Journal of High Energy Physics

, 2019:44 | Cite as

Thermal correlation functions of KdV charges in 2D CFT

  • Alexander Maloney
  • Gim Seng Ng
  • Simon F. RossEmail author
  • Ioannis Tsiares
Open Access
Regular Article - Theoretical Physics


Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give an expression for an arbitrary correlation function which is determined up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m + 1, 2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level.


Conformal Field Theory Integrable Hierarchies 


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.


  1. [1]
    A. Maloney, S.G. Ng, S.F. Ross and I. Tsiares, Generalized Gibbs Ensemble and the Statistics of KdV Charges in 2D CFT, arXiv:1810.11054 [INSPIRE].
  2. [2]
    R. Sasaki and I. Yamanaka, Virasoro Algebra, Vertex Operators, Quantum Sine-Gordon and Solvable Quantum Field Theories, Adv. Stud. Pure Math. 16 (1988) 271 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T. Eguchi and S.-K. Yang, Deformations of Conformal Field Theories and Soliton Equations, Phys. Lett. B 224 (1989) 373 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum Quench in the Transverse Field Ising Chain, Phys. Rev. Lett. 106 (2011) 227203 [arXiv:1104.0154] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S. Sotiriadis and P. Calabrese, Validity of the GGE for quantum quenches from interacting to noninteracting models, J. Stat. Mech. 1407 (2014) P07024 [arXiv:1403.7431] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F.H.L. Essler and T. Prosen, Complete generalized gibbs ensembles in an interacting theory, Phys. Rev. Lett. 115 (2015) 157201.ADSCrossRefGoogle Scholar
  8. [8]
    L. Vidmar and M. Rigol, Generalized gibbs ensemble in integrable lattice models, J. Stat. Mech. 6 (2016) 064007 [arXiv:1604.03990].MathSciNetCrossRefGoogle Scholar
  9. [9]
    B. Pozsgay, E. Vernier and M.A. Werner, On generalized gibbs ensembles with an infinite set of conserved charges, J. Stat. Mech. 9 (2017) 093103 [arXiv:1703.09516].MathSciNetCrossRefGoogle Scholar
  10. [10]
    T. Langen et al., Experimental observation of a generalized gibbs ensemble, Science 348 (2015) 207.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    T. Kinoshita, T. Wenger and D.S. Weiss, A quantum newton’s cradle, Nature 440 (2006) 900.ADSCrossRefGoogle Scholar
  12. [12]
    J. de Boer and D. Engelhardt, Remarks on thermalization in 2D CFT, Phys. Rev. D 94 (2016) 126019 [arXiv:1604.05327] [INSPIRE].ADSGoogle Scholar
  13. [13]
    A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for General Relativity on AdS 3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Higher level eigenvalues of Q operators and Schroedinger equation, Adv. Theor. Math. Phys. 7 (2003) 711 [hep-th/0307108] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSCrossRefGoogle Scholar
  16. [16]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.ADSGoogle Scholar
  17. [17]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].ADSCrossRefGoogle Scholar
  18. [18]
    M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A 32 (1999) 1163.ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate Thermalization Hypothesis in Conformal Field Theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    A. Dymarsky, N. Lashkari and H. Liu, Subsystem ETH, Phys. Rev. E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].ADSGoogle Scholar
  23. [23]
    N. Lashkari, A. Dymarsky and H. Liu, Universality of Quantum Information in Chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP 06 (2018) 123 [arXiv:1712.03464] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. He, F.-L. Lin and J.-j. Zhang, Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis, JHEP 12 (2017) 073 [arXiv:1708.05090] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    P. Basu, D. Das, S. Datta and S. Pal, Thermality of eigenstates in conformal field theories, Phys. Rev. E 96 (2017) 022149 [arXiv:1705.03001] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].ADSGoogle Scholar
  28. [28]
    A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].ADSGoogle Scholar
  30. [30]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. Dymarsky and K. Pavlenko, Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP 01 (2019) 098 [arXiv:1810.11025] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B 493 (1997) 588 [hep-th/9609022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum field theories in finite volume: Excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    N.J. Iles and G.M.T. Watts, Modular properties of characters of the W 3 algebra, JHEP 01 (2016) 089 [arXiv:1411.4039] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S.D. Mathur, S. Mukhi and A. Sen, On the Classification of Rational Conformal Field Theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    H.R. Hampapura and S. Mukhi, On 2d Conformal Field Theories with Two Characters, JHEP 01 (2016) 005 [arXiv:1510.04478] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    M.R. Gaberdiel, H.R. Hampapura and S. Mukhi, Cosets of Meromorphic CFTs and Modular Differential Equations, JHEP 04 (2016) 156 [arXiv:1602.01022] [INSPIRE].ADSzbMATHGoogle Scholar
  39. [39]
    Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Am. Math. Soc. 9 (1996) 237.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    M.R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP 04 (2012) 103 [arXiv:1203.0015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M. Leitner, An algebraic approach to minimal models in CFTs, arXiv:1705.08294 [INSPIRE].
  42. [42]
    P. Kraus and E. Perlmutter, Partition functions of higher spin black holes and their CFT duals, JHEP 11 (2011) 061 [arXiv:1108.2567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    N.J. Iles and G.M.T. Watts, Characters of the W 3 algebra, JHEP 02 (2014) 009 [arXiv:1307.3771] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    P.G.O. Freund, T.R. Klassen and E. Melzer, S Matrices for Perturbations of Certain Conformal Field Theories, Phys. Lett. B 229 (1989) 243 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    P. Di Francesco and P. Mathieu, Singular vectors and conservation laws of quantum KdV type equations, Phys. Lett. B 278 (1992) 79 [hep-th/9109042] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    S. Negro, Integrable structures in quantum field theory, J. Phys. A 49 (2016) 323006 [arXiv:1606.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  47. [47]
    L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, September 26–October 6, 1995, pp. 149–219 (1996) [hep-th/9605187] [INSPIRE].
  48. [48]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    N. Beisert, Lecture Notes: Integrability in QFT and AdS/CFT, (2014) [].
  50. [50]
    A. Torrielli, Lectures on Classical Integrability, J. Phys. A 49 (2016) 323001 [arXiv:1606.02946] [INSPIRE].MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Alexander Maloney
    • 1
  • Gim Seng Ng
    • 2
    • 3
  • Simon F. Ross
    • 4
    Email author
  • Ioannis Tsiares
    • 1
  1. 1.Department of PhysicsMcGill UniversityMontréalCanada
  2. 2.School of MathematicsTrinity College DublinDublin D2Ireland
  3. 3.Hamilton Mathematical InstituteTrinity College DublinDublin D2Ireland
  4. 4.Centre for Particle Theory, Department of Mathematical SciencesDurham UniversityDurhamU.K.

Personalised recommendations