Journal of High Energy Physics

, 2018:52 | Cite as

Marginal deformations of heterotic G2 sigma models

  • Marc-Antoine Fiset
  • Callum Quigley
  • Eirik Eik Svanes
Open Access
Regular Article - Theoretical Physics


Recently, the infinitesimal moduli space of heterotic G2 compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST operator. This BRST operator is nilpotent if and only if the target space geometry satisfies the heterotic supersymmetry conditions. We relate this to the supergravity approach by showing that the corresponding cohomologies are indeed isomorphic. We work at tree-level in α′ perturbation theory and study general geometries, in particular with non-vanishing torsion.


Conformal Field Models in String Theory Superspaces BRST Quantization Differential and Algebraic Geometry 


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]
    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    C.M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D. Nemeschansky and A. Sen, Conformal invariance of supersymmetric σ models on Calabi-Yau manifolds, Phys. Lett. B 178 (1986) 365 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Witten, New issues in manifolds of SU(3) holonomy, Nucl. Phys. B 268 (1986) 79 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    K. Becker, D. Robbins and E. Witten, The α expansion on a compact manifold of exceptional holonomy, JHEP 06 (2014) 051 [arXiv:1404.2460] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    L.B. Anderson, J. Gray and E. Sharpe, Algebroids, heterotic moduli spaces and the Strominger system, JHEP 07 (2014) 037 [arXiv:1402.1532] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    X. de la Ossa and E.E. Svanes, Holomorphic bundles and the moduli space of N = 1 supersymmetric heterotic compactifications, JHEP 10 (2014) 123 [arXiv:1402.1725] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    M. Garcia-Fernandez, R. Rubio and C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, arXiv:1503.07562 [INSPIRE].
  10. [10]
    A. Clarke, M. Garcia-Fernandez and C. Tipler, Moduli of G 2 structures and the Strominger system in dimension 7, arXiv:1607.01219 [INSPIRE].
  11. [11]
    X. de la Ossa, M. Larfors and E.E. Svanes, Infinitesimal moduli of G 2 holonomy manifolds with instanton bundles, JHEP 11 (2016) 016 [arXiv:1607.03473] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  12. [12]
    X. de la Ossa, M. Larfors and E.E. Svanes, The infinitesimal moduli space of heterotic G 2 systems, Commun. Math. Phys. (2017) [arXiv:1704.08717] [INSPIRE].
  13. [13]
    X. de la Ossa, M. Larfors and E.E. Svanes, Restrictions of heterotic G 2 structures and instanton connections, arXiv:1709.06974 [INSPIRE].
  14. [14]
    S. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The geometric universe, Oxford U.K., (1996) [INSPIRE].
  15. [15]
    C. Lewis, Spin(7) instantons, Ph.D. thesis, University of Oxford, Oxford U.K., (1999).Google Scholar
  16. [16]
    S. Brendle, On the construction of solutions to the Yang-Mills equations in higher dimensions, math.DG/0302093 [INSPIRE].
  17. [17]
    S. Donaldson and E. Segal, Gauge theory in higher dimensions, II, arXiv:0902.3239 [INSPIRE].
  18. [18]
    H.N.S. Earp, Instantons on G 2 -manifolds, Ph.D. thesis, Imperial College London, London U.K., (2009).Google Scholar
  19. [19]
    D. Harland, T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Yang-Mills flows on nearly Kähler manifolds and G 2 -instantons, Commun. Math. Phys. 300 (2010) 185 [arXiv:0909.2730] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  20. [20]
    K.-P. Gemmer, O. Lechtenfeld, C. Nolle and A.D. Popov, Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds, JHEP 09 (2011) 103 [arXiv:1108.3951] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  21. [21]
    T. Walpuski, G 2 -instantons on generalised Kummer constructions, Geom. Topol. 17 (2013) 2345 [arXiv:1109.6609] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    T. Walpuski, Spin(7)-instantons, Cayley submanifolds and Fueter sections, Commun. Math. Phys. 352 (2017) 1 [arXiv:1409.6705] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    T. Walpuski, G 2 -instantons over twisted connected sums: an example, arXiv:1505.01080.
  24. [24]
    A.S. Haupt, Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with G 2 -structure, JHEP 03 (2016) 038 [arXiv:1512.07254] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    H.N.S. Earp, G 2 -instantons on Kovalev manifolds, Geom. Topol. 19 (2015) 61 [arXiv:1101.0880] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    H.N.S. Earp and T. Walpuski, G 2 -instantons over twisted connected sums, Geom. Topol. 19 (2015) 1263.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    G. Menet, J. Nordström and H.N.S. Earp, Construction of G 2 -instantons via twisted connected sums, arXiv:1510.03836.
  28. [28]
    D. Joyce, Conjectures on counting associative 3-folds in G 2 -manifolds, arXiv:1610.09836 [INSPIRE].
  29. [29]
    A. Haydys and T. Walpuski A compactness theorem for the Seiberg-Witten equation with multiple spinors in dimension three, Geom. Funct. Anal. 25 (2015) 1799 [arXiv:1406.5683].
  30. [30]
    V. Muñoz and C.S. Shahbazi, Construction of the moduli space of Spin(7)-instantons, arXiv:1611.04127 [INSPIRE].
  31. [31]
    M. Becker, L.-S. Tseng and S.-T. Yau, Moduli space of torsional manifolds, Nucl. Phys. B 786 (2007) 119 [hep-th/0612290] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing the complex structure in heterotic Calabi-Yau vacua, JHEP 02 (2011) 088 [arXiv:1010.0255] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing all geometric moduli in heterotic Calabi-Yau vacua, Phys. Rev. D 83 (2011) 106011 [arXiv:1102.0011] [INSPIRE].ADSMATHGoogle Scholar
  34. [34]
    M. Cicoli, S. de Alwis and A. Westphal, Heterotic moduli stabilisation, JHEP 10 (2013) 199 [arXiv:1304.1809] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    I.V. Melnikov and E. Sharpe, On marginal deformations of (0, 2) non-linear σ-models, Phys. Lett. B 705 (2011) 529 [arXiv:1110.1886] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    A. Adams, J. Distler and M. Ernebjerg, Topological heterotic rings, Adv. Theor. Math. Phys. 10 (2006) 657 [hep-th/0506263] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    R. Donagi, J. Guffin, S. Katz and E. Sharpe, A mathematical theory of quantum sheaf cohomology, Asian J. Math. 18 (2014) 387 [arXiv:1110.3751] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    R. Donagi, J. Guffin, S. Katz and E. Sharpe, Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, Adv. Theor. Math. Phys. 17 (2013) 1255 [arXiv:1110.3752] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    S.L. Shatashvili and C. Vafa, Superstrings and manifold of exceptional holonomy, Selecta Math. 1 (1995) 347 [hep-th/9407025] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    J. de Boer, A. Naqvi and A. Shomer, The topological G 2 string, Adv. Theor. Math. Phys. 12 (2008) 243 [hep-th/0506211] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    D.D. Joyce, Compact manifolds with special holonomy, Oxford University Press, Oxford U.K., (2000).MATHGoogle Scholar
  42. [42]
    S. Grigorian, Moduli spaces of G 2 manifolds, Rev. Math. Phys. 22 (2010) 1061 [arXiv:0911.2185] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    T. Friedrich and S. Ivanov, Killing spinor equations in dimension 7 and geometry of integrable G 2 manifolds, J. Geom. Phys. 48 (2003) 1 [math.DG/0112201] [INSPIRE].
  44. [44]
    J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    U. Gran, J. Gutowski and G. Papadopoulos, The G 2 spinorial geometry of supersymmetric IIB backgrounds, Class. Quant. Grav. 23 (2006) 143 [hep-th/0505074] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    A. Lukas and C. Matti, G-structures and domain walls in heterotic theories, JHEP 01 (2011) 151 [arXiv:1005.5302] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    J. Gray, M. Larfors and D. Lüst, Heterotic domain wall solutions and SU(3) structure manifolds, JHEP 08 (2012) 099 [arXiv:1205.6208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    C.M. Hull and P.K. Townsend, World sheet supersymmetry and anomaly cancellation in the heterotic string, Phys. Lett. B 178 (1986) 187 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    E.A. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    R. Blumenhagen, Covariant construction of N = 1 super W -algebras, Nucl. Phys. B 381 (1992) 641 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    J.M. Figueroa-O’Farrill, A note on the extended superconformal algebras associated with manifolds of exceptional holonomy, Phys. Lett. B 392 (1997) 77 [hep-th/9609113] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    P.S. Howe and G. Papadopoulos, Holonomy groups and W symmetries, Commun. Math. Phys. 151 (1993) 467 [hep-th/9202036] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    I.V. Melnikov, R. Minasian and S. Sethi, Spacetime supersymmetry in low-dimensional perturbative heterotic compactifications, arXiv:1707.04613 [INSPIRE].
  54. [54]
    P. Di Francesco, D. Sénéchal and P. Mathieu, Conformal field theory, Springer-Verlag, New York U.S.A., (1997) [INSPIRE].
  55. [55]
    E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    J. Polchinski, String theory: volume 2, superstring theory and beyond, Cambridge University Press, Cambridge U.K., (1998) [INSPIRE].
  57. [57]
    C. Beasley and E. Witten, New instanton effects in string theory, JHEP 02 (2006) 060 [hep-th/0512039] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    P.S. Howe and G. Papadopoulos, Holonomy groups and W -symmetries, Commun. Math. Phys. 151 (1993) 467 [hep-th/9202036] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    R. Reyes Carrión, A generalization of the notion of instanton, Diff. Geom. Appl. 8 (1998) 1.MathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    M. Fernandez and L. Ugarte, Dolbeault cohomology for G 2 -manifolds, Geometriae Dedicata 70 (1998) 57.MathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181.MathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    I.V. Melnikov, R. Minasian and S. Sethi, Non-duality in three dimensions, JHEP 10 (2017) 053 [arXiv:1702.08537] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    G. Papadopoulos and P.K. Townsend, Compactification of D = 11 supergravity on spaces of exceptional holonomy, Phys. Lett. B 357 (1995) 300 [hep-th/9506150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    B.S. Acharya, Dirichlet Joyce manifolds, discrete torsion and duality, Nucl. Phys. B 492 (1997) 591 [hep-th/9611036] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  65. [65]
    B.S. Acharya, On mirror symmetry for manifolds of exceptional holonomy, Nucl. Phys. B 524 (1998) 269 [hep-th/9707186] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    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].ADSCrossRefMATHGoogle Scholar
  67. [67]
    A.P. Braun and S. Schäfer-Nameki, Compact, singular G 2 -holonomy manifolds and M/heterotic/F-theory duality, arXiv:1708.07215 [INSPIRE].
  68. [68]
    I. Melnikov, S. Sethi and E. Sharpe, Recent developments in (0, 2) mirror symmetry, SIGMA 8 (2012) 068 [arXiv:1209.1134] [INSPIRE].MathSciNetMATHGoogle Scholar
  69. [69]
    E. Sharpe, A few recent developments in 2d (2, 2) and (0, 2) theories, Proc. Symp. Pure Math. 93 (2015) 67 [arXiv:1501.01628] [INSPIRE].MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordU.K.
  2. 2.Department of PhysicsUniversity of TorontoTorontoCanada
  3. 3.Department of PhysicsKing’s College LondonLondonU.K.
  4. 4.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  5. 5.Laboratoire de Physique Théorique et Hautes Énergies, UPMC Paris 6, Sorbonne Universités, CNRSParisFrance
  6. 6.Institut Lagrange de ParisParisFrance

Personalised recommendations