Connections, field redefinitions and heterotic supergravity

  • Xenia de la Ossa
  • Eirik E. Svanes
Open Access
Regular Article - Theoretical Physics


We study heterotic supergravity at \( \mathcal{O}\left(\alpha^{\prime}\right) \), first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss the ambiguity of a connection choice on the tangent bundle. It is well known that in order to have a consistent supergravity with supersymmetry transformations given in the usual way, this connection must be the Hull connection at \( \mathcal{O}\left(\alpha^{\prime}\right) \). We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We also discuss possible extensions of this theory to higher orders in α′. We are interested in the moduli space of such field redefinitions which allow for supersymmetric solutions to the equations of motion. We show that for solutions on M 4 × X, where M 4 is Minkowski and X is compact, this is given by H (0,1)(X, End(TX)). This space corresponds to infinitesimally close connections for which the equations of motion are satisfied. The setup suggests a symmetry between the gauge connection and the tangent bundle connection, as also employed by Bergshoeff and de Roo. We propose that this symmetry should be kept to higher orders in α′, and propose a natural choice for the corresponding tangent bundle connection used in curvature computations.


Superstrings and Heterotic Strings Superstring Vacua Supergravity Models Flux compactifications 


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]
    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].ADSCrossRefGoogle Scholar
  2. [2]
    A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    C.M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    D. Lüst, Compactification of ten-dimensional superstring theories over Ricci flat coset spaces, Nucl. Phys. B 276 (1986) 220 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    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
  6. [6]
    C.M. Hull and P.K. Townsend, World sheet supersymmetry and anomaly cancellation in the heterotic string, Phys. Lett. B 178 (1986) 187 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Sen, (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory, Nucl. Phys. B 278 (1986) 289 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    P.S. Howe and G. Papadopoulos, Anomalies in two-dimensional supersymmetric nonlinear σ models, Class. Quant. Grav. 4 (1987) 1749 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    I.V. Melnikov, R. Minasian and S. Theisen, Heterotic flux backgrounds and their IIA duals, JHEP 07 (2014) 023 [arXiv:1206.1417] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    I.V. Melnikov, C. Quigley, S. Sethi and M. Stern, Target spaces from chiral gauge theories, JHEP 02 (2013) 111 [arXiv:1212.1212] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    C.M. Hull, Anomalies, ambiguities and superstrings, Phys. Lett. B 167 (1986) 51 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    K. Becker and S. Sethi, Torsional heterotic geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    I.V. Melnikov, R. Minasian and S. Sethi, Heterotic fluxes and supersymmetry, JHEP 06 (2014) 174 [arXiv:1403.4298] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    L. Witten and E. Witten, Large radius expansion of superstring compactifications, Nucl. Phys. B 281 (1987) 109 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    E. Bergshoeff and M. de Roo, Supersymmetric Chern-Simons terms in ten-dimensions, Phys. Lett. B 218 (1989) 210 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    E.A. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    J. Gillard, G. Papadopoulos and D. Tsimpis, Anomaly, fluxes and (2,0) heterotic string compactifications, JHEP 06 (2003) 035 [hep-th/0304126] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    L. Anguelova, C. Quigley and S. Sethi, The leading quantum corrections to stringy Kähler potentials, JHEP 10 (2010) 065 [arXiv:1007.4793] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    O.A. Bedoya, D. Marques and C. Núñez, Heterotic α’-corrections in double field theory, arXiv:1407.0365 [INSPIRE].
  20. [20]
    O. Hohm and B. Zwiebach, Green-Schwarz mechanism and α-deformed Courant brackets, arXiv:1407.0708 [INSPIRE].
  21. [21]
    O. Hohm and B. Zwiebach, Double field theory at order α′, arXiv:1407.3803 [INSPIRE].
  22. [22]
    A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, arXiv:1407.7542 [INSPIRE].
  23. [23]
    O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    O. Hohm, W. Siegel and B. Zwiebach, Doubled α-geometry, JHEP 02 (2014) 065 [arXiv:1306.2970] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D. Martelli and J. Sparks, Non-Kähler heterotic rotations, Adv. Theor. Math. Phys. 15 (2011) 131 [arXiv:1010.4031] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    D. Harland and C. Nolle, Instantons and Killing spinors, JHEP 03 (2012) 082 [arXiv:1109.3552] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    K.-P. Gemmer, A.S. Haupt, O. Lechtenfeld, C. Nölle and A.D. Popov, Heterotic string plus five-brane systems with asymptotic AdS3, Adv. Theor. Math. Phys. 17 (2013) 771 [arXiv:1202.5046] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    A. Chatzistavrakidis, O. Lechtenfeld and A.D. Popov, Nearly Kähler heterotic compactifications with fermion condensates, JHEP 04 (2012) 114 [arXiv:1202.1278] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    M. Klaput, A. Lukas, C. Matti and E.E. Svanes, Moduli stabilising in heterotic nearly Kähler compactifications, JHEP 01 (2013) 015 [arXiv:1210.5933] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    A.S. Haupt, O. Lechtenfeld and E.T. Musaev, Order αheterotic domain walls with warped nearly Kähler geometry, arXiv:1409.0548 [INSPIRE].
  32. [32]
    J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, Five-branes wrapped on SLAG three cycles and related geometry, JHEP 11 (2001) 018 [hep-th/0110034] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    J. Held, D. Lüst, F. Marchesano and L. Martucci, DWSB in heterotic flux compactifications, JHEP 06 (2010) 090 [arXiv:1004.0867] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S.-T. Yau and J. Li, Hermitian-Yang-Mills connections on non-Kähler manifolds, World Scient. Publ., London U.K. (1987), pg. 560.Google Scholar
  35. [35]
    S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.CrossRefMATHMathSciNetGoogle Scholar
  36. [36]
    K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) S257.CrossRefMATHMathSciNetGoogle Scholar
  37. [37]
    D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge U.K. (2010).Google Scholar
  38. [38]
    D. Andriot, Heterotic string from a higher dimensional perspective, Nucl. Phys. B 855 (2012) 222 [arXiv:1102.1434] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    A.P. Foakes, N. Mohammedi and D.A. Ross, Three loop β-functions for the superstring and heterotic string, Nucl. Phys. B 310 (1988) 335 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    J.P. Gauntlett, D. Martelli, S. Pakis and D. Waldram, G structures and wrapped NS5-branes, Commun. Math. Phys. 247 (2004) 421 [hep-th/0205050] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    D.J. Gross and J.H. Sloan, The quartic effective action for the heterotic string, Nucl. Phys. B 291 (1987) 41 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    Y. Cai and C.A. Núñez, Heterotic string covariant amplitudes and low-energy effective action, Nucl. Phys. B 287 (1987) 279 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A. Gray and L.M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980) 35.CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 structures, in Differential geometry. Proceedings of the international conference held in honour of the 60th birthday of A.M. Naveira, Valencia Spain July 8-14 2001, O. Gil-Medrano et al. eds., World Scientific, Singapore (2002), pg 115.Google Scholar
  45. [45]
    G. Lopes Cardoso et al., Non-Kähler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5 [hep-th/0211118] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].ADSMathSciNetGoogle Scholar
  47. [47]
    G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 10 (2003) 004 [hep-th/0306088] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of Oxford, Andrew Wiles Building, Radcliffe Observatory QuarterOxfordU.K.
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordU.K.
  3. 3.LPTHE, Sorbonne Universités, UPMC Univ Paris 06, UMR 7589ParisFrance
  4. 4.CNRS, LPTHE, UMR 7589ParisFrance
  5. 5.Institut Lagrange de ParisSorbonne UniversitésParisFrance

Personalised recommendations