Skip to main content
Download PDF

Poisson-Lie duals of the η-deformed AdS2 × S2 × T6 superstring

Journal of High Energy Physics volume 2018, Article number: 107 (2018) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We investigate Poisson-Lie duals of the η-deformed AdS2 ×S2 ×T6 superstring. The η-deformed background satisfies a generalisation of the type II supergravity equations. We discuss three Poisson-Lie duals, with respect to (i) the full \( \mathfrak{p}\mathfrak{s}\mathfrak{u}\left(1,\left.1\right|2\right) \) superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard type II supergravity equations. The metrics and B-fields for the first two cases are the same and given by an analytic continuation of the λ-deformed model on AdS2 × S2 × T6 with the torus undeformed. However, the RR fluxes and dilaton will differ. Focusing on the second case we explicitly derive the background and show agreement with an analytic continuation of a known embedding of the λ-deformed model on AdS2 × S2 in type II supergravity.

References

  1. B. Hoare and F.K. Seibold, Poisson-Lie duals of the η deformed symmetric space σ-model, JHEP 11 (2017) 014 [arXiv:1709.01448] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. C. Klimčík and P. Ševera, Dual non-Abelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].

  3. C. Klimčík, Poisson-Lie T-duality, Nucl. Phys. Proc. Suppl. 46 (1996) 116 [hep-th/9509095] [INSPIRE].

  4. C. Klimčík, Yang-Baxter σ-models and dS/AdS T-duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].

  5. C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].

  6. F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. H. Eichenherr and M. Forger, On the dual symmetry of the non-linear σ-models, Nucl. Phys. B 155 (1979) 381 [INSPIRE].

    ADS  Article  Google Scholar 

  8. H. Eichenherr and M. Forger, More about non-linear σ-models on symmetric spaces, Nucl. Phys. B 164 (1980) 528 [Erratum ibid. B 282 (1987) 745] [INSPIRE].

  9. R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  10. J.-G. Zhou, Super 0-brane and GS superstring actions on AdS 2 × S 2, Nucl. Phys. B 559 (1999) 92 [hep-th/9906013] [INSPIRE].

    ADS  Article  Google Scholar 

  11. N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Zwiebach, Superstring theory on AdS 2 × S 2 as a coset supermanifold, Nucl. Phys. B 567 (2000) 61 [hep-th/9907200] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  12. K. Zarembo, Strings on semisymmetric superspaces, JHEP 05 (2010) 002 [arXiv:1003.0465] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].

    ADS  Article  Google Scholar 

  14. F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS 5 × S 5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  15. M.B. Green and J.H. Schwarz, Covariant description of superstrings, Phys. Lett. B 136 (1984) 367 [INSPIRE].

    ADS  Article  Google Scholar 

  16. M.B. Green and J.H. Schwarz, Properties of the covariant formulation of superstring theories, Nucl. Phys. B 243 (1984) 285 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. E. Witten, Twistor-like transform in ten-dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. M.T. Grisaru, P.S. Howe, L. Mezincescu, B. Nilsson and P.K. Townsend, N = 2 superstrings in a supergravity background, Phys. Lett. 162B (1985) 116 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. D. Sorokin, A. Tseytlin, L. Wulff and K. Zarembo, Superstrings in AdS 2 × S 2 × T 6, J. Phys. A 44 (2011) 275401 [arXiv:1104.1793] [INSPIRE].

    ADS  MATH  Google Scholar 

  20. B. Hoare and S.J. van Tongeren, Non-split and split deformations of AdS5, J. Phys. A 49 (2016) 484003 [arXiv:1605.03552] [INSPIRE].

    MATH  Google Scholar 

  21. T. Araujo, E. Ó Colgáin, J. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, I in generalized supergravity, Eur. Phys. J. C 77 (2017) 739 [arXiv:1708.03163] [INSPIRE].

  22. F. Delduc, S. Lacroix, M. Magro and B. Vicedo, On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models, J. Phys. A 49 (2016) 415402 [arXiv:1606.01712] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  23. F. Delduc, T. Kameyama, M. Magro and B. Vicedo, Affine q-deformed symmetry and the classical Yang-Baxter σ-model, JHEP 03 (2017) 126 [arXiv:1701.03691] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS 5 × S 5, JHEP 04 (2014) 002 [arXiv:1312.3542] [INSPIRE].

    ADS  Article  Google Scholar 

  25. C. Klimčík and P. Ševera, Poisson-Lie T duality and loop groups of Drinfeld doubles, Phys. Lett. B 372 (1996) 65 [hep-th/9512040] [INSPIRE].

  26. C. Klimčík and P. Ševera, NonAbelian momentum winding exchange, Phys. Lett. B 383 (1996) 281 [hep-th/9605212] [INSPIRE].

  27. C. Klimčík and P. Ševera, Dressing cosets, Phys. Lett. B 381 (1996) 56 [hep-th/9602162] [INSPIRE].

  28. R. Squellari, Dressing cosets revisited, Nucl. Phys. B 853 (2011) 379 [arXiv:1105.0162] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. K. Sfetsos, Duality invariant class of two-dimensional field theories, Nucl. Phys. B 561 (1999) 316 [hep-th/9904188] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. D. Lüst and D. Osten, Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T-duality, JHEP 05 (2018) 165 [arXiv:1803.03971] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  31. B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys. A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  32. B. Hoare and A.A. Tseytlin, On integrable deformations of superstring σ-models related to AdS n × S n supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. K. Sfetsos, K. Siampos and D.C. Thompson, Generalised integrable λ- and η-deformations and their relation, Nucl. Phys. B 899 (2015) 489 [arXiv:1506.05784] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. C. Klimčík, η and λ deformations as ε-models, Nucl. Phys. B 900 (2015) 259 [arXiv:1508.05832] [INSPIRE].

  35. S. Driezen, A. Sevrin and D.C. Thompson, D-branes in λ-deformations, arXiv:1806.10712 [INSPIRE].

  36. T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An integrable deformation of the AdS 5 × S 5 superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].

    MATH  Google Scholar 

  37. K. Sfetsos, Integrable interpolations: from exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable deformations of strings on symmetric spaces, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. E. Alvarez, L. Álvarez-Gaumé and Y. Lozano, On non-Abelian duality, Nucl. Phys. B 424 (1994) 155 [hep-th/9403155] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. S. Elitzur, A. Giveon, E. Rabinovici, A. Schwimmer and G. Veneziano, Remarks on non-Abelian duality, Nucl. Phys. B 435 (1995) 147 [hep-th/9409011] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. E. Tyurin and R. von Unge, Poisson-Lie T-duality: the path integral derivation, Phys. Lett. B 382 (1996) 233 [hep-th/9512025] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. A. Bossard and N. Mohammedi, Poisson-Lie duality in the string effective action, Nucl. Phys. B 619 (2001) 128 [hep-th/0106211] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  43. R. Von Unge, Poisson-Lie T-plurality, JHEP 07 (2002) 014 [hep-th/0205245] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  44. L. Hlavatý and L. Šnobl, Poisson-Lie T plurality of three-dimensional conformally invariant σ-models, JHEP 05 (2004) 010 [hep-th/0403164] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. L. Hlavatý and L. Šnobl, Poisson-Lie T-plurality of three-dimensional conformally invariant σ-models. II. Nondiagonal metrics and dilaton puzzle, JHEP 10 (2004) 045 [hep-th/0408126] [INSPIRE].

  46. B. Jurčo and J. Vysoký, Poisson-Lie T-duality of string effective actions: a new approach to the dilaton puzzle, J. Geom. Phys. 130 (2018) 1 [arXiv:1708.04079] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  47. G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS 5 × S 5 superstring, T-duality and modified type-II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  48. L. Wulff and A.A. Tseytlin, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].

  49. B. Hoare and A.A. Tseytlin, Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS 5 σ-model, J. Phys. A 49 (2016) 494001 [arXiv:1609.02550] [INSPIRE].

    MATH  Google Scholar 

  50. M. Hong, Y. Kim and E. Ó Colgáin, On non-Abelian T-duality for non-semisimple groups, arXiv:1801.09567 [INSPIRE].

  51. L. Wulff, Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly, Phys. Lett. B 781 (2018) 417 [arXiv:1803.07391] [INSPIRE].

    ADS  Article  Google Scholar 

  52. Y. Sakatani, S. Uehara and K. Yoshida, Generalized gravity from modified DFT, JHEP 04 (2017) 123 [arXiv:1611.05856] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. A. Baguet, M. Magro and H. Samtleben, Generalized IIB supergravity from exceptional field theory, JHEP 03 (2017) 100 [arXiv:1612.07210] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  54. J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Weyl invariance for generalized supergravity backgrounds from the doubled formalism, PTEP 2017 (2017) 053B07 [arXiv:1703.09213] [INSPIRE].

  55. J.J. Fernandez-Melgarejo, J.-i. Sakamoto, Y. Sakatani and K. Yoshida, T-folds from Yang-Baxter deformations, JHEP 12 (2017) 108 [arXiv:1710.06849] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. J.-I. Sakamoto and Y. Sakatani, Local β-deformations and Yang-Baxter σ-model, JHEP 06 (2018) 147 [arXiv:1803.05903] [INSPIRE].

    ADS  Article  Google Scholar 

  57. B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].

    ADS  Article  Google Scholar 

  58. V.A. Fateev, E. Onofri and A.B. Zamolodchikov, The sausage model (integrable deformations of O(3) σ-model), Nucl. Phys. B 406 (1993) 521 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  59. R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. I. Bakhmatov, Ö. Kelekci, E. Ó Colgáin and M.M. Sheikh-Jabbari, Classical Yang-Baxter equation from supergravity, Phys. Rev. D 98 (2018) 021901 [arXiv:1710.06784] [INSPIRE].

  61. T. Araujo, E. O Colgáin and H. Yavartanoo, Embedding the modified CYBE in Supergravity, arXiv:1806.02602 [INSPIRE].

  62. R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS 2 × S 2 supercoset, Nucl. Phys. B 905 (2016) 264 [arXiv:1601.08192] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  63. B. Hoare and A.A. Tseytlin, Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring, JHEP 10 (2015) 060 [arXiv:1508.01150] [INSPIRE].

    ADS  Article  Google Scholar 

  64. K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].

    ADS  Article  Google Scholar 

  65. O. Lunin, R. Roiban and A.A. Tseytlin, Supergravity backgrounds for deformations of AdS n × S n supercoset string models, Nucl. Phys. B 891 (2015) 106 [arXiv:1411.1066] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  66. G. Arutyunov, R. Borsato and S. Frolov, Puzzles of η-deformed AdS 5 × S 5, JHEP 12 (2015) 049 [arXiv:1507.04239] [INSPIRE].

    ADS  Google Scholar 

  67. C.M. Hull, Timelike T-duality, de Sitter space, large N gauge theories and topological field theory, JHEP 07 (1998) 021 [hep-th/9806146] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: squashing coset CFTs and AdS 5 × S 5, JHEP 07 (2015) 019 [arXiv:1504.02781] [INSPIRE].

    ADS  Article  Google Scholar 

  69. P. Ševera, On integrability of 2-dimensional σ-models of Poisson-Lie type, JHEP 11 (2017) 015 [arXiv:1709.02213] [INSPIRE].

    MathSciNet  Google Scholar 

  70. L. Wulff, Superisometries and integrability of superstrings, JHEP 05 (2014) 115 [arXiv:1402.3122] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  71. L. Wulff, On integrability of strings on symmetric spaces, JHEP 09 (2015) 115 [arXiv:1505.03525] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  72. N. Beisert and P. Koroteev, Quantum deformations of the one-dimensional Hubbard model, J. Phys. A 41 (2008) 255204 [arXiv:0802.0777] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  73. B. Hoare, T.J. Hollowood and J.L. Miramontes, q-deformation of the AdS 5 × S 5 Superstring S-matrix and its Relativistic Limit, JHEP 03 (2012) 015 [arXiv:1112.4485] [INSPIRE].

  74. B. Hoare, A. Pittelli and A. Torrielli, Integrable S-matrices, massive and massless modes and the AdS 2 × S 2 superstring, JHEP 11 (2014) 051 [arXiv:1407.0303] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  75. G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The quantum deformed mirror TBA I, JHEP 10 (2012) 090 [arXiv:1208.3478] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  76. G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The quantum deformed mirror TBA II, JHEP 02 (2013) 012 [arXiv:1210.8185] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  77. G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS 5 × S 5)η superstring, Theor. Math. Phys. 182 (2015) 23 [Teor. Mat. Fiz. 182 (2014) 28] [arXiv:1403.6104] [INSPIRE].

  78. R. Klabbers and S.J. van Tongeren, Quantum spectral curve for the eta-deformed AdS 5 × S 5 superstring, Nucl. Phys. B 925 (2017) 252 [arXiv:1708.02894] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  79. C. Appadu, T.J. Hollowood, J.L. Miramontes, D. Price and D.M. Schmidtt, Giant magnons of string theory in the λ background, JHEP 07 (2017) 098 [arXiv:1704.05437] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  80. G. Arutyunov and S.J. van Tongeren, AdS 5 × S 5 mirror model as a string σ-model, Phys. Rev. Lett. 113 (2014) 261605 [arXiv:1406.2304] [INSPIRE].

    ADS  Article  Google Scholar 

  81. A. Pachoł and S.J. van Tongeren, Quantum deformations of the flat space superstring, Phys. Rev. D 93 (2016) 026008 [arXiv:1510.02389] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  82. A. Yu. Alekseev, C. Klimčík and A.A. Tseytlin, Quantum Poisson-Lie T-duality and WZNW model, Nucl. Phys. B 458 (1996) 430 [hep-th/9509123] [INSPIRE].

  83. G. Valent, C. Klimčík and R. Squellari, One loop renormalizability of the Poisson-Lie σ-models, Phys. Lett. B 678 (2009) 143 [arXiv:0902.1459] [INSPIRE].

  84. K. Sfetsos and K. Siampos, Quantum equivalence in Poisson-Lie T-duality, JHEP 06 (2009) 082 [arXiv:0904.4248] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  85. S.D. Avramis, J.-P. Derendinger and N. Prezas, Conformal chiral boson models on twisted doubled tori and non-geometric string vacua, Nucl. Phys. B 827 (2010) 281 [arXiv:0910.0431] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  86. K. Sfetsos, K. Siampos and D.C. Thompson, Renormalization of Lorentz non-invariant actions and manifest T-duality, Nucl. Phys. B 827 (2010) 545 [arXiv:0910.1345] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  87. L. Hlavatý, J. Navrátil and L. Šnobl, On renormalization of Poisson-Lie T-plural σ-models, Acta Polytech. 53 (2013) 433 [arXiv:1212.5936] [INSPIRE].

    Article  Google Scholar 

  88. F. Hassler, Poisson-Lie T-duality in double field theory, arXiv:1707.08624 [INSPIRE].

Download references

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.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich, Wolfgang-Pauli-Strasse 27, 8093, Zürich, Switzerland

    Ben Hoare & Fiona K. Seibold

Authors
  1. Ben Hoare
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fiona K. Seibold
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fiona K. Seibold.

Additional information

ArXiv ePrint: 1807.04608

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hoare, B., Seibold, F.K. Poisson-Lie duals of the η-deformed AdS2 × S2 × T6 superstring. J. High Energ. Phys. 2018, 107 (2018). https://doi.org/10.1007/JHEP08(2018)107

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2018)107

Keywords

  • String Duality
  • Sigma Models
  • Integrable Field Theories