Journal of High Energy Physics

, 2019:59 | Cite as

Positive geometry in the diagonal limit of the conformal bootstrap

  • Kallol Sen
  • Aninda Sinha
  • Ahmadullah ZahedEmail author
Open Access
Regular Article - Theoretical Physics


We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in d = 1, the geometric understanding of the boot- strap equations for unitary theories leads to cyclic polytopes for which the faces can all be written down and, in principle, the intersection between the unitarity polytope and the crossing plane can be systematically explored. We find that in higher dimensions, the natural structure that emerges, due to the inclusion of spin, is the weighted Minkowski sum of cyclic polytopes. While it can be explicitly shown that for physical theories, the weighted Minkowski sum of cyclic polytopes is not a cyclic polytope, it also turns out that in the large conformal dimension limit it is indeed a cyclic polytope. We write down several analytic formulae in this limit and show that remarkably, in many cases, this works out to be very good approximation even for O (1) conformal dimensions. Furthermore, we initiate a comparison between usual numerics obtained using linear programming and what arises from positive geometry considerations.


Conformal Field Theory Scattering Amplitudes 


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]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    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].ADSCrossRefGoogle Scholar
  3. [3]
    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].
  4. [4]
    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].
  5. [5]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N ) models, JHEP08 (2016) 036 [arXiv:1603.04436] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    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].
  7. [7]
    V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev.D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
  8. [8]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [Sov. Phys. JETP39 (1974) 9] [INSPIRE].
  9. [9]
    K. Sen and A. Sinha, On critical exponents without Feynman diagrams, J. Phys.A 49 (2016) 445401 [arXiv:1510.07770] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal bootstrap in Mellin space, Phys. Rev. Lett.118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
  11. [11]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP05 (2017) 027 [arXiv:1611.08407] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Gopakumar and A. Sinha, On the Polyakov-Mellin bootstrap, JHEP12 (2018) 040 [arXiv:1809.10975] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
  14. [14]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
  15. [15]
    A. Kaviraj and M.F. Paulos, The functional bootstrap for boundary CFT, arXiv:1812.04034 [INSPIRE].
  16. [16]
    N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the positive geometry of conformal field theory, JHEP06 (2019) 124 [arXiv:1812.07739] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP08 (2013) 014 [arXiv:1305.1321] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean bootstrap, arXiv:1808.03212 [INSPIRE].
  19. [19]
    C. Weibel, Minkowski sum of polytopes: combinatorics and computation, Ph.D. thesis, EPFL, Lausanne, Switzerland (2007).Google Scholar
  20. [20]
    Y.L. Luke, The special functions and their approximations, volume 1, Academic Press, U.S.A. (1969).Google Scholar
  21. [21]
    S. Caron-Huot, Analyticity in spin in conformal theories, JHEP09 (2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP03 (2017) 086 [arXiv:1612.08471] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    Y. Huang, A. Sinha, L. Wei and A. Zahed, Polytopes and conformal bootstrap, in preparation.Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the Universe (WPI)University of TokyoChibaJapan
  2. 2.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations