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Integrability of classical strings dual for noncommutative gauge theories

A preprint version of the article is available at arXiv.

Abstract

We derive the gravity duals of noncommutative gauge theories from the Yang-Baxter sigma model description of the AdS5 × S5 superstring with classical r-matrices. The corresponding classical r-matrices are 1) solutions of the classical Yang-Baxter equation (CYBE), 2) skew-symmetric, 3) nilpotent and 4) abelian. Hence these should be called abelian Jordanian deformations. As a result, the gravity duals are shown to be integrable deformations of AdS5 × S5. Then, abelian twists of AdS5 are also investigated. These results provide a support for the gravity/CYBE correspondence proposed in arXiv:1404.1838.

References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].

  2. N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  3. 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  MathSciNet  Google Scholar 

  4. I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  5. R. Roiban and W. Siegel, Superstrings on AdS 5 × S 5 supertwistor space, JHEP 11 (2000) 024 [hep-th/0010104] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  6. M. Hatsuda and K. Yoshida, Classical integrability and super Yangian of superstring on AdS 5 × S 5, Adv. Theor. Math. Phys. 9 (2005) 703 [hep-th/0407044] [INSPIRE].

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Hatsuda and K. Yoshida, Super Yangian of superstring on AdS 5 × S 5 revisited, Adv. Theor. Math. Phys. 15 (2011) 1485 [arXiv:1107.4673] [INSPIRE].

    Article  MATH  MathSciNet  Google Scholar 

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

    ADS  Article  MathSciNet  Google Scholar 

  9. L. Wulff, Superisometries and integrability of superstrings, arXiv:1402.3122 [INSPIRE].

  10. S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset construction for duals of non-relativistic CFTs, JHEP 05 (2009) 038 [arXiv:0903.4245] [INSPIRE].

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  Article  MathSciNet  Google Scholar 

  13. C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, arXiv:1402.2105 [INSPIRE].

  14. R. Squellari, Yang-Baxter σ model: quantum aspects, Nucl. Phys. B 881 (2014) 502 [arXiv:1401.3197] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

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

    ADS  Article  MathSciNet  Google Scholar 

  16. I. Kawaguchi and K. Yoshida, Hybrid classical integrability in squashed σ-models, Phys. Lett. B 705 (2011) 251 [arXiv:1107.3662] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  17. I. Kawaguchi and K. Yoshida, Hybrid classical integrable structure of squashed σ-models: a short summary, J. Phys. Conf. Ser. 343 (2012) 012055 [arXiv:1110.6748] [INSPIRE].

    ADS  Article  Google Scholar 

  18. I. Kawaguchi, T. Matsumoto and K. Yoshida, The classical origin of quantum affine algebra in squashed σ-models, JHEP 04 (2012) 115 [arXiv:1201.3058] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  19. I. Kawaguchi, T. Matsumoto and K. Yoshida, On the classical equivalence of monodromy matrices in squashed σ-model, JHEP 06 (2012) 082 [arXiv:1203.3400] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  20. 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 

  21. V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].

    Google Scholar 

  22. V.G. Drinfel’d, Quantum groups, J. Sov. Math. 41 (1988) 898 [Zap. Nauchn. Semin. 155 (1986) 18] [INSPIRE].

  23. M. Jimbo, A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63 [INSPIRE].

    ADS  Article  MATH  MathSciNet  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. B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, arXiv:1403.5517 [INSPIRE].

  26. G. Arutyunov, M. de Leeuw and S.J. van Tongeren, On the exact spectrum and mirror duality of the (AdS 5 × S 5) η superstring, arXiv:1403.6104 [INSPIRE].

  27. I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 × S 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].

    ADS  Article  Google Scholar 

  28. N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990) 331 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  29. A. Stolin and P.P. Kulish, New rational solutions of Yang-Baxter equation and deformed Yangians, Czech. J. Phys. 47 (1997) 123 [q-alg/9608011].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  30. P.P. Kulish, V.D. Lyakhovsky and A.I. Mudrov, Extended Jordanian twists for Lie algebras, J. Math. Phys. 40 (1999) 4569 [math.QA/9806014] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  31. I. Kawaguchi, T. Matsumoto and K. Yoshida, A Jordanian deformation of AdS space in type IIB supergravity, arXiv:1402.6147 [INSPIRE].

  32. V.E. Hubeny, M. Rangamani and S.F. Ross, Causal structures and holography, JHEP 07 (2005) 037 [hep-th/0504034] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  33. O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  34. S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].

    ADS  Article  Google Scholar 

  35. T. Matsumoto and K. Yoshida, Lunin-Maldacena backgrounds from the classical Yang-Baxter equationtowards the gravity/CYBE correspondence, arXiv:1404.1838 [INSPIRE].

  36. A. Hashimoto and N. Itzhaki, Noncommutative Yang-Mills and the AdS/CFT correspondence, Phys. Lett. B 465 (1999) 142 [hep-th/9907166] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  37. J.M. Maldacena and J.G. Russo, Large-N limit of noncommutative gauge theories, JHEP 09 (1999) 025 [hep-th/9908134] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  38. D. Dhokarh, S.S. Haque and A. Hashimoto, Melvin twists of global AdS 5 × S 5 and their non-commutative field theory dual, JHEP 08 (2008) 084 [arXiv:0801.3812] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  39. T. McLoughlin and I. Swanson, Integrable twists in AdS/CFT, JHEP 08 (2006) 084 [hep-th/0605018] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  40. I. Kawaguchi and K. Yoshida, Classical integrability of Schrödinger σ-models and q-deformed Poincaré symmetry, JHEP 11 (2011) 094 [arXiv:1109.0872] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  41. I. Kawaguchi and K. Yoshida, Exotic symmetry and monodromy equivalence in Schrödinger σ-models, JHEP 02 (2013) 024 [arXiv:1209.4147] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  42. I. Kawaguchi, T. Matsumoto and K. Yoshida, Schrödinger σ-models and Jordanian twists, JHEP 08 (2013) 013 [arXiv:1305.6556] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  43. T. Kameyama and K. Yoshida, String theories on warped AdS backgrounds and integrable deformations of spin chains, JHEP 05 (2013) 146 [arXiv:1304.1286] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  44. M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157 [q-alg/9709040] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  45. G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring: I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

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Correspondence to Kentaroh Yoshida.

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ArXiv ePrint: 1404.3657

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Matsumoto, T., Yoshida, K. Integrability of classical strings dual for noncommutative gauge theories. J. High Energ. Phys. 2014, 163 (2014). https://doi.org/10.1007/JHEP06(2014)163

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  • DOI: https://doi.org/10.1007/JHEP06(2014)163

Keywords

  • AdS-CFT Correspondence
  • Sigma Models
  • Integrable Field Theories
  • Non-Commutative Geometry