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Weight shifting operators and conformal blocks. (English) Zbl 1387.81323
Summary: 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.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
Software:
SDPB; CFTs4D
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References:
[1] Rattazzi, R.; Rychkov, VS; Tonni, E.; Vichi, A., Bounding scalar operator dimensions in 4D CFT, JHEP, 12, 031, (2008) · Zbl 1329.81324
[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]. · Zbl 1390.81546
[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]. · Zbl 1206.81116
[5] Poland, D.; Simmons-Duffin, D., Bounds on 4D conformal and superconformal field theories, JHEP, 05, 017, (2011) · Zbl 1296.81067
[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]. · Zbl 1206.81116
[7] Vichi, A., Improved bounds for CFT’s with global symmetries, JHEP, 01, 162, (2012) · Zbl 1306.81289
[8] D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP05 (2012) 110 [arXiv:1109.5176] [INSPIRE]. · Zbl 1380.81362
[9] S. Rychkov, Conformal Bootstrap in Three Dimensions?, arXiv:1111.2115 [INSPIRE]. · Zbl 1365.81007
[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]. · Zbl 1310.82013
[11] Liendo, P.; Rastelli, L.; Rees, BC, The bootstrap program for boundary CFT_{d}, JHEP, 07, 113, (2013) · Zbl 1342.81504
[12] El-Showk, S.; Paulos, MF, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett., 111, 241601, (2013)
[13] Gliozzi, F., More constraining conformal bootstrap, Phys. Rev. Lett., 111, 161602, (2013)
[14] F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(\(N\)) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE]. · Zbl 1392.81202
[15] L.F. Alday and A. Bissi, The superconformal bootstrap for structure constants, JHEP09 (2014) 144 [arXiv:1310.3757] [INSPIRE]. · Zbl 1388.81382
[16] D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP03 (2014) 100 [arXiv:1310.5078] [INSPIRE]. · Zbl 1390.81546
[17] M. Berkooz, R. Yacoby and A. Zait, Bounds on\( \mathcal{N}=1 \)superconformal theories with global symmetries, JHEP08 (2014) 008 [Erratum ibid.01 (2015) 132] [arXiv:1402.6068] [INSPIRE]. · Zbl 1333.81361
[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]. · Zbl 1310.82013
[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]. · Zbl 1383.81205
[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]. · Zbl 1382.81173
[21] S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The\( \mathcal{N}=8 \)superconformal bootstrap in three dimensions, JHEP09 (2014) 143 [arXiv:1406.4814] [INSPIRE]. · Zbl 1380.81320
[22] Kos, F.; Poland, D.; Simmons-Duffin, D., Bootstrapping mixed correlators in the 3D Ising model, JHEP, 11, 109, (2014)
[23] Caracciolo, F.; Castedo Echeverri, A.; Harling, B.; Serone, M., Bounds on OPE coefficients in 4D conformal field theories, JHEP, 10, 020, (2014) · Zbl 1333.81365
[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]. · Zbl 1383.81172
[25] M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE]. · Zbl 1383.81231
[26] J.-B. Bae and S.-J. Rey, Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions, arXiv:1412.6549 [INSPIRE]. · Zbl 1388.81693
[27] C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The\( \mathcal{N}=2 \)superconformal bootstrap, JHEP03 (2016) 183 [arXiv:1412.7541] [INSPIRE]. · Zbl 1388.81482
[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]. · Zbl 1383.81231
[29] Alday, LF; Bissi, A.; Lukowski, T., Large spin systematics in CFT, JHEP, 11, 101, (2015) · Zbl 1388.81752
[30] D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
[31] Bobev, N.; El-Showk, S.; Mazac, D.; Paulos, MF, Bootstrapping SCFTs with four supercharges, JHEP, 08, 142, (2015) · Zbl 1388.81638
[32] Kos, F.; Poland, D.; Simmons-Duffin, D.; Vichi, A., Bootstrapping the O(N) archipelago, JHEP, 11, 106, (2015) · Zbl 1388.81054
[33] Chester, SM; Giombi, S.; Iliesiu, LV; Klebanov, IR; Pufu, SS; Yacoby, R., Accidental symmetries and the conformal bootstrap, JHEP, 01, 110, (2016) · Zbl 1388.81382
[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]. · Zbl 1388.81482
[35] Iliesiu, L.; Kos, F.; Poland, D.; Pufu, SS; Simmons-Duffin, D.; Yacoby, R., Bootstrapping 3D fermions, JHEP, 03, 120, (2016)
[36] Rejon-Barrera, F.; Robbins, D., Scalar-vector bootstrap, JHEP, 01, 139, (2016) · Zbl 1388.81693
[37] D. Poland and A. Stergiou, Exploring the Minimal 4D\( \mathcal{N}=1 \)SCFT, JHEP12 (2015) 121 [arXiv:1509.06368] [INSPIRE]. · Zbl 1388.81691
[38] M. Lemos and P. Liendo, Bootstrapping\( \mathcal{N}=2 \)chiral correlators, JHEP01 (2016) 025 [arXiv:1510.03866] [INSPIRE]. · Zbl 1388.81056
[39] Kim, H.; Kravchuk, P.; Ooguri, H., Reflections on conformal spectra, JHEP, 04, 184, (2016)
[40] Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \)superconformal bootstrap of the K3 CFT, JHEP05 (2017) 126 [arXiv:1511.04065] [INSPIRE]. · Zbl 1377.81182
[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, JHEP05 (2016) 103 [arXiv:1511.07552] [INSPIRE]. · Zbl 0384.22004
[42] Alday, LF; Zhiboedov, A., Conformal bootstrap with slightly broken higher spin symmetry, JHEP, 06, 091, (2016) · Zbl 1388.81753
[43] L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP04 (2017) 157 [arXiv:1510.08091] [INSPIRE]. · Zbl 1378.81097
[44] Chester, SM; Pufu, SS, Towards bootstrapping QED_{3}, JHEP, 08, 019, (2016) · Zbl 1390.81498
[45] Behan, C., Pycftboot: A flexible interface for the conformal bootstrap, Commun. Comput. Phys., 22, 1, (2017)
[46] P. Dey, A. Kaviraj and K. Sen, More on analytic bootstrap for O(N) models, JHEP06 (2016) 136 [arXiv:1602.04928] [INSPIRE]. · Zbl 1388.83223
[47] Y. Nakayama, Bootstrap bound for conformal multi-flavor QCD on lattice, JHEP07 (2016) 038 [arXiv:1605.04052] [INSPIRE]. · Zbl 1342.81492
[48] S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, arXiv:1605.08087 [INSPIRE]. · Zbl 1388.81656
[49] Li, Z.; Su, N., Bootstrapping mixed correlators in the five dimensional critical O(N) models, JHEP, 04, 098, (2017) · Zbl 1378.81120
[50] Y. Pang, J. Rong and N. Su, \( ϕ \)\^{}{3}theory with F_{4}flavor symmetry in 6 − 2ϵ dimensions: 3-loop renormalization and conformal bootstrap, JHEP12 (2016) 057 [arXiv:1609.03007] [INSPIRE]. · Zbl 1390.81366
[51] Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2\(,\) 2) superconformal bootstrap in two dimensions, JHEP05 (2017) 112 [arXiv:1610.05371] [INSPIRE]. · Zbl 1388.81048
[52] Alday, LF; Bissi, A., Crossing symmetry and higher spin towers, JHEP, 12, 118, (2017) · Zbl 1383.81172
[53] L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett.119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
[54] Alday, LF, Solving CFTs with weakly broken higher spin symmetry, JHEP, 10, 161, (2017) · Zbl 1383.81149
[55] M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping\( \mathcal{N}=3 \)superconformal theories, JHEP04 (2017) 032 [arXiv:1612.01536] [INSPIRE]. · Zbl 1378.81142
[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, JHEP03 (2017) 086 [arXiv:1612.08471] [INSPIRE]. · Zbl 1377.81184
[58] D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D\( \mathcal{N}=1 \)SCFTs, JHEP07 (2017) 029 [arXiv:1702.00404] [INSPIRE]. · Zbl 1380.81405
[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]. · Zbl 1388.81798
[60] Cornagliotto, M.; Lemos, M.; Schomerus, V., Long multiplet bootstrap, JHEP, 10, 119, (2017) · Zbl 1383.81287
[61] Gliozzi, F.; Guerrieri, AL; Petkou, AC; Wen, C., The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points, JHEP, 04, 056, (2017) · Zbl 1378.81119
[62] Gopakumar, R.; Kaviraj, A.; Sen, K.; Sinha, A., A Mellin space approach to the conformal bootstrap, JHEP, 05, 027, (2017) · Zbl 1380.81320
[63] Qiao, J.; Rychkov, S., Cut-touching linear functionals in the conformal bootstrap, JHEP, 06, 076, (2017) · Zbl 1380.81357
[64] Y. Nakayama, Bootstrap experiments on higher dimensional CFTs, arXiv:1705.02744 [INSPIRE]. · Zbl 1388.81051
[65] Chang, C-M; Lin, Y-H, Carving out the end of the world or (superconformal bootstrap in six dimensions), JHEP, 08, 128, (2017) · Zbl 1381.83122
[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]. · Zbl 1333.83125
[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]. · Zbl 1097.81734
[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]. · Zbl 1097.81735
[69] Costa, MS; Penedones, J.; Poland, D.; Rychkov, S., Spinning conformal correlators, JHEP, 11, 071, (2011) · Zbl 1306.81207
[70] M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP11 (2011)154 [arXiv:1109.6321] [INSPIRE]. · Zbl 1306.81148
[71] Simmons-Duffin, D., Projectors, shadows and conformal blocks, JHEP, 04, 146, (2014) · Zbl 1333.83125
[72] Iliesiu, L.; Kos, F.; Poland, D.; Pufu, SS; Simmons-Duffin, D.; Yacoby, R., Fermion-scalar conformal blocks, JHEP, 04, 074, (2016) · Zbl 1388.81051
[73] Costa, MS; Hansen, T.; Penedones, J.; Trevisani, E., Radial expansion for spinning conformal blocks, JHEP, 07, 057, (2016)
[74] Costa, MS; Hansen, T.; Penedones, J.; Trevisani, E., Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP, 07, 018, (2016) · Zbl 1388.81798
[75] Cuomo, GF; Karateev, D.; Kravchuk, P., General bootstrap equations in 4D cfts, JHEP, 01, 130, (2018) · Zbl 1384.81094
[76] Castedo Echeverri, A.; Elkhidir, E.; Karateev, D.; Serone, M., Deconstructing conformal blocks in 4D CFT, JHEP, 08, 101, (2015) · Zbl 1388.81409
[77] Castedo Echeverri, A.; Elkhidir, E.; Karateev, D.; Serone, M., Seed conformal blocks in 4D CFT, JHEP, 02, 183, (2016) · Zbl 1388.81745
[78] A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d Stress-Tensor Bootstrap, arXiv:1708.05718 [INSPIRE]. · Zbl 1387.81313
[79] Hartman, T.; Jain, S.; Kundu, S., Causality constraints in conformal field theory, JHEP, 05, 099, (2016)
[80] Hartman, T.; Jain, S.; Kundu, S., A new spin on causality constraints, JHEP, 10, 141, (2016) · Zbl 1390.83114
[81] Li, D.; Meltzer, D.; Poland, D., Conformal collider physics from the lightcone bootstrap, JHEP, 02, 143, (2016)
[82] Hofman, DM; Li, D.; Meltzer, D.; Poland, D.; Rejon-Barrera, F., A proof of the conformal collider bounds, JHEP, 06, 111, (2016) · Zbl 1388.81048
[83] Hartman, T.; Kundu, S.; Tajdini, A., Averaged null energy condition from causality, JHEP, 07, 066, (2017) · Zbl 1380.81327
[84] Afkhami-Jeddi, N.; Hartman, T.; Kundu, S.; Tajdini, A., Einstein gravity 3-point functions from conformal field theory, JHEP, 12, 049, (2017) · Zbl 1383.83024
[85] Li, D.; Meltzer, D.; Poland, D., Conformal bootstrap in the Regge limit, JHEP, 12, 013, (2017)
[86] M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, arXiv:1705.02934 [INSPIRE]. · Zbl 1395.83012
[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]. · Zbl 0407.43010
[88] L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
[89] Costa, MS; Goncalves, V.; Penedones, J., Conformal Regge theory, JHEP, 12, 091, (2012) · Zbl 1397.81297
[90] Caron-Huot, S., Analyticity in spin in conformal theories, JHEP, 09, 078, (2017) · Zbl 1382.81173
[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] Hogervorst, M.; Rees, BC, Crossing symmetry in alpha space, JHEP, 11, 193, (2017) · Zbl 1383.81231
[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/. · Zbl 1296.81067
[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, JHEP04 (2016) 001 [arXiv:1601.06768] [INSPIRE]. · Zbl 1388.81067
[97] Murugan, J.; Stanford, D.; Witten, E., More on supersymmetric and 2d analogs of the SYK model, JHEP, 08, 146, (2017) · Zbl 1381.81121
[98] Hijano, E.; Kraus, P.; Perlmutter, E.; Snively, R., Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP, 01, 146, (2016) · Zbl 1388.81047
[99] M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field, PTEP2017 (2017) 053B06 [arXiv:1609.04563] [INSPIRE].
[100] Dyer, E.; Freedman, DZ; Sully, J., Spinning geodesic Witten diagrams, JHEP, 11, 060, (2017) · Zbl 1383.81205
[101] H.-Y. Chen, E.-J. Kuo and H. Kyono, Anatomy of Geodesic Witten Diagrams, JHEP05 (2017) 070 [arXiv:1702.08818] [INSPIRE]. · Zbl 1380.81303
[102] Sleight, C.; Taronna, M., Spinning Witten diagrams, JHEP, 06, 100, (2017) · Zbl 1380.81362
[103] Castro, A.; Llabrés, E.; Rejon-Barrera, F., Geodesic diagrams, gravitational interactions & OPE structures, JHEP, 06, 099, (2017) · Zbl 1380.81301
[104] Giombi, S.; Prakash, S.; Yin, X., A note on CFT correlators in three dimensions, JHEP, 07, 105, (2013) · Zbl 1342.81492
[105] Penedones, J.; Trevisani, E.; Yamazaki, M., Recursion relations for conformal blocks, JHEP, 09, 070, (2016) · Zbl 1390.81533
[106] T.Y. Thomas, On conformal geometry, Proc. Natl. Acad. Sci. U.S.A.12 (1926) 352. · JFM 52.0769.13
[107] Bailey, T.; Eastwood, M.; Gover, A., Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math., 24, 1191, (1994) · Zbl 0828.53012
[108] Dighton, K., An introduction to the theory of local twistors, Int. J. Theor. Phys., 11, 31, (1974)
[109] Penrose, R.; MacCallum, MAH, Twistor theory: an approach to the quantization of fields and space-time, Phys. Rept., 6, 241, (1972)
[110] H. Friedrich, Twistor connection and normal conformal cartan connection, Gen. Rel. Grav.8 (1977) 303. · Zbl 0465.53039
[111] Dirac, PAM, Wave equations in conformal space, Annals Math., 37, 429, (1936) · Zbl 0014.08004
[112] Mack, G.; Salam, A., Finite component field representations of the conformal group, Annals Phys., 53, 174, (1969)
[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] Ferrara, S.; Gatto, R.; Grillo, AF, Conformal algebra in space-time and operator product expansion, Springer Tracts Mod. Phys., 67, 1, (1973)
[115] Ferrara, S.; Grillo, AF; Gatto, R., Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys., 76, 161, (1973)
[116] Cornalba, L.; Costa, MS; Penedones, J., Deep inelastic scattering in conformal QCD, JHEP, 03, 133, (2010) · Zbl 1271.81170
[117] S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev.D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE]. · Zbl 1271.81170
[118] Zuckerman, G., Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Annals Math., 106, 295, (1977) · Zbl 0384.22004
[119] J.C. Jantzen, Moduln mit einem höchsten Gewicht, Lect. Notes Math.750, Springer (1979). · Zbl 0426.17001
[120] Isachenkov, M.; Schomerus, V., Superintegrability of d-dimensional conformal blocks, Phys. Rev. Lett., 117, (2016)
[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] Schomerus, V.; Sobko, E.; Isachenkov, M., Harmony of spinning conformal blocks, JHEP, 03, 085, (2017) · Zbl 1377.81182
[123] Zamolodchikov, AB, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys., 96, 419, (1984)
[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. · Zbl 1342.81492
[127] Yamazaki, M., Comments on determinant formulas for general cfts, JHEP, 10, 035, (2016) · Zbl 1390.81546
[128] Brauer, R., Sur la multiplication des caractéristiques des groupes continus et semi-simples, C. R. Acad. Sci. Paris, 204, 1784, (1937) · Zbl 0016.29502
[129] Klimyk, AU, Multiplicities of weights of representations and multiplicities of representations of semisimple Lie algebras, Dokl. Akad. Nauk SSSR, 177, 1001, (1967) · Zbl 0239.17005
[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]. · Zbl 1359.81165
[131] P. Etingof and F. Latour, The dynamical yang-baxter equation, representation theory, and quantum integrable systems, Oxford Lecture Series in Mathematics and Its Applications29 (2005). · Zbl 1071.17012
[132] Arnaudon, D.; Buffenoir, E.; Ragoucy, E.; Roche, P., Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys., 44, 201, (1998) · Zbl 0973.81047
[133] M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP02 (2015) 151 [arXiv:1411.7351] [INSPIRE]. · Zbl 1388.53102
[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]. · Zbl 1380.81327
[135] W. Siegel, Embedding versus 6D twistors, arXiv:1204.5679 [INSPIRE]. · Zbl 1329.81324
[136] Elkhidir, E.; Karateev, D.; Serone, M., General three-point functions in 4D CFT, JHEP, 01, 133, (2015)
[137] Mack, G., Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys., 53, 155, (1977)
[138] P. Kravchuk and D. Simmons-Duffin, Counting Conformal Correlators, arXiv:1612.08987 [INSPIRE]. · Zbl 1378.81119
[139] Czech, B.; Lamprou, L.; McCandlish, S.; Mosk, B.; Sully, J., A stereoscopic look into the bulk, JHEP, 07, 129, (2016) · Zbl 1390.83101
[140] F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE]. · Zbl 1390.81501
[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]. · Zbl 1383.83024
[142] M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev.D 87 (2013)106004 [arXiv:1303.1111] [INSPIRE]. · Zbl 1342.81497
[143] Oshima, Y.; Yamazaki, M., Determinant formula for parabolic Verma modules of Lie superalgebras, J. Algebra, 495, 51, (2018) · Zbl 1425.17010
[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]. · Zbl 1111.81078
[146] C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, arXiv:1612.00809 [INSPIRE]. · Zbl 1390.81500
[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]. · Zbl 1388.81382
[148] C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, arXiv:1705.03025 [INSPIRE]. · Zbl 1387.81264
[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,. · Zbl 0911.53020
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