Advertisement

On the massless modes of the AdS3/CFT2 integrable systems

  • Olof Ohlsson Sax
  • Bogdan StefanskiJr
  • Alessandro Torrielli
Article

Abstract

We make a proposal for incorporating massless modes into the spin-chain of the AdS 3 /CF T 2 integrable system. We do this by considering the α → 0 limit of the alternating \( \mathfrak{d}{{\left( {2,1;\alpha } \right)}^2} \) spin-chain constructed in arXiv:1106.2558. In the process we encounter integrable spin-chains with non-irreducible representations at some of thei r sites. We investigate their properties and construct their R-matrices in terms of Yangians.

Keywords

AdS-CFT Correspondence Lattice Integrable Models 

References

  1. [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].MathSciNetADSzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  3. [3]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    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].MathSciNetADSGoogle Scholar
  8. [8]
    K. Zoubos, Review of AdS/CFT integrability, chapter IV.2: deformations, orbifolds and open boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [9]
    D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, arXiv:0904.4466 [INSPIRE].
  10. [10]
    R. Reid-Edwards and B. Stefanski Jr., On type IIA geometries dual to N = 2 SCFTs, Nucl. Phys. B 849 (2011) 549 [arXiv:1011.0216] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    E. O Colgain and B. Stefanski Jr., A search for AdS 5 × S 2 IIB supergravity solutions dual to N =2 SCFTs, JHEP 10(2011) 061 [arXiv:1107.5763] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    O. Aharony, L. Berdichevsky and M. Berkooz, 4d N = 2 superconformal linear quivers with type IIA duals, JHEP 08 (2012) 131 [arXiv:1206.5916] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A. Gadde, E. Pomoni and L. Rastelli, The Veneziano limit of N = 2 superconformal QCD: towards the string dual of N = 2 SU(N c) SYM with N f = 2N c, arXiv:0912.4918 [INSPIRE].
  14. [14]
    A. Gadde, E. Pomoni and L. Rastelli, Spin chains in \( \mathcal{N}=2 \) superconformal theories: from the Z 2 quiver to superconformal QCD, JHEP 06 (2012) 107 [arXiv:1006.0015] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Gadde and L. Rastelli, Twisted magnons, JHEP 04 (2012) 053 [arXiv:1012.2097] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    P. Liendo, E. Pomoni and L. Rastelli, The complete one-loop dilation operator of N = 2 superconformal QCD, JHEP 07 (2012) 003 [arXiv:1105.3972] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    P. Liendo and L. Rastelli, The complete one-loop spin chain of N = 1 SQCD, JHEP 10 (2012) 117 [arXiv:1111.5290] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    J. Minahan and K. Zarembo, The Bethe ansatz for superconformal Chern-Simons, JHEP 09 (2008) 040 [arXiv:0806.3951] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    D. Gaiotto, S. Giombi and X. Yin, Spin chains in N = 6 superconformal Chern-Simons-matter theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    N. Gromov and P. Vieira, The all loop AdS 4 /CFT 3 Bethe ansatz, JHEP 01 (2009) 016 [arXiv:0807.0777] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    G. Arutyunov and S. Frolov, Superstrings on AdS 4 × CP 3 as a coset σ-model, JHEP 09 (2008) 129 [arXiv:0806.4940] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    B. Stefanski Jr., Green-Schwarz action for type IIA strings on AdS 4 × CP 3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    J. Gomis, D. Sorokin and L. Wulff, The complete AdS 4 × CP 3 superspace for the type IIA superstring and D-branes, JHEP 03 (2009) 015 [arXiv:0811.1566] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    T. Klose, Review of AdS/CFT integrability, chapter IV.3: N = 6 Chern-Simons and strings on AdS 4 × CP 3, Lett. Math. Phys. 99 (2012) 401 [arXiv:1012.3999] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. [27]
    J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS 3 /CF T 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    J.R. David and B. Sahoo, Giant magnons in the D1-D5 system, JHEP 07 (2008) 033 [arXiv:0804.3267] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    J.R. David and B. Sahoo, S-matrix for magnons in the D1-D5 system, JHEP 10 (2010) 112 [arXiv:1005.0501] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    O. Ohlsson Sax and B. Stefanski Jr., Integrability, spin-chains and the AdS 3 /CFT 2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].MathSciNetGoogle Scholar
  33. [33]
    N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A. Cagnazzo and K. Zarembo, B-field in AdS 3 /CF T 2 correspondence and integrability, JHEP 11 (2012) 133 [arXiv:1209.4049] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
  38. [38]
    J.P. Gauntlett, R.C. Myers and P. Townsend, Supersymmetry of rotating branes, Phys. Rev. D 59 (1999) 025001 [hep-th/9809065] [INSPIRE].MathSciNetADSGoogle Scholar
  39. [39]
    A. Pakman, L. Rastelli and S.S. Razamat, A spin chain for the symmetric product CFT 2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    S. Gukov, E. Martinec, G.W. Moore and A. Strominger, The search for a holographic dual to AdS 3 × S 3 × S 3 × S 1, Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  41. [41]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N =4 super Yang-Mills,JHEP 04(2002) 013[hep-th/0202021][INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    J. Russo and A.A. Tseytlin, On solvable models of type 2B superstring in NS-NS and RR plane wave backgrounds, JHEP 04 (2002) 021 [hep-th/0202179] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    H. Lü and J.F. Vazquez-Poritz, Penrose limits of nonstandard brane intersections, Class. Quant. Grav. 19 (2002) 4059 [hep-th/0204001] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  44. [44]
    Y. Hikida and Y. Sugawara, Superstrings on PP wave backgrounds and symmetric orbifolds, JHEP 06 (2002) 037 [hep-th/0205200] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    J. Gomis, L. Motl and A. Strominger, PP wave/CFT 2 duality, JHEP 11 (2002) 016 [hep-th/0206166] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    E. Gava and K. Narain, Proving the PP wave/CFT 2 duality, JHEP 12 (2002) 023 [hep-th/0208081] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    L. Sommovigo, Penrose limit of AdS 3 × S 3 × S 3 × S 1 and its associated σ-model, JHEP 07 (2003) 035 [hep-th/0305151] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    A. Perelomov, Generalized coherent states and their applications, Springer, Germany Berlin (1986).zbMATHCrossRefGoogle Scholar
  49. [49]
    W.-M. Zhang, D.H. Feng and R. Gilmore, Coherent states: theory and some applications, Rev. Mod. Phys. 62 (1990) 867 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58 (1940) 1098 [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  51. [51]
    J.M. Maldacena and H. Ooguri, Strings in AdS 3 and \( \mathrm{SL}\left( {2,\mathbb{R}} \right) \) WZW model 1: the spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. [52]
    G. Korchemsky, Review of AdS/CFT integrability, chapter IV.4: integrability in QCD and N <4 SYM,Lett. Math. Phys. 99(2012) 425[arXiv:1012.4000][INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. [53]
    V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge U.K. (1994).zbMATHGoogle Scholar
  54. [54]
    A.I. Molev, Yangians and their applications, in Handbook of algebra, volume 3, Elsevier, The Netherlands (2003), pg. 907 [math.QA/0211288].
  55. [55]
    A. Torrielli, Yangians, S-matrices and AdS/CFT, J. Phys. A 44 (2011) 263001 [arXiv:1104.2474] [INSPIRE].ADSGoogle Scholar
  56. [56]
    N. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].MathSciNetADSGoogle Scholar
  57. [57]
    V. Drinfeld, A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988)212 [INSPIRE].MathSciNetGoogle Scholar
  58. [58]
    S. Khoroshkin and V. Tolstoi, Yangian double and rational R matrix, Lett. Math. Phys. 36 (1994)373 [hep-th/9406194] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    G. Arutyunov, M. de Leeuw and A. Torrielli, Universal blocks of the AdS/CFT scattering matrix, JHEP 05 (2009) 086 [arXiv:0903.1833] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    S.E. Derkachov, D. Karakhanian and R. Kirschner, Heisenberg spin chains based on sl(2|1) symmetry, Nucl. Phys. B 583 (2000) 691 [nlin/0003029] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    N. Beisert, The complete one loop dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory, J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [INSPIRE].MathSciNetGoogle Scholar
  63. [63]
    G. Korchemsky, Bethe ansatz for QCD pomeron, Nucl. Phys. B 443 (1995) 255 [hep-ph/9501232] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M.R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593 [hep-th/0111260] [INSPIRE].MathSciNetADSGoogle Scholar
  65. [65]
    D. Fioravanti and M. Rossi, A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations, J. Phys. A 35 (2002) 3647 [hep-th/0104002] [INSPIRE].MathSciNetADSGoogle Scholar
  66. [66]
    D. Fioravanti and M. Rossi, From the braided to the usual Yang-Baxter relation, J. Phys. A 34 (2001) L567 [hep-th/0107050] [INSPIRE].MathSciNetADSGoogle Scholar
  67. [67]
    V. Tarasov, L. Takhtajan and L. Faddeev, Local hamiltonians for integrable quantum models on a lattice, Theor. Math. Phys. 57 (1983) 1059 [Teor. Mat. Fiz. 57 (1983) 163] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  68. [68]
    L. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187 [INSPIRE].
  69. [69]
    W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  70. [70]
    C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  71. [71]
    G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  72. [72]
    A. Rej and F. Spill, The Yangian of \( \mathfrak{s}\mathfrak{l}\left( {\left. n \right|m} \right) \) and the universal R-matrix, JHEP 05 (2011) 012 [arXiv:1008.0872] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  73. [73]
    L. Faddeev and G. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Olof Ohlsson Sax
    • 1
  • Bogdan StefanskiJr
    • 2
  • Alessandro Torrielli
    • 3
  1. 1.Institute for Theoretical PhysicsUtrechtThe Netherlands
  2. 2.Centre for Mathematical ScienceCity University of LondonLondonU.K
  3. 3.Department of MathematicsUniversity of SurreyGuildfordU.K

Personalised recommendations