Extremal bootstrapping: go with the flow

Open Access
Regular Article - Theoretical Physics


The extremal functional method determines approximate solutions to the constraints of crossing symmetry, which saturate bounds on the space of unitary CFTs. We show that such solutions are characterized by extremality conditions, which may be used to flow continuously along the boundaries of parameter space. Along the flow there is generically no further need for optimization, which dramatically reduces computational requirements, bringing calculations from the realm of computing clusters to laptops. Conceptually, extremality sheds light on possible ways to bootstrap without positivity, extending the method to non-unitary theories, and implies that theories saturating bounds, and especially those sitting at kinks, have unusually sparse spectra. We discuss several applications, including the first high-precision bootstrap of a non-unitary CFT.


Conformal and W Symmetry Conformal Field Theory 


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.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  2. [2]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
  3. [3]
    V.S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
  4. [4]
    F. Caracciolo and V.S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
  5. [5]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
  6. [6]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
  7. [7]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
  8. [8]
    K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179 (1969) 1499 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE Convergence in Conformal Field Theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
  10. [10]
    Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    J.D. Qualls, Lectures on Conformal Field Theory, arXiv:1511.04074 [INSPIRE].
  15. [15]
    S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics. 2016,  https://doi.org/10.1007/978-3-319-43626-5 [arXiv:1601.05000] [INSPIRE].
  16. [16]
    D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, in Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder U.S.A. (2015), pg. 1 [arXiv:1602.07982] [INSPIRE].
  17. [17]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
  18. [18]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
  20. [20]
    R. Hettich and K.O. Kortanek, Semi-infinite programming: theory, methods, and applications, SIAM Rev. 35 (1993) 380.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    R. Reemtsen and J.-J. Rückmann, Nonconvex Optimization and Its Applications. Vol. 25: Semi-infinite programming, Springer Science & Business Media, New York U.S.A. (1998).Google Scholar
  22. [22]
    M. López and G. Still, Semi-infinite programming, Eur. J. Oper. Res. 180 (2007) 491.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
  24. [24]
    M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
  25. [25]
    H. Kim, P. Kravchuk and H. Ooguri, Reflections on Conformal Spectra, JHEP 04 (2016) 184 [arXiv:1510.08772] [INSPIRE].ADSGoogle Scholar
  26. [26]
    F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and Interface CFTs from the Conformal Bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    F. Gliozzi and A. Rago, Critical exponents of the 3d Ising and related models from Conformal Bootstrap, JHEP 10 (2014) 042 [arXiv:1403.6003] [INSPIRE].
  28. [28]
    S. Rychkov and P. Yvernay, Remarks on the Convergence Properties of the Conformal Block Expansion, Phys. Lett. B 753 (2016) 682 [arXiv:1510.08486] [INSPIRE].
  29. [29]
    S. El-Showk and M. Paulos, Extremal flows in higher dimensions, to appear.Google Scholar
  30. [30]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
  31. [31]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) Archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
  32. [32]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589Université Pierre et Marie CurieParis Cedex 05France
  2. 2.Theoretical Physics Department, CERNGenevaSwitzerland

Personalised recommendations