Extended Riemannian geometry II: local heterotic double field theory

  • Andreas Deser
  • Marc Andre Heller
  • Christian Sämann
Open Access
Regular Article - Theoretical Physics


We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α′-corrections are interpreted naturally in our framework.


Differential and Algebraic Geometry Superstrings and Heterotic Strings 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]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    C.M. Hull and R.A. Reid-Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 09 (2009) 014 [arXiv:0902.4032] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    I. Vaisman, On the geometry of double field theory, J. Math. Phys. 53 (2012) 033509 [arXiv:1203.0836] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    I. Vaisman, Towards a double field theory on para-Hermitian manifolds, J. Math. Phys. 54 (2013) 123507 [arXiv:1209.0152] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Deser and J. Stasheff, Even symplectic supermanifolds and double field theory, Commun. Math. Phys. 339 (2015) 1003 [arXiv:1406.3601] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    L. Freidel, F.J. Rudolph and D. Svoboda, Generalised kinematics for double field theory, JHEP 11 (2017) 175 [arXiv:1706.07089] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  12. [12]
    N. Hitchin, Brackets, forms and invariant functionals, math/0508618 [INSPIRE].
  13. [13]
    M. Gualtieri, Generalized complex geometry, Ph.D. Thesis, Oxford University, Oxford U.K. (2003) [math/0401221] [INSPIRE].
  14. [14]
    D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Workshop on Quantization, Deformations and New Homological and Categorical Methods in Mathematical Physics, Manchester U.K. (2002) [math/0203110] [INSPIRE].
  15. [15]
    A. Deser and C. Sämann, Extended Riemannian geometry I: local double field theory, arXiv:1611.02772 [INSPIRE].
  16. [16]
    M.A. Heller, N. Ikeda and S. Watamura, Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds, JHEP 02 (2017) 078 [arXiv:1611.08346] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    O. Hohm and B. Zwiebach, L algebras and field theory, Fortsch. Phys. 65 (2017) 1700014 [arXiv:1701.08824] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    O. Hohm, W. Siegel and B. Zwiebach, Doubled α-geometry, JHEP 02 (2014) 065 [arXiv:1306.2970] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    O. Hohm and B. Zwiebach, Double field theory at order α′, JHEP 11 (2014) 075 [arXiv:1407.3803] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    O. Hohm and B. Zwiebach, Green-Schwarz mechanism and α-deformed Courant brackets, JHEP 01 (2015) 012 [arXiv:1407.0708] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    R. Blumenhagen and R. Sun, T-duality, non-geometry and Lie algebroids in heterotic double field theory, JHEP 02 (2015) 097 [arXiv:1411.3167] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Garcia-Fernandez, Torsion-free generalized connections and Heterotic Supergravity, Commun. Math. Phys. 332 (2014) 89 [arXiv:1304.4294] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    D. Baraglia and P. Hekmati, Transitive Courant algebroids, string structures and T-duality, Adv. Theor. Math. Phys. 19 (2015) 613 [arXiv:1308.5159] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    H. Sati, U. Schreiber and J. Stasheff, Differential twisted string and fivebrane structures, Commun. Math. Phys. 315 (2012) 169 [arXiv:0910.4001] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D. Fiorenza, H. Sati and U. Schreiber, Multiple M5-branes, string 2-connections and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229 [arXiv:1201.5277] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  27. [27]
    C. Sämann and L. Schmidt, The non-Abelian self-dual string and the (2, 0)-theory, arXiv:1705.02353 [INSPIRE].
  28. [28]
    D. Roytenberg, A note on quasi Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002) 123 [math/0112152] [INSPIRE].
  29. [29]
    A. Deser, Star products on graded manifolds and α-corrections to Courant algebroids from string theory, J. Math. Phys. 56 (2015) 092302 [arXiv:1412.5966] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    O.A. Bedoya, D. Marques and C. Núñez, Heterotic α-corrections in double field theory, JHEP 12 (2014) 074 [arXiv:1407.0365] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 1. The free heterotic string, Nucl. Phys. B 256 (1985) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 2. The interacting heterotic string, Nucl. Phys. B 267 (1986) 75 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Ten-dimensional Maxwell-Einstein supergravity, its currents and the issue of its auxiliary fields, Nucl. Phys. B 195 (1982) 97 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    G.F. Chapline and N.S. Manton, Unification of Yang-Mills theory and supergravity in ten-dimensions, Phys. Lett. B 120 (1983) 105 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. 149B (1984) 117 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    H. Sati, U. Schreiber and J. Stasheff, L algebra connections and applications to String- and Chern-Simons n-transport, in Quantum Field Theory, B. Fauser, J. Tolksdorf and E. Zeidler eds., Birkhäuser, Boston U.S.A. (2009), pg. 303 [arXiv:0801.3480] [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    K. Gawedzki, Topological actions in two-dimensional quantum field theories, in Proceedings of Nonperturbative quantum field theory, Cargèse France (1987), [NATO Sci. Ser. B 185 (1988)101].Google Scholar
  39. [39]
    D.S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999) 819 [hep-th/9907189] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  40. [40]
    J.C. Baez and A.S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theor. Appl. Categor. 12 (2004) 492 [math/0307263] [INSPIRE].
  41. [41]
    A. Henriques, Integrating L -algebras, Comput. Math. 144 (2008) 1017 [math/0603563].
  42. [42]
    J.C. Baez, D. Stevenson, A.S. Crans and U. Schreiber, From loop groups to 2-groups, Homol. Homot. Appl. 9 (2007) 101 [math/0504123] [INSPIRE].
  43. [43]
    G.A. Demessie and C. Sämann, Higher gauge theory with string 2-groups, Adv. Theor. Math. Phys. 21 (2017) 1895 [arXiv:1602.03441] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  44. [44]
    P. Severa, Letter No. 3 to Alan Weinstein (1998), available at
  45. [45]
    M. Garcia-Fernandez, Ricci flow, Killing spinors and T-duality in generalized geometry, arXiv:1611.08926 [INSPIRE].
  46. [46]
    B. Jurčo and J. Vysoký, Heterotic reduction of Courant algebroid connections and Einstein-Hilbert actions, Nucl. Phys. B 909 (2016) 86 [arXiv:1512.08522] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    B. Jurčo and J. Vysoky, Courant algebroid connections and string effective actions, in Proceedings of Workshop on Strings, Membranes and Topological Field Theory, Tohoku Japan (2017), pg. 211 [arXiv:1612.01540] [INSPIRE].
  48. [48]
    B. Jurčo and J. Vysoky, Poisson-Lie T-duality of string effective actions: a new approach to the dilaton puzzle, arXiv:1708.04079 [INSPIRE].
  49. [49]
    O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring double field theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    G. Papadopoulos, Seeking the balance: patching double and exceptional field theories, JHEP 10 (2014) 089 [arXiv:1402.2586] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    O. Hohm, D. Lüst and B. Zwiebach, The spacetime of double field theory: review, remarks and outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087 [hep-th/9209099] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  54. [54]
    D. Roytenberg and A. WEinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys. 46 (1998) 81 [math/9802118] [INSPIRE].
  55. [55]
    D. Fiorenza and M. Manetti, L structures on mapping cones, Alg. Numb. Theor. 1 (2007) 301 [math/0601312].MathSciNetCrossRefGoogle Scholar
  56. [56]
    E. Getzler, Higher derived brackets, arXiv:1010.5859.

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Andreas Deser
    • 1
  • Marc Andre Heller
    • 2
  • Christian Sämann
    • 3
  1. 1.Istituto Nationale di Fisica NucleareTorinoItaly
  2. 2.Particle Theory and Cosmology Group, Department of Physics, Graduate School of ScienceTohoku UniversitySendaiJapan
  3. 3.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghU.K.

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