Skip to main content

Poisson-Lie duals of the η deformed symmetric space sigma model

A preprint version of the article is available at arXiv.

Abstract

Poisson-Lie dualising the η deformation of the G/H symmetric space sigma model with respect to the simple Lie group G is conjectured to give an analytic continuation of the associated λ deformed model. In this paper we investigate when the η deformed model can be dualised with respect to a subgroup G0 of G. Starting from the first-order action on the complexified group and integrating out the degrees of freedom associated to different subalgebras, we find it is possible to dualise when G0 is associated to a sub-Dynkin diagram. Additional U1 factors built from the remaining Cartan generators can also be included. The resulting construction unifies both the Poisson-Lie dual with respect to G and the complete abelian dual of the η deformation in a single framework, with the integrated algebras unimodular in both cases. We speculate that extending these results to the path integral formalism may provide an explanation for why the η deformed AdS5 × S5 superstring is not one-loop Weyl invariant, that is the couplings do not solve the equations of type IIB supergravity, yet its complete abelian dual and the λ deformed model are.

References

  1. 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].

  2. 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].

  3. 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].

  4. M.B. Green and J.H. Schwarz, Covariant Description of Superstrings, Phys. Lett. B 136 (1984) 367 [INSPIRE].

    ADS  Article  Google Scholar 

  5. 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 

  6. E. Witten, Twistor-Like Transform in Ten Dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. M.T. Grisaru, P.S. Howe, L. Mezincescu, B. Nilsson and P.K. Townsend, N = 2 Superstrings in a Supergravity Background, Phys. Lett. B 162 (1985) 116 [INSPIRE].

  8. R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  9. 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].

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

    ADS  MathSciNet  Article  Google Scholar 

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

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

  13. 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 

  14. 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].

  15. 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 

  16. 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].

  17. 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].

  18. 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].

  19. 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].

  20. B. Hoare and S.J. van Tongeren, Non-split and split deformations of AdS 5, J. Phys. A 49 (2016) 484003 [arXiv:1605.03552] [INSPIRE].

  21. 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 

  22. T. Araujo, E. Ó. Colgáin, J. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, I in generalized supergravity, arXiv:1708.03163 [INSPIRE].

  23. 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 

  24. 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 

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

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

  27. 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 

  28. 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].

  29. Y. Chervonyi and O. Lunin, Generalized λ-deformations of AdS p × S p, Nucl. Phys. B 913 (2016) 912 [arXiv:1608.06641] [INSPIRE].

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

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

  33. 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].

  34. 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 

  35. 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].

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

  37. 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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

  39. A.A. Tseytlin, Duality Symmetric Formulation of String World Sheet Dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

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

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

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. K. Zarembo, Strings on Semisymmetric Superspaces, JHEP 05 (2010) 002 [arXiv:1003.0465] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  47. L. Wulff, Integrability of the superstring in AdS 3 × S 2 × S 2 × T 3, J. Phys. A 50 (2017) 23LT01 [arXiv:1702.08788] [INSPIRE].

  48. L. Wulff, All symmetric AdS n>2 solutions of type-II supergravity, arXiv:1706.02118 [INSPIRE].

  49. L. Wulff, Condition on Ramond-Ramond fluxes for factorization of worldsheet scattering in anti-de Sitter space, arXiv:1708.09673 [INSPIRE].

  50. E. Tyurin and R. von Unge, Poisson-lie T duality: The Path integral derivation, Phys. Lett. B 382 (1996) 233 [hep-th/9512025] [INSPIRE].

  51. 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 

  52. 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 [arXiv:1403.6104] [INSPIRE].

  53. 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].

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

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

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

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

    MathSciNet  Article  Google Scholar 

  58. A.Y. 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].

  59. 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].

  60. 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 

  61. 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 

  62. 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].

  63. 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].

  64. F. Hassler, Poisson-Lie T-duality in Double Field Theory, arXiv:1707.08624 [INSPIRE].

  65. B. Jurčo and J. Vysoký, Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle, arXiv:1708.04079 [INSPIRE].

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

  67. 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 

  68. P. Ševera, On integrability of 2-dimensional σ-models of Poisson-Lie type, arXiv:1709.02213 [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

Authors

Corresponding author

Correspondence to Ben Hoare.

Additional information

ArXiv ePrint: 1709.01448

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 symmetric space sigma model. J. High Energ. Phys. 2017, 14 (2017). https://doi.org/10.1007/JHEP11(2017)014

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2017)014

Keywords

  • Sigma Models
  • Field Theories in Lower Dimensions
  • Integrable Field Theories