DFT in supermanifold formulation and group manifold as background geometry

  • Ursula Carow-Watamura
  • Noriaki IkedaEmail author
  • Tomokazu Kaneko
  • Satoshi Watamura
Open Access
Regular Article - Theoretical Physics


We develop the formulation of DFT on pre-QP-manifold. The consistency conditions like section condition and closure constraint are unified by a weak master equation. The Bianchi identities are also characterized by the pre-Bianchi identity. Then, the background metric and connections are formulated by using covariantized pre-QP-manifold. An application to the analysis of the DFT on group manifold is given.


Differential and Algebraic Geometry Field Theories in Higher Dimensions String Duality 


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]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
  2. [2]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
  3. [3]
    C.M. Hull, Doubled geometry and T-folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math.DG/0209099] [INSPIRE].
  10. [10]
    M. Gualtieri, Generalized complex geometry, math.DG/0401221 [INSPIRE].
  11. [11]
    N. Hitchin, Brackets, forms and invariant functionals, math.DG/0508618 [INSPIRE].
  12. [12]
    N. Hitchin, Lectures on generalized geometry, arXiv:1008.0973 [INSPIRE].
  13. [13]
    O. Hohm and B. Zwiebach, Towards an invariant geometry of double field theory, J. Math. Phys. 54 (2013) 032303 [arXiv:1212.1736] [INSPIRE].
  14. [14]
    D.S. Berman, M. Cederwall and M.J. Perry, Global aspects of double geometry, JHEP 09 (2014) 066 [arXiv:1401.1311] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. Scherk and J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979) 61 [INSPIRE].
  16. [16]
    G. Aldazabal, W. Baron, D. Marques and C. Núñez, The effective action of double field theory, JHEP 11 (2011) 052 [Erratum ibid. 11 (2011) 109] [arXiv:1109.0290] [INSPIRE].
  17. [17]
    M. Graña and D. Marques, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D.S. Berman and K. Lee, Supersymmetry for gauged double field theory and generalised Scherk-Schwarz reductions, Nucl. Phys. B 881 (2014) 369 [arXiv:1305.2747] [INSPIRE].
  19. [19]
    N. Ikeda, Lectures on AKSZ σ-models for physicists, in Proceedings, Workshop on Strings, Membranes and Topological Field Theory, WSPC, World Scientific, Singapore (2017), pg. 79 [arXiv:1204.3714] [INSPIRE].
  20. [20]
    T. Bessho, M.A. Heller, N. Ikeda and S. Watamura, Topological membranes, current algebras and H-flux-R-flux duality based on Courant algebroids, JHEP 04 (2016) 170 [arXiv:1511.03425] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    A.J. Bruce and J. Grabowski, Pre-Courant algebroids, arXiv:1608.01585.
  22. [22]
    A. Deser and C. Sämann, Extended Riemannian geometry I: local double field theory, arXiv:1611.02772 [INSPIRE].
  23. [23]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Deser and J. Stasheff, Even symplectic supermanifolds and double field theory, Commun. Math. Phys. 339 (2015) 1003 [arXiv:1406.3601] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M.A. Heller, N. Ikeda and S. Watamura, Courant algebroids from double field theory in supergeometry, in Proceedings, Workshop on Strings, Membranes and Topological Field Theory, WSPC, World Scientific, Singapore (2017), pg. 315 [arXiv:1703.00638] [INSPIRE].
  26. [26]
    D. Geissbuhler, Double field theory and N = 4 gauged supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [INSPIRE].
  27. [27]
    D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring double field theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, math.DG/9910078.
  29. [29]
    A. Chatzistavrakidis, L. Jonke, F.S. Khoo and R.J. Szabo, Double field theory and membrane σ-models, JHEP 07 (2018) 015 [arXiv:1802.07003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    P. du Bosque, F. Hassler and D. Lüst, Flux formulation of DFT on group manifolds and generalized Scherk-Schwarz compactifications, JHEP 02 (2016) 039 [arXiv:1509.04176] [INSPIRE].
  31. [31]
    R. Blumenhagen, F. Hassler and D. Lüst, Double field theory on group manifolds, JHEP 02 (2015) 001 [arXiv:1410.6374] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    O. Hohm and D. Marques, Perturbative double field theory on general backgrounds, Phys. Rev. D 93 (2016) 025032 [arXiv:1512.02658] [INSPIRE].
  33. [33]
    F. Hassler, Poisson-Lie T-duality in double field theory, arXiv:1707.08624 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Ursula Carow-Watamura
    • 1
  • Noriaki Ikeda
    • 2
    Email author
  • Tomokazu Kaneko
    • 1
  • Satoshi Watamura
    • 1
  1. 1.Particle Theory and Cosmology Group, Department of Physics, Graduate School of ScienceTohoku UniversitySendaiJapan
  2. 2.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan

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