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Weight shifting operators and conformal blocks

  • Regular Article - Theoretical Physics
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  • Published: 14 February 2018
  • Volume 2018, article number 81, (2018)
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Weight shifting operators and conformal blocks
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  • Denis Karateev1,
  • Petr Kravchuk2 &
  • David Simmons-Duffin2,3 

  • 838 Accesses

  • 125 Citations

  • 4 Altmetric

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A preprint version of the article is available at arXiv.

Abstract

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for “seed conformal blocks” in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including deriving recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/“crossing kernels”) of the conformal group.

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References

  1. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. V.S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].

  3. 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].

  4. R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].

  5. D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].

  7. A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  8. D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].

  9. S. Rychkov, Conformal Bootstrap in Three Dimensions?, arXiv:1111.2115 [INSPIRE].

  10. 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].

  11. P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFT d , JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  13. F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].

    Article  ADS  Google Scholar 

  14. F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].

  15. L.F. Alday and A. Bissi, The superconformal bootstrap for structure constants, JHEP 09 (2014) 144 [arXiv:1310.3757] [INSPIRE].

  16. D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].

  17. M. Berkooz, R. Yacoby and A. Zait, Bounds on \( \mathcal{N}=1 \) superconformal theories with global symmetries, JHEP 08 (2014) 008 [Erratum ibid. 01 (2015) 132] [arXiv:1402.6068] [INSPIRE].

  18. 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].

  19. Y. Nakayama and T. Ohtsuki, Approaching the conformal window of O(n) × O(m) symmetric Landau-Ginzburg models using the conformal bootstrap, Phys. Rev. D 89 (2014) 126009 [arXiv:1404.0489] [INSPIRE].

  20. Y. Nakayama and T. Ohtsuki, Five dimensional O(N)-symmetric CFTs from conformal bootstrap, Phys. Lett. B 734 (2014) 193 [arXiv:1404.5201] [INSPIRE].

  21. S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The \( \mathcal{N}=8 \) superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].

  22. F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].

    Article  ADS  Google Scholar 

  23. F. Caracciolo, A. Castedo Echeverri, B. von Harling and M. Serone, Bounds on OPE Coefficients in 4D Conformal Field Theories, JHEP 10 (2014) 020 [arXiv:1406.7845] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Y. Nakayama and T. Ohtsuki, Bootstrapping phase transitions in QCD and frustrated spin systems, Phys. Rev. D 91 (2015) 021901 [arXiv:1407.6195] [INSPIRE].

  25. M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].

  26. J.-B. Bae and S.-J. Rey, Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions, arXiv:1412.6549 [INSPIRE].

  27. C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=2 \) superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].

  28. S.M. Chester, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models in 4 < d < 6, Phys. Rev. D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE].

  29. L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].

    ADS  MATH  Google Scholar 

  30. D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].

  31. N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with Four Supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  32. F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) Archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  33. S.M. Chester, S. Giombi, L.V. Iliesiu, I.R. Klebanov, S.S. Pufu and R. Yacoby, Accidental Symmetries and the Conformal Bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].

  35. L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].

    Article  ADS  Google Scholar 

  36. F. Rejon-Barrera and D. Robbins, Scalar-Vector Bootstrap, JHEP 01 (2016) 139 [arXiv:1508.02676] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  37. D. Poland and A. Stergiou, Exploring the Minimal 4D \( \mathcal{N}=1 \) SCFT, JHEP 12 (2015) 121 [arXiv:1509.06368] [INSPIRE].

  38. M. Lemos and P. Liendo, Bootstrapping \( \mathcal{N}=2 \) chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [INSPIRE].

  39. H. Kim, P. Kravchuk and H. Ooguri, Reflections on Conformal Spectra, JHEP 04 (2016) 184 [arXiv:1510.08772] [INSPIRE].

    ADS  Google Scholar 

  40. Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \) superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].

  41. S.M. Chester, L.V. Iliesiu, S.S. Pufu and R. Yacoby, Bootstrapping O(N ) Vector Models with Four Supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103 [arXiv:1511.07552] [INSPIRE].

  42. L.F. Alday and A. Zhiboedov, Conformal Bootstrap With Slightly Broken Higher Spin Symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  43. L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].

  44. S.M. Chester and S.S. Pufu, Towards bootstrapping QED 3, JHEP 08 (2016) 019 [arXiv:1601.03476] [INSPIRE].

    Article  ADS  Google Scholar 

  45. C. Behan, PyCFTBoot: A flexible interface for the conformal bootstrap, Commun. Comput. Phys. 22 (2017) 1 [arXiv:1602.02810] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  46. P. Dey, A. Kaviraj and K. Sen, More on analytic bootstrap for O(N) models, JHEP 06 (2016) 136 [arXiv:1602.04928] [INSPIRE].

  47. Y. Nakayama, Bootstrap bound for conformal multi-flavor QCD on lattice, JHEP 07 (2016) 038 [arXiv:1605.04052] [INSPIRE].

  48. S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, arXiv:1605.08087 [INSPIRE].

  49. Z. Li and N. Su, Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models, JHEP 04 (2017) 098 [arXiv:1607.07077] [INSPIRE].

    Article  ADS  Google Scholar 

  50. Y. Pang, J. Rong and N. Su, \( \phi \) 3 theory with F 4 flavor symmetry in 6 − 2ϵ dimensions: 3-loop renormalization and conformal bootstrap, JHEP 12 (2016) 057 [arXiv:1609.03007] [INSPIRE].

  51. Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].

  52. L.F. Alday and A. Bissi, Crossing symmetry and Higher spin towers, JHEP 12 (2017) 118 [arXiv:1603.05150] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  53. L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].

  54. L.F. Alday, Solving CFTs with Weakly Broken Higher Spin Symmetry, JHEP 10 (2017) 161 [arXiv:1612.00696] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  55. M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping \( \mathcal{N}=3 \) superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].

  56. C. Beem, L. Rastelli and B.C. van Rees, More \( \mathcal{N}=4 \) superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].

  57. D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].

  58. D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D \( \mathcal{N}=1 \) SCFTs, JHEP 07 (2017) 029 [arXiv:1702.00404] [INSPIRE].

  59. S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ, arXiv:1702.00423 [INSPIRE].

  60. M. Cornagliotto, M. Lemos and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  61. F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points, JHEP 04 (2017) 056 [arXiv:1702.03938] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP 06 (2017) 076 [arXiv:1705.01357] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Y. Nakayama, Bootstrap experiments on higher dimensional CFTs, arXiv:1705.02744 [INSPIRE].

  65. C.-M. Chang and Y.-H. Lin, Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions), JHEP 08 (2017) 128 [arXiv:1705.05392] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, arXiv:1705.04278 [INSPIRE].

  67. 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].

  68. F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].

  69. M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011)154 [arXiv:1109.6321] [INSPIRE].

  71. D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Fermion-Scalar Conformal Blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  73. M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  74. M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  75. G.F. Cuomo, D. Karateev and P. Kravchuk, General Bootstrap Equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  76. A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing Conformal Blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  77. A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed Conformal Blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  78. A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d Stress-Tensor Bootstrap, arXiv:1708.05718 [INSPIRE].

  79. T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].

    Article  ADS  Google Scholar 

  80. T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP 10 (2016) 141 [arXiv:1601.07904] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  81. D. Li, D. Meltzer and D. Poland, Conformal Collider Physics from the Lightcone Bootstrap, JHEP 02 (2016) 143 [arXiv:1511.08025] [INSPIRE].

    Article  ADS  Google Scholar 

  82. D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  83. T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  84. N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  85. D. Li, D. Meltzer and D. Poland, Conformal Bootstrap in the Regge Limit, JHEP 12 (2017) 013 [arXiv:1705.03453] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  86. M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, arXiv:1705.02934 [INSPIRE].

  87. V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].

  88. L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].

  89. M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  90. S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  91. A. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].

  92. M. Hogervorst, Crossing Kernels for Boundary and Crosscap CFTs, arXiv:1703.08159 [INSPIRE].

  93. M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  94. A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/; A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.

  95. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].

  96. J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].

  97. J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  98. E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  99. M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field, PTEP 2017 (2017) 053B06 [arXiv:1609.04563] [INSPIRE].

  100. E. Dyer, D.Z. Freedman and J. Sully, Spinning Geodesic Witten Diagrams, JHEP 11 (2017) 060 [arXiv:1702.06139] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  101. H.-Y. Chen, E.-J. Kuo and H. Kyono, Anatomy of Geodesic Witten Diagrams, JHEP 05 (2017) 070 [arXiv:1702.08818] [INSPIRE].

  102. C. Sleight and M. Taronna, Spinning Witten Diagrams, JHEP 06 (2017) 100 [arXiv:1702.08619] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  103. A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic Diagrams, Gravitational Interactions & OPE Structures, JHEP 06 (2017) 099 [arXiv:1702.06128] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  104. S. Giombi, S. Prakash and X. Yin, A Note on CFT Correlators in Three Dimensions, JHEP 07 (2013) 105 [arXiv:1104.4317] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  105. J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  106. T.Y. Thomas, On conformal geometry, Proc. Natl. Acad. Sci. U.S.A. 12 (1926) 352.

  107. T. Bailey, M. Eastwood and A. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994) 1191.

    Article  MathSciNet  MATH  Google Scholar 

  108. K. Dighton, An introduction to the theory of local twistors, Int. J. Theor. Phys. 11 (1974) 31.

    Article  MathSciNet  Google Scholar 

  109. R. Penrose and M.A.H. MacCallum, Twistor theory: An approach to the quantization of fields and space-time, Phys. Rept. 6 (1972) 241 [INSPIRE].

    Article  ADS  Google Scholar 

  110. H. Friedrich, Twistor connection and normal conformal cartan connection, Gen. Rel. Grav. 8 (1977) 303.

  111. P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.

    Article  MathSciNet  MATH  Google Scholar 

  112. G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  113. D.G. Boulware, L.S. Brown and R.D. Peccei, Deep-inelastic electroproduction and conformal symmetry, Phys. Rev. D 2 (1970) 293 [INSPIRE].

  114. S. Ferrara, R. Gatto and A.F. Grillo, Conformal algebra in space-time and operator product expansion, Springer Tracts Mod. Phys. 67 (1973) 1. [INSPIRE]

    Article  ADS  Google Scholar 

  115. S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  116. L. Cornalba, M.S. Costa and J. Penedones, Deep Inelastic Scattering in Conformal QCD, JHEP 03 (2010) 133 [arXiv:0911.0043] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  117. S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].

  118. G. Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple lie groups, Annals Math. 106 (1977) 295.

    Article  MathSciNet  MATH  Google Scholar 

  119. J.C. Jantzen, Moduln mit einem höchsten Gewicht, Lect. Notes Math. 750, Springer (1979).

  120. M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  121. H.-Y. Chen and J.D. Qualls, Quantum Integrable Systems from Conformal Blocks, Phys. Rev. D 95 (2017) 106011 [arXiv:1605.05105] [INSPIRE].

  122. V. Schomerus, E. Sobko and M. Isachenkov, Harmony of Spinning Conformal Blocks, JHEP 03 (2017) 085 [arXiv:1612.02479] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  123. A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  124. Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.

  125. M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arXiv:1703.09805 [INSPIRE].

  126. D. Karateev, P. Kravchuk and D. Simmons-Duffin, in progress.

  127. M. Yamazaki, Comments on Determinant Formulas for General CFTs, JHEP 10 (2016) 035 [arXiv:1601.04072] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  128. R. Brauer, Sur la multiplication des caractéristiques des groupes continus et semi-simples, C. R. Acad. Sci. Paris 204 (1937) 1784.

    MATH  Google Scholar 

  129. A.U. Klimyk, Multiplicities of weights of representations and multiplicities of representations of semisimple Lie algebras, Dokl. Akad. Nauk SSSR 177 (1967) 1001.

    MathSciNet  MATH  Google Scholar 

  130. D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, U.S.A., June 1-26, 2015, (2017) pp. 1-74, arXiv:1602.07982 [INSPIRE].

  131. P. Etingof and F. Latour, The dynamical yang-baxter equation, representation theory, and quantum integrable systems, Oxford Lecture Series in Mathematics and Its Applications 29 (2005).

  132. D. Arnaudon, E. Buffenoir, E. Ragoucy and P. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998) 201.

    Article  MathSciNet  MATH  Google Scholar 

  133. M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].

  134. V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].

  135. W. Siegel, Embedding versus 6D twistors, arXiv:1204.5679 [INSPIRE].

  136. E. Elkhidir, D. Karateev and M. Serone, General Three-Point Functions in 4D CFT, JHEP 01 (2015) 133 [arXiv:1412.1796] [INSPIRE].

    Article  ADS  Google Scholar 

  137. G. Mack, Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory, Commun. Math. Phys. 53 (1977) 155 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  138. P. Kravchuk and D. Simmons-Duffin, Counting Conformal Correlators, arXiv:1612.08987 [INSPIRE].

  139. B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  140. F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].

  141. 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].

  142. M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013)106004 [arXiv:1303.1111] [INSPIRE].

  143. Y. Oshima and M. Yamazaki, Determinant Formula for Parabolic Verma Modules of Lie Superalgebras, J. Algebra 495 (2018) 51 [arXiv:1603.06705] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  144. J. Slovák, Natural operators on conformal manifolds, in Differential geometry and its applications (Opava, 1992), Math. Publ. 1 pp. 335-349, Silesian Univ. Opava, Opava, Czech Republic (1993).

  145. K. Krasnov and J. Louko, SO(1, d + 1) Racah coefficients: Type I representations, J. Math. Phys. 47 (2006) 033513 [math-ph/0502017] [INSPIRE].

  146. C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, arXiv:1612.00809 [INSPIRE].

  147. V. Cardoso, T. Houri and M. Kimura, Mass Ladder Operators from Spacetime Conformal Symmetry, Phys. Rev. D 96 (2017) 024044 [arXiv:1706.07339] [INSPIRE].

  148. C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, arXiv:1705.03025 [INSPIRE].

  149. S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].

  150. M. Eastwood, Notes on conformal differential geometry, in The Proceedings of the 15th Winter School “Geometry and Physics” (Srnís, 1995), no. 43, (1996), pp. 57-76,.

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  1. SISSA and INFN, Via Bonomea 265, I-34136, Trieste, Italy

    Denis Karateev

  2. Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California, 91125, U.S.A.

    Petr Kravchuk & David Simmons-Duffin

  3. School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, U.S.A.

    David Simmons-Duffin

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  1. Denis Karateev
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Correspondence to David Simmons-Duffin.

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ArXiv ePrint: 1706.07813

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Karateev, D., Kravchuk, P. & Simmons-Duffin, D. Weight shifting operators and conformal blocks. J. High Energ. Phys. 2018, 81 (2018). https://doi.org/10.1007/JHEP02(2018)081

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  • Received: 06 December 2017

  • Accepted: 22 January 2018

  • Published: 14 February 2018

  • DOI: https://doi.org/10.1007/JHEP02(2018)081

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Keywords

  • Conformal and W Symmetry
  • Conformal Field Theory
  • Space-Time Symmetries
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