Advertisement

On the positive geometry of conformal field theory

  • Nima Arkani-Hamed
  • Yu-tin HuangEmail author
  • Shu-Heng Shao
Open Access
Regular Article - Theoretical Physics

Abstract

It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2, R) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of UX. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection UX by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.

Keywords

Conformal Field Theory Conformal and W Symmetry 

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]
    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]
    S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, Springer Briefs in Physics, Springer, Germany (2016).Google Scholar
  4. [4]
    D. Simmons-Duffin, The conformal bootstrap, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), June 1-26, Boulder U.S.A. (2015), arXiv:1602.07982 [INSPIRE].
  5. [5]
    D. Poland and D. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 (2016) 535.ADSCrossRefGoogle Scholar
  6. [6]
    D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys. 91 (2019) 15002 [arXiv:1805.04405] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
  8. [8]
    B. Czech et al., A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
  11. [11]
    M.F. Paulos et al., The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].zbMATHGoogle Scholar
  12. [12]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  13. [13]
    M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP 06 (2017) 076 [arXiv:1705.01357] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I. 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
  17. [17]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
  18. [18]
    D. Mazáč, A crossing-symmetric OPE inversion formula, arXiv:1812.02254 [INSPIRE].
  19. [19]
    P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077 [arXiv:1806.01862] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
  22. [22]
    D. Mazac, Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
  23. [23]
    B. Grunbaum et al., Convex polytopes, Springer, Germany (2003).Google Scholar
  24. [24]
    N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge U.K. (2016).Google Scholar
  25. [25]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    S. Karlin and W. Studden, Tchebycheff systems: with applications in analysis and statistics, Pure and applied mathematics, Interscience Publishers, Geneva, Switzerland (1966).Google Scholar
  27. [27]
    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the amplituhedron in binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Ferrara, R. Gatto and A.F. Grillo, Properties of partial wave amplitudes in conformal invariant field theories, Nuovo Cim. A 26 (1975) 226 [INSPIRE].
  30. [30]
    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].
  31. [31]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Nima Arkani-Hamed
    • 1
  • Yu-tin Huang
    • 2
    • 3
    Email author
  • Shu-Heng Shao
    • 1
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Department of Physics and AstronomyNational Taiwan UniversityTaipeiTaiwan
  3. 3.Physics Division, National Center for Theoretical SciencesNational Tsing-Hua UniversityHsinchuTaiwan

Personalised recommendations