Leading multi-stress tensors and conformal bootstrap

  • 13 Accesses


Near lightcone correlators are dominated by operators with the lowest twist. We consider the contributions of such leading lowest twist multi-stress tensor operators to a heavy-heavy-light-light correlator in a CFT of any even dimensionality with a large central charge. An infinite number of such operators contribute, but their sum is described by a simple ansatz. We show that the coefficients in this ansatz can be determined recursively, thereby providing an operational procedure to compute them. This is achieved by bootstrapping the corresponding near lightcone correlator: conformal data for any minimal­ twist determines that for the higher minimal-twist and so on. To illustrate this procedure in four spacetime dimensions we determine the contributions of double- and triple-stress tensors. We compute the OPE coefficients; whenever results are available in the literature, we observe complete agreement. We also compute the contributions of double-stress tensors in six spacetime dimensions and determine the corresponding OPE coefficients. In all cases the results are consistent with the exponentiation of the near lightcone correlator. This is similar to the situation in two spacetime dimensions for the Virasoro vacuum block.

A preprint version of the article is available at ArXiv.


  1. [1]

    A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett.43 (1986) 730 [Pisma Zh. Eksp. Tear. Fiz.43 (1986) 565] [INSPIRE].

  2. [2]

    Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP12 (2011) 099 [arXiv:1107.3987] [INSPIRE].

  3. [3]

    D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

  4. [4]

    T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP05 (2016) 099 [arXiv:1509.00014] [INSPIRE].

  5. [5]

    D. Li, D. Meltzer and D. Poland, Conformal Collider Physics from the Lightcone Bootstrap, JHEP02 (2016) 143 [arXiv:1511.08025] [INSPIRE].

  6. [6]

    Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal Field Theories and Deep Inelastic Scattering, Phys. Rev.D 95 (2017) 065011 [arXiv:1601.05453] [INSPIRE].

  7. [7]

    T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP10 (2016) 141 [arXiv:1601.07904] [INSPIRE].

  8. [8]

    D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP06 (2016) 111 [arXiv:1603. 03771] [INSPIRE].

  9. [9]

    T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP09 (2016) 038 [arXiv:1605.08072] [INSPIRE].

  10. [10]

    M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, The light-ray OPE and conformal colliders, arXiv:1905.01311 [INSPIRE].

  11. [11]

    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].

  12. [12]

    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP11 (2015) 200 [arXiv:1501.05315] [INSPIRE].

  13. [13]

    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP07 (2015) 131 [arXiv:1501.02260] [INSPIRE].

  14. [14]

    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3gravity, JHEP12 (2015) 077 [arXiv:1508.04987] [INSPIRE].

  15. [15]

    A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Hawking from Catalan, JHEP05 (2016) 069 [arXiv:1510.00014] [INSPIRE].

  16. [16]

    J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3gravity, JHEP02 (2019) 079 [arXiv:1808.03263] [INSPIRE].

  17. [17]

    S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP05 (2019) 212 [arXiv:1811.05710] [INSPIRE].

  18. [18]

    A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP05 (2016) 075 [arXiv:1512.03052] [INSPIRE].

  19. [19]

    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3/CFT 2, JHEP05 (2016) 109 [arXiv:1603. 08925] [INSPIRE].

  20. [20]

    T. Anous, T. Hartman, A. Rovai and J. Sonner, Black Hole Collapse in the 1/c Expansion, JHEP07 (2016) 123 [arXiv:1603.04856] [INSPIRE].

  21. [21]

    A.L. Fitzpatrick and J. Kaplan, On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles, JHEP04 (2017) 072 [arXiv:1609.07153] [INSPIRE].

  22. [22]

    H. Chen, C. Hussong, J. Kaplan and D. Li, A Numerical Approach to Virasoro Blocks and the Information Paradox, JHEP09 (2017) 102 [arXiv:1703.09727] [INSPIRE].

  23. [23]

    T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP06 (2018) 123 [arXiv:1712.03464] [INSPIRE].

  24. [24]

    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heav y States and Local Quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].

  25. [25]

    P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum Entanglement of Localized Excited States at Finite Temperature, JHEP01 (2015) 102 [arXiv:1410.2287] [INSPIRE].

  26. [26]

    B. Chen and J.-q. Wu, Holographic Entanglement Entropy For a Large Class of States in 2D CFT, JHEP09 (2016) 015 [arXiv:1605.06753] [INSPIRE].

  27. [27]

    B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP10 (2016) 110 [arXiv:1609.00801] [INSPIRE].

  28. [28]

    T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].

  29. [29]

    T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].

  30. [30]

    M. Kulaxizi, G.S. Ng and A. Parnachev, Black Holes, Heavy States, Phase Shift and Anomalous Dimensions, SciPost Phys.6 (2019) 065 [arXiv:1812.03120] [INSPIRE].

  31. [31]

    M. Beccaria, A. Fachechi and G. Macorini, Virasoro vacuum block at next-to-leading order in the heavy-light limit, JHEP02 (2016) 072 [arXiv:1511.05452] [INSPIRE].

  32. [32]

    M. Kulaxizi, G.S. Ng and A. Parnachev, Subleading Eikonal, AdS/CFT and Double Stress Tensors, JHEP10 (2019) 107 [arXiv:1907.00867] [INSPIRE].

  33. [33]

    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].

  34. [34]

    S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics Series, Springer, Cham Switzerland (2017) [arXiv:1601.05000] [INSPIRE].

  35. [35]

    D. Simmons-Duffin, The Conformal Bootstrap, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TAS Boulder, CO, U.S.A., 1–26 June 2015, pp. 1–74 [arXiv:1602.07982] [INSPIRE].

  36. [36]

    D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, Rev. Mod. Phys.91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].

  37. [37]

    A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from Holography, JHEP08 (2019) 138 [arXiv:1903.05306] [INSPIRE].

  38. [38]

    Y.-Z. Li, Z.-F. Mai and H. Lü, Holographic OPE Coefficients from AdS Black Holes with Matters, JHEP09 (2019) 001 [arXiv:1905.09302] [INSPIRE].

  39. [39]

    A.L. Fitzpatrick, K.-W. Huang and D. Li, Probing universalities in d > 2 CFTs: from black holes to shockwaves, JHEP11 (2019) 139 [arXiv:1907.10810] [INSPIRE].

  40. [40]

    K.-W. Huang, Stress-tensor commutators in conformal field theories near the lightcone, Phys. Rev.D 100 (2019) 061701 [arXiv:1907.00599] [INSPIRE].

  41. [41]

    R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Black Holes and Conformal Regge Bootstrap, J HEP10 (2019) 046 [arXiv:1904.00060] [INSPIRE].

  42. [42]

    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].

  43. [43]

    Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].

  44. [44]

    A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP10 (2012) 032 [arXiv:1112.4845] [INSPIRE].

  45. [45]

    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP08 (2007) 019 [hep-th/0611122] [INSPIRE].

  46. [46]

    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys.B 767 (2007) 327 [hep-th/0611123] [INSPIRE].

  47. [47]

    L. Cornalba, M.S. Costa and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP09 (2007) 037 [arXiv:0707.0120] [INSPIRE].

  48. [48]

    L. Cornalba, M.S. Costa and J. Penedones, Eikonal Methods in AdS/CFT: BFKL Pomeron at Weak Coupling, JHEP06 (2008) 048 [arXiv:0801.3002] [INSPIRE].

  49. [49]

    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP12 (2012) 091 [arXiv:1209.4355] [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

Correspondence to Robin Karlsson.

Additional information

ArXiv ePrint: 1909.05775

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Karlsson, R., Kulaxizi, M., Parnachev, A. et al. Leading multi-stress tensors and conformal bootstrap. J. High Energ. Phys. 2020, 76 (2020).

Download citation


  • AdS-CFT Correspondence
  • Conformal Field Theory