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Consistent four-derivative heterotic truncations and the Kerr-Sen solution

Minghao Xia1, Liang Ma1, Yi Pang1,2,*, and Robert J. Saskowski1,†

  • *Contact author: pangyi1@tju.edu.cn
  • †Contact author: robert_saskowski@tju.edu.cn
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Phys. Rev. D 113, 046019 – Published 24 February, 2026

DOI: https://doi.org/10.1103/h7kw-7cym

Abstract

Four-derivative heterotic supergravity (without gauge fields) reduced on a p-dimensional torus leads to half-maximal supergravity coupled to p vector multiplets, and it is known that removing the vector multiplets is a consistent truncation of the theory. We find a new consistent truncation of four-derivative heterotic supergravity on a torus that keeps the vector multiplets and precisely reproduces the bosonic action of heterotic supergravity (with heterotic gauge fields). We show that both truncations have an O(d+p,d) symmetry when reduced on a d-dimensional torus and demonstrate how this embeds in the O(d+p,d+p) symmetry that one gets from reducing on a (d+p)-dimensional torus without truncation. We then use our new truncation to obtain four-derivative corrections to the Kerr-Sen solution and compute thermodynamic quantities and multipole moments. Finally, we compare the Kerr-Sen solutions of the actions corresponding to the two different choices of truncation with the Kerr solution, the Kerr-Newman solution, and each other, and show that they have distinct four-derivative multipole structures.

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References (87)

  1. G. Veneziano, Scale factor duality for classical and quantum strings, Phys. Lett. B 265, 287 (1991).
  2. K. A. Meissner and G. Veneziano, Symmetries of cosmological superstring vacua, Phys. Lett. B 267, 33 (1991).
  3. A. Sen, O(d) ⊗ O(d) symmetry of the space of cosmological solutions in string theory, scale factor duality and two-dimensional black holes, Phys. Lett. B 271, 295 (1991).
  4. A. Sen, Twisted black p-brane solutions in string theory, Phys. Lett. B 274, 34 (1992).
  5. M. Gasperini, J. Maharana, and G. Veneziano, From trivial to nontrivial conformal string backgrounds via O(d,d) transformations, Phys. Lett. B 272, 277 (1991).
  6. S. F. Hassan and A. Sen, Twisting classical solutions in heterotic string theory, Nucl. Phys. B375, 103 (1992).
  7. A. Sen, Rotating charged black hole solution in heterotic string theory, Phys. Rev. Lett. 69, 1006 (1992).
  8. M. Cvetic and D. Youm, All the four-dimensional static, spherically symmetric solutions of Abelian Kaluza-Klein theory, Phys. Rev. Lett. 75, 4165 (1995).
  9. M. Cvetic and D. Youm, All the static spherically symmetric black holes of heterotic string on a six torus, Nucl. Phys. B472, 249 (1996).
  10. M. Cvetic and D. Youm, General rotating five-dimensional black holes of toroidally compactified heterotic string, Nucl. Phys. B476, 118 (1996).
  11. O. Hohm, A. Sen, and B. Zwiebach, Heterotic effective action and duality symmetries revisited, J. High Energy Phys. 02 (2015) 079.
  12. P.-J. Hu, L. Ma, Y. Pang, and R. J. Saskowski, Higher-derivative heterotic Kerr-Sen black holes, arXiv:2506.20077.
  13. H. Godazgar and M. Godazgar, Duality completion of higher derivative corrections, J. High Energy Phys. 09 (2013) 140.
  14. O. Hohm and B. Zwiebach, T-duality constraints on higher derivatives revisited, J. High Energy Phys. 04 (2016) 101.
  15. D. Marques and C. A. Nunez, T-duality and α’-corrections, J. High Energy Phys. 10 (2015) 084.
  16. W. H. Baron, J. J. Fernandez-Melgarejo, D. Marques, and C. Nunez, The odd story of α’-corrections, J. High Energy Phys. 04 (2017) 078.
  17. M. R. Garousi, Four-derivative couplings via the T-duality invariance constraint, Phys. Rev. D 99, 126005 (2019).
  18. M. R. Garousi, Effective action of bosonic string theory at order α′2, Eur. Phys. J. C 79, 827 (2019).
  19. M. R. Garousi, Effective action of type II superstring theories at order α′3: NS-NS couplings, J. High Energy Phys. 02 (2021) 157.
  20. T. Codina, O. Hohm, and D. Marques, String dualities at order α′3, Phys. Rev. Lett. 126, 171602 (2021).
  21. M. David and J. T. Liu, T duality and hints of generalized geometry in string α’ corrections, Phys. Rev. D 106, 106008 (2022).
  22. M. David and J. T. Liu, T-duality building blocks for α’ string corrections, Phys. Rev. D 107, 046008 (2023).
  23. M. Ozkan, Y. Pang, and E. Sezgin, Higher derivative supergravities in diverse dimensions, Phys. Rep. 1086, 1 (2024).
  24. O. Hohm, W. Siegel, and B. Zwiebach, Doubled α′-geometry, J. High Energy Phys. 02 (2014) 065.
  25. O. A. Bedoya, D. Marques, and C. Nunez, Heterotic α’-corrections in double field theory, J. High Energy Phys. 12 (2014) 074.
  26. O. Hohm and B. Zwiebach, Double field theory at order α′, J. High Energy Phys. 11 (2014) 075.
  27. A. Coimbra, R. Minasian, H. Triendl, and D. Waldram, Generalised geometry for string corrections, J. High Energy Phys. 11 (2014) 160.
  28. K. Lee, Quadratic α’-corrections to heterotic double field theory, Nucl. Phys. B899, 594 (2015).
  29. E. Lescano, C. A. Núñez, and J. A. Rodríguez, Supersymmetry, T-duality and heterotic α’-corrections, J. High Energy Phys. 07 (2021) 092.
  30. S. Hronek and L. Wulff, O(D,D) and the string α′ expansion: An obstruction, J. High Energy Phys. 04 (2021) 013.
  31. S. W. Hsia, A. R. Kamal, and L. Wulff, No manifest T duality at order α’3, Phys. Rev. D 111, L061904 (2025).
  32. O. Lunin and P. Shah, Double field theory and α’ corrections: Explicit examples, Nucl. Phys. B1017, 116932 (2025).
  33. E. Bergshoeff and M. de Roo, Supersymmetric Chern-Simons terms in ten dimensions, Phys. Lett. B 218, 210 (1989).
  34. E. A. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B328, 439 (1989).
  35. J. T. Liu and R. J. Saskowski, Consistent truncations in higher derivative supergravity, J. High Energy Phys. 09 (2023) 136.
  36. M. Rogatko, Uniqueness theorem for stationary axisymmetric black holes in Einstein-Maxwell-axion-dilaton gravity, Phys. Rev. D 82, 044017 (2010).
  37. C. Eloy, O. Hohm, and H. Samtleben, Duality invariance and higher derivatives, Phys. Rev. D 101, 126018 (2020).
  38. Z. Elgood and T. Ortin, T duality and Wald entropy formula in the heterotic superstring effective action at first-order in α’, J. High Energy Phys. 10 (2020) 097; 06 (2021) 105(E).
  39. T. Ortin, O(n, n) invariance and Wald entropy formula in the heterotic superstring effective action at first order in α′, J. High Energy Phys. 01 (2021) 187.
  40. S. Jayaprakash and J. T. Liu, Higher derivative heterotic supergravity on a torus and supersymmetry, J. High Energy Phys. 12 (2024) 076.
  41. J. Sonner and P. K. Townsend, Recurrent acceleration in dilaton-axion cosmology, Phys. Rev. D 74, 103508 (2006).
  42. R. Catena and J. Moller, Axion-dilaton cosmology and dark energy, J. Cosmol. Astropart. Phys. 03 (2008) 012.
  43. A. D. Linde, Axions in inflationary cosmology, Phys. Lett. B 259, 38 (1991).
  44. B. A. Campbell, N. Kaloper, R. Madden, and K. A. Olive, Physical properties of four-dimensional superstring gravity black hole solutions, Nucl. Phys. B399, 137 (1993).
  45. S. Dastan, R. Saffari, and S. Soroushfar, Shadow of a Kerr-Sen dilaton-axion black hole, arXiv:1610.09477.
  46. M. Cvetic, G. W. Gibbons, and C. N. Pope, STU black holes and SgrA, J. Cosmol. Astropart. Phys. 08 (2017) 016.
  47. M. Guo, S. Song, and H. Yan, Observational signature of a near-extremal Kerr-Sen black hole in the heterotic string theory, Phys. Rev. D 101, 024055 (2020).
  48. A. Narang, S. Mohanty, and A. Kumar, Test of Kerr-Sen metric with black hole observations, arXiv:2002.12786.
  49. S. V. M. C. B. Xavier, P. V. P. Cunha, L. C. B. Crispino, and C. A. R. Herdeiro, Shadows of charged rotating black holes: Kerr–Newman versus Kerr–Sen, Int. J. Mod. Phys. D 29, 2041005 (2020).
  50. M. Shahzadi, M. Kološ, Z. Stuchlík, and Y. Habib, Testing alternative theories of gravity by fitting the hot-spot data of Sgr A*, Eur. Phys. J. C 82, 407 (2022).
  51. R. Della Monica, I. de Martino, and M. de Laurentis, Testing space–time geometries and theories of gravity at the Galactic centre with pulsar’s time delay, Mon. Not. R. Astron. Soc. 524, 3782 (2023).
  52. S. K. Sahoo, N. Yadav, and I. Banerjee, Imprints of Einstein-Maxwell-dilaton-axion gravity in the observed shadows of Sgr A* and M87*, Phys. Rev. D 109, 044008 (2024).
  53. H. Feng, R.-J. Yang, and W.-Q. Chen, Thin accretion disk and shadow of Kerr–Sen black hole in Einstein–Maxwell-dilaton–axion gravity, Astropart. Phys. 166, 103075 (2025).
  54. P. Amaro-Seoane et al. (LISA Collaboration), Laser Interferometer Space Antenna, arXiv:1702.00786.
  55. L. Barack and C. Cutler, Using LISA EMRI sources to test off-Kerr deviations in the geometry of massive black holes, Phys. Rev. D 75, 042003 (2007).
  56. V. Cardoso and L. Gualtieri, Testing the black hole ‘no-hair’ hypothesis, Classical Quantum Gravity 33, 174001 (2016).
  57. S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. L. Berry, E. Berti, P. Amaro-Seoane, A. Petiteau, and A. Klein, Science with the space-based interferometer LISA. V: Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012 (2017).
  58. F. D. Ryan, Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments, Phys. Rev. D 52, 5707 (1995).
  59. N. V. Krishnendu, C. K. Mishra, and K. G. Arun, Spin-induced deformations and tests of binary black hole nature using third-generation detectors, Phys. Rev. D 99, 064008 (2019).
  60. Y. Cai, S. Jayaprakash, J. T. Liu, and R. J. Saskowski, Multicenter higher-derivative BPS black holes, J. High Energy Phys. 10 (2025) 014.
  61. J. Maharana and J. H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B390, 3 (1993).
  62. A. Sen, Strong—weak coupling duality in four-dimensional string theory, Int. J. Mod. Phys. A 09, 3707 (1994).
  63. O. Hohm and S. K. Kwak, Double field theory formulation of heterotic strings, J. High Energy Phys. 06 (2011) 096.
  64. G. W. Gibbons, Antigravitating black hole solitons with scalar hair in N=4 supergravity, Nucl. Phys. B207, 337 (1982).
  65. G. W. Gibbons and K.-i. Maeda, Black holes and membranes in higher dimensional theories with dilaton fields, Nucl. Phys. B298, 741 (1988).
  66. D. Garfinkle, G. T. Horowitz, and A. Strominger, Charged black holes in string theory, Phys. Rev. D 43, 3140 (1991); 45, 3888(E) (1992).
  67. R. Kallosh, A. D. Linde, T. Ortin, A. W. Peet, and A. Van Proeyen, Supersymmetry as a cosmic censor, Phys. Rev. D 46, 5278 (1992).
  68. J. M. Bardeen and G. T. Horowitz, The extreme Kerr throat geometry: A vacuum analog of AdS2×S2, Phys. Rev. D 60, 104030 (1999).
  69. G. W. Gibbons and C. M. Hull, A Bogomolny Bound for general relativity and solitons in N=2 supergravity, Phys. Lett. 109B, 190 (1982).
  70. K. P. Tod, All metrics admitting supercovariantly constant spinors, Phys. Lett. 121B, 241 (1983).
  71. P. Pani and V. Cardoso, Are black holes in alternative theories serious astrophysical candidates? The case for Einstein-Dilaton-Gauss-Bonnet black holes, Phys. Rev. D 79, 084031 (2009).
  72. N. Yunes and F. Pretorius, Dynamical Chern-Simons modified gravity. I. Spinning black holes in the slow-rotation approximation, Phys. Rev. D 79, 084043 (2009).
  73. K. Konno, T. Matsuyama, and S. Tanda, Rotating black hole in extended Chern-Simons modified gravity, Prog. Theor. Phys. 122, 561 (2009).
  74. K. Yagi, N. Yunes, and T. Tanaka, Slowly rotating black holes in dynamical Chern-Simons gravity: Deformation quadratic in the spin, Phys. Rev. D 86, 044037 (2012); 89, 049902(E) (2014).
  75. S. Mignemi and N. R. Stewart, Dilaton axion hair for slowly rotating Kerr black holes, Phys. Lett. B 298, 299 (1993).
  76. P. A. Cano and A. Ruipérez, Leading higher-derivative corrections to Kerr geometry, J. High Energy Phys. 05 (2019) 189; 03 (2020) 187(E).
  77. K. S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52, 299 (1980).
  78. P. A. Cano, B. Ganchev, D. R. Mayerson, and A. Ruipérez, Black hole multipoles in higher-derivative gravity, J. High Energy Phys. 12 (2022) 120.
  79. I. Bena and D. R. Mayerson, Black holes lessons from multipole ratios, J. High Energy Phys. 03 (2021) 114.
  80. L. Ma, Y. Pang, and H. Lu, Leading higher derivative corrections to multipole moments of Kerr-Newman black hole, J. High Energy Phys. 02 (2025) 079.
  81. B. Chen and L. C. Stein, Deformation of extremal black holes from stringy interactions, Phys. Rev. D 97, 084012 (2018).
  82. P. A. Cano and M. David, The extremal Kerr entropy in higher-derivative gravities, J. High Energy Phys. 05 (2023) 219.
  83. D. Cassani, A. Ruipérez, and E. Turetta, Boundary terms and conserved charges in higher-derivative gauged supergravity, J. High Energy Phys. 06 (2023) 203.
  84. P. A. Cano and M. David, Near-horizon geometries and black hole thermodynamics in higher-derivative AdS5 supergravity, J. High Energy Phys. 03 (2024) 036.
  85. A. Van Proeyen, Tools for supersymmetry, Ann. U. Craiova Phys. 9, 1 (1999).
  86. D. Z. Freedman and A. Van Proeyen, Supergravity (Cambridge University Press, Cambridge, England, 2012).
  87. E. Cremmer, S. Ferrara, L. Girardello, and A. Van Proeyen, Yang-Mills theories with local supersymmetry: Lagrangian, transformation laws and superHiggs effect, Nucl. Phys. B212, 413 (1983).

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Phys. Rev. D 113, 046019– Published 24 February, 2026
  • Received 11 November 2025
  • Accepted 2 February 2026
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