, D. Lüst, S. Massai and C. Mayrhofer, Ubiquity of non-geometry in heterotic compactifications, JHEP 03 (2017) 046 [arXiv:1611.10291] [INSPIRE].
  52. [52]
    P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, in Mirror Symmetry II, B. Greene and S.-T. Yau eds., International Press (1997), hep-th/9404151 [INSPIRE].
  53. [53]
    P.S. Aspinwall, K3 surfaces and string duality, in the proceedings of Fields, strings and duality. Theoretical Advanced Study Institute in Elementary Particle Physics (TASI’96), June 2-28, Boulder, U.S.A. (1996), hep-th/9611137 [INSPIRE].
  54. [54]
    N. Elkies and A. Kumar, K 3 surfaces and equations for Hilbert modular surfaces, Alg. Number Theor. 8 (2014) 2297.MathSciNetCrossRefGoogle Scholar
  55. [55]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
  56. [56]
    D. Joyce, Conjectures on counting associative 3-folds in G 2 -manifolds, arXiv:1610.09836 [INSPIRE].
  57. [57]
    M. Bianchi, A. Collinucci and L. Martucci, Magnetized E3-brane instantons in F-theory, JHEP 12 (2011) 045 [arXiv:1107.3732] [INSPIRE].
  58. [58]
    J.J. Heckman et al., Instantons and SUSY breaking in F-theory, arXiv:0808.1286 [INSPIRE].
  59. [59]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, On fluxed instantons and moduli stabilisation in IIB orientifolds and F-theory, Phys. Rev. D 84 (2011) 066001 [arXiv:1105.3193] [INSPIRE].
  60. [60]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, G-flux, M 5 instantons and U(1) symmetries in F-theory, Phys. Rev. D 87 (2013) 066007 [arXiv:1107.1718] [INSPIRE].
  61. [61]
    M. Kerstan and T. Weigand, Fluxed M 5-instantons in F-theory, Nucl. Phys. B 864 (2012) 597 [arXiv:1205.4720] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    L. Martucci and T. Weigand, Hidden selection rules, M 5-instantons and fluxes in F-theory, JHEP 10 (2015) 131 [arXiv:1507.06999] [INSPIRE].
  63. [63]
    E.I. Buchbinder, R. Donagi and B.A. Ovrut, Vector bundle moduli superpotentials in heterotic superstrings and M-theory, JHEP 07 (2002) 066 [hep-th/0206203] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    E.I. Buchbinder, R. Donagi and B.A. Ovrut, Superpotentials for vector bundle moduli, Nucl. Phys. B 653 (2003) 400 [hep-th/0205190] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    M. Cvetič, I. García Etxebarria and J. Halverson, Three looks at instantons in F-theory — New insights from anomaly inflow, string junctions and heterotic duality, JHEP 11 (2011) 101 [arXiv:1107.2388] [INSPIRE].
  66. [66]
    M. Berg, M. Haack and B. Körs, Loop corrections to volume moduli and inflation in string theory, Phys. Rev. D 71 (2005) 026005 [hep-th/0404087] [INSPIRE].
  67. [67]
    D. Baumann et al., On D3-brane potentials in compactifications with fluxes and wrapped D-branes, JHEP 11 (2006) 031 [hep-th/0607050] [INSPIRE].
  68. [68]
    R. Blumenhagen, M. Cvetič and T. Weigand, Spacetime instanton corrections in 4D string vacua: the seesaw mechanism for D-brane models, Nucl. Phys. B 771 (2007) 113 [hep-th/0609191] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    L.E. Ibáñez and A.M. Uranga, Neutrino Majorana masses from string theory instanton effects, JHEP 03 (2007) 052 [hep-th/0609213] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    B. Florea, S. Kachru, J. McGreevy and N. Saulina, Stringy instantons and quiver gauge theories, JHEP 05 (2007) 024 [hep-th/0610003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  71. [71]
    A.P. Braun et al., The Hodge numbers of divisors of Calabi-Yau threefold hypersurfaces, arXiv:1712.04946 [INSPIRE].
  72. [72]
    B.S. Acharya, M theory, Joyce orbifolds and super-Yang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227 [hep-th/9812205] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  73. [73]
    B.S. Acharya, On realizing N = 1 super-Yang-Mills in M-theory, hep-th/0011089 [INSPIRE].
  74. [74]
    J. Eckhard, S. Schäfer-Nameki and J.-M. Wong, An \( \mathcal{N}=1 \) 3d-3d correspondence, JHEP 07 (2018) 052 [arXiv:1804.02368] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Andreas P. Braun
    • 1
    Email author
  • Michele Del Zotto
    • 2
  • James Halverson
    • 3
  • Magdalena Larfors
    • 4
  • David R. Morrison
    • 5
  • Sakura Schäfer-Nameki
    • 1
  1. 1.Mathematical Institute, Oxford UniversityOxfordU.K.
  2. 2.Simons Center for Geometry and Physics, SUNYStony BrookU.S.A.
  3. 3.Department of PhysicsNortheastern UniversityBostonU.S.A.
  4. 4.Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  5. 5.Department of MathematicsUniversity of CaliforniaSanta BarbaraU.S.A.

Personalised recommendations