Download PDF

Journal of High Energy Physics

, 2015:187 | Cite as

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

  • Nikolay Gromov
  • Vladimir Kazakov
  • Sébastien Leurent
  • Dmytro Volin
Open Access
Regular Article - Theoretical Physics
First Online:
Received:
Revised:
Accepted:

Abstract

We give a derivation of quantum spectral curve (QSC) — a finite set of Riemann-Hilbert equations for exact spectrum of planar \( \mathcal{N}=4 \) SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system — a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

Keywords

AdS-CFT Correspondence Integrable Field Theories 
Download to read the full article text

Notes

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.

References

  1. [1]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  2. [2]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N =4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory: konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].ADSMATHGoogle Scholar
  5. [5]
    F. Levkovich-Maslyuk, Numerical results for the exact spectrum of planar AdS4/CFT3, JHEP 05 (2012) 142 [arXiv:1110.5869] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  7. [7]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].ADSMATHGoogle Scholar
  8. [8]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5 /CF T 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  10. [10]
    N. Gromov, V. Kazakov and Z. Tsuboi, PSU(2, 2|4) character of quasiclassical AdS/CFT, JHEP 07 (2010) 097 [arXiv:1002.3981] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  11. [11]
    N. Gromov, V. Kazakov, S. Leurent and Z. Tsuboi, Wronskian solution for AdS/CFT Y-system, JHEP 01 (2011) 155 [arXiv:1010.2720] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, JHEP 07 (2012) 023 [arXiv:1110.0562] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett. 109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar N = 4 SYM, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  15. [15]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N} \) = super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    D. Volin, Quantum spectral curve for AdS5/CF T 4 spectral problem, talk given at Integrability in gauge and string theory, August 19-23, Utrecht, The Netherlands (2013).Google Scholar
  17. [17]
    D. Volin and C. Marboe, Quantum spectral curve as a tool for a perturbative quantum field theory, Nucl. Phys. B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE].ADSMATHGoogle Scholar
  18. [18]
    N. Gromov, F. Levkovich-Maslyuk, G. Sizov and S. Valatka, Quantum spectral curve at work: from small spin to strong coupling in \( \mathcal{N}=4 \) SYM, JHEP 07 (2014) 156 [arXiv:1402.0871] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum spectral curve of the \( \mathcal{N}=6 \) supersymmetric Chern-Simons theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    N. Gromov and G. Sizov, Exact slope and interpolating functions in N = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  22. [22]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
  23. [23]
    N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe ansatze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  24. [24]
    J. Balog and A. Hegedus, AdS 5 × S 5 mirror TBA equations from Y-system and discontinuity relations, JHEP 08 (2011) 095 [arXiv:1104.4054] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  25. [25]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  27. [27]
    Z. Bajnok et al., The spectrum of tachyons in AdS/CFT, JHEP 03 (2014) 055 [arXiv:1312.3900] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    P. Kulish and N.Y. Reshetikhin, Generalized Heisenberg ferromagnet and the Gross-Neveu model, J. Exp. Theor. Phys. 53 (1981) 108.MATHGoogle Scholar
  29. [29]
    P. Kulish, Integrable graded magnets, J. Sov. Math. 35 (1986) 2648.CrossRefMATHGoogle Scholar
  30. [30]
    B. Sutherland, Low-lying eigenstates of the one-dimensional Heisenberg ferromagnet for any magnetization and momentum, Phys. Rev. Lett. 74 (1995) 816 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys. 188 (1997) 267 [hep-th/9604080] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  32. [32]
    V. Kazakov, A.S. Sorin and A. Zabrodin, Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics, Nucl. Phys. B 790 (2008) 345 [hep-th/0703147] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  33. [33]
    Z. Tsuboi, Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r + 1|s + 1), J. Phys. A 30 (1997) 7975 [arXiv:0911.5386] [INSPIRE].ADSMATHGoogle Scholar
  34. [34]
    V. Kazakov, S. Leurent, and D. Volin, T-system on T-hook: grassmannian solution and twisted quantum spectral curve, in preparation.Google Scholar
  35. [35]
    Z. Tsuboi, Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M |N ), Nucl. Phys. B 870 (2013) 92 [arXiv:1109.5524] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  36. [36]
    Z. Tsuboi, Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r + 1|s + 1), Physica A 252 (1998) 565 [arXiv:0911.5387] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP 04 (2008) 046 [arXiv:0709.3487] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  38. [38]
    G.P. Pronko and Yu.G. Stroganov, Bethe equations ‘on the wrong side of equator’, J. Phys. A 32 (1999) 2333 [hep-th/9808153] [INSPIRE].ADSMATHGoogle Scholar
  39. [39]
    Z. Tsuboi, Solutions of the T-system and Baxter equations for supersymmetric spin chains, Nucl. Phys. B 826 (2010) 399 [arXiv:0906.2039] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  40. [40]
    N. Gromov and V. Kazakov, Review of AdS/CFT integrability, chapter III.7: Hirota dynamics for quantum integrability, Lett. Math. Phys. 99 (2012) 321 [arXiv:1012.3996] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  41. [41]
    N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    D. Volin, Minimal solution of the AdS/CFT crossing equation, J. Phys. A 42 (2009) 372001 [arXiv:0904.4929] [INSPIRE].MATHGoogle Scholar
  43. [43]
    R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].ADSGoogle Scholar
  44. [44]
    P. Vieira and D. Volin, Review of AdS/CFT integrability. Chapter III.3: the dressing factor, Lett. Math. Phys. 99 (2012) 231 [arXiv:1012.3992] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  45. [45]
    M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].CrossRefMATHGoogle Scholar
  47. [47]
    A. Santambrogio and D. Zanon, Exact anomalous dimensions of N = 4 Yang-Mills operators with large R charge, Phys. Lett. B 545 (2002) 425 [hep-th/0206079] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  48. [48]
    N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    N. Gromov and A. Sever, Analytic solution of Bremsstrahlung TBA, JHEP 11 (2012) 075 [arXiv:1207.5489] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic solution of Bremsstrahlung TBA II: turning on the sphere angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  51. [51]
    D. Bombardelli, D. Fioravanti and R. Tateo, TBA and Y-system for planar AdS 4 /CF T 3, Nucl. Phys. B 834 (2010) 543 [arXiv:0912.4715] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  52. [52]
    N. Gromov and F. Levkovich-Maslyuk, Y-system, TBA and quasi-classical strings in AdS 4 × CP 3, JHEP 06 (2010) 088 [arXiv:0912.4911] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  53. [53]
    A. Cavaglia, D. Fioravanti and R. Tateo, Discontinuity relations for the AdS 4 /CF T 3 correspondence, Nucl. Phys. B 877 (2013) 852 [arXiv:1307.7587] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  54. [54]
    N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].ADSMATHGoogle Scholar
  55. [55]
    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].CrossRefMATHGoogle Scholar
  56. [56]
    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].ADSCrossRefMATHGoogle Scholar
  57. [57]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Towards the all-loop worldsheet S matrix for AdS 3 × S 3 × T 4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  58. [58]
    S. Negro and F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B 875 (2013) 166 [arXiv:1306.1476] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  59. [59]
    N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    V. Kazakov and S. Leurent, Finite size spectrum of SU(N ) principal chiral field from discrete Hirota dynamics, arXiv:1007.1770 [INSPIRE].
  61. [61]
    S.E. Derkachov, Baxter’s Q-operator for the homogeneous XXX spin chain, J. Phys. A 32 (1999) 5299 [solv-int/9902015] [INSPIRE].
  62. [62]
    S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Separation of variables for the quantum \( \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) \) spin chain, JHEP 07 (2003) 047 [hep-th/0210216] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baxter Q-operator for graded SL(2|1) spin chain, J. Stat. Mech. (2007) P01005 [hep-th/0610332] [INSPIRE].
  64. [64]
    R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Baxter operators and hamiltonians for ‘nearly all’ integrable closed \( \mathfrak{g}\mathfrak{l}(n) \) spin chains, Nucl. Phys. B 874 (2013) 620 [arXiv:1112.3600] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  65. [65]
    R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Oscillator construction of SU(n|m) Q-operators, Nucl. Phys. B 850 (2011) 175 [arXiv:1012.6021] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  66. [66]
    V. Kazakov and P. Vieira, From characters to quantum (super)spin chains via fusion, JHEP 10 (2008) 050 [arXiv:0711.2470] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  67. [67]
    V. Kazakov, S. Leurent and Z. Tsuboi, Baxter’s Q-operators and operatorial Backlund flow for quantum (super)-spin chains, Commun. Math. Phys. 311 (2012) 787 [arXiv:1010.4022] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  68. [68]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. 2. Q operator and DDV equation, Commun. Math. Phys. 190 (1997) 247 [hep-th/9604044] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  69. [69]
    S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett. 109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  71. [71]
    N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  72. [72]
    N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  73. [73]
    M. Gunaydin and D. Volin, in preparation.Google Scholar
  74. [74]
    D. Volin, String hypothesis for gl(n|m) spin chains: a particle/hole democracy, Lett. Math. Phys. 102 (2012) 1 [arXiv:1012.3454] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  75. [75]
    H.P. Jakobsen, The full set of unitarizable highest weight modules of basic classical lie superalgebras, Memoirs of American Mathematical Society volume 532, American Mathematical Society, U.S.A. (1994).Google Scholar
  76. [76]
    H. Furutsu and K. Nishiyama, Classification of irreducible super-unitary representations of SU(p, q|n), Commun. Math. Phys. 141 (1991) 475.ADSCrossRefMATHGoogle Scholar
  77. [77]
    A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math. J 9 (1996) 239.Google Scholar
  78. [78]
    A. Molev, Factorial supersymmetric schur functions and super capelli identities, Amer. Math. Soc. Transl. Ser 2 (1997) 109.MATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Nikolay Gromov
    • 1
    • 2
  • Vladimir Kazakov
    • 3
    • 4
    • 5
  • Sébastien Leurent
    • 6
  • Dmytro Volin
    • 7
    • 8
  1. 1.Mathematics DepartmentKing’s College LondonLondonUnited Kingdom
  2. 2.St.Petersburg INPGatchinaRussia
  3. 3.LPT, École Normale SuperieureParisFrance
  4. 4.Université Paris-VIParisFrance
  5. 5.School of Natural SciencesInstitute for Advanced StudyPrincetonUnited States
  6. 6.Institut de Mathématiques de Bourgogne, UMR 5584 du CNRSUniversité de BourgogneDIJONFrance
  7. 7.Nordita KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
  8. 8.School of MathematicsTrinity College DublinDublin 2Ireland

Personalised recommendations