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Chen, Bin; Liu, Reiko

The shadow formalism of Galilean \(\mathrm{CFT}_2\). (English) Zbl 07702039

J. High Energy Phys. 2023, No. 5, Paper No. 224, 77 p. (2023).
Summary: In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (\(\mathrm{GCFT}_2\)). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the \(\mathrm{GCFT}_2\) can be regarded as null defect in higher dimensional CFTs.

Cited in 6 Documents

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T20 Quantum field theory on curved space or space-time backgrounds

Keywords:

conformal and W symmetry; field theories in lower dimensions; scale and conformal symmetries; spacetime symmetries
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References:

[1] 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].
[2] A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [INSPIRE].
[3] Poland, D.; Rychkov, S.; Vichi, A., The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys., 91 (2019) · doi:10.1103/RevModPhys.91.015002
[4] Rattazzi, R.; Rychkov, VS; Tonni, E.; Vichi, A., Bounding scalar operator dimensions in 4D CFT, JHEP, 12, 031 (2008) · Zbl 1329.81324 · doi:10.1088/1126-6708/2008/12/031
[5] El-Showk, S., Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D, 86 (2012) · doi:10.1103/PhysRevD.86.025022
[6] El-Showk, S., Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys., 157, 869 (2014) · Zbl 1310.82013 · doi:10.1007/s10955-014-1042-7
[7] Komargodski, Z.; Zhiboedov, A., Convexity and Liberation at Large Spin, JHEP, 11, 140 (2013) · doi:10.1007/JHEP11(2013)140
[8] Fitzpatrick, AL; Kaplan, J.; Poland, D.; Simmons-Duffin, D., The Analytic Bootstrap and AdS Superhorizon Locality, JHEP, 12, 004 (2013) · Zbl 1342.83239 · doi:10.1007/JHEP12(2013)004
[9] Kaviraj, A.; Sen, K.; Sinha, A., Analytic bootstrap at large spin, JHEP, 11, 083 (2015) · Zbl 1390.81703 · doi:10.1007/JHEP11(2015)083
[10] Kaviraj, A.; Sen, K.; Sinha, A., Universal anomalous dimensions at large spin and large twist, JHEP, 07, 026 (2015) · Zbl 1388.83275 · doi:10.1007/JHEP07(2015)026
[11] Alday, LF, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett., 119 (2017) · doi:10.1103/PhysRevLett.119.111601
[12] Caron-Huot, S., Analyticity in Spin in Conformal Theories, JHEP, 09, 078 (2017) · Zbl 1382.81173 · doi:10.1007/JHEP09(2017)078
[13] Simmons-Duffin, D.; Stanford, D.; Witten, E., A spacetime derivation of the Lorentzian OPE inversion formula, JHEP, 07, 085 (2018) · Zbl 1395.81246 · doi:10.1007/JHEP07(2018)085
[14] Mazáč, D., Analytic bounds and emergence of AdS_2physics from the conformal bootstrap, JHEP, 04, 146 (2017) · Zbl 1378.81121 · doi:10.1007/JHEP04(2017)146
[15] Mazáč, D.; Paulos, MF, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP, 02, 162 (2019) · doi:10.1007/JHEP02(2019)162
[16] Mazáč, D.; Paulos, MF, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP, 02, 163 (2019) · doi:10.1007/JHEP02(2019)163
[17] Mazáč, D.; Rastelli, L.; Zhou, X., A basis of analytic functionals for CFTs in general dimension, JHEP, 08, 140 (2021) · Zbl 1469.81064 · doi:10.1007/JHEP08(2021)140
[18] Paulos, MF, Analytic functional bootstrap for CFTs in d > 1, JHEP, 04, 093 (2020) · Zbl 1436.81119 · doi:10.1007/JHEP04(2020)093
[19] Bissi, A.; Sinha, A.; Zhou, X., Selected topics in analytic conformal bootstrap: A guided journey, Phys. Rept., 991, 1 (2022) · Zbl 1516.81165 · doi:10.1016/j.physrep.2022.09.004
[20] Heemskerk, I.; Penedones, J.; Polchinski, J.; Sully, J., Holography from Conformal Field Theory, JHEP, 10, 079 (2009) · doi:10.1088/1126-6708/2009/10/079
[21] Penedones, J., Writing CFT correlation functions as AdS scattering amplitudes, JHEP, 03, 025 (2011) · Zbl 1301.81154 · doi:10.1007/JHEP03(2011)025
[22] Fitzpatrick, AL, A Natural Language for AdS/CFT Correlators, JHEP, 11, 095 (2011) · Zbl 1306.81225 · doi:10.1007/JHEP11(2011)095
[23] Alday, LF; Bissi, A., Unitarity and positivity constraints for CFT at large central charge, JHEP, 07, 044 (2017) · Zbl 1380.81281 · doi:10.1007/JHEP07(2017)044
[24] M.F. Paulos et al., The S-matrix bootstrap. Part I: QFT in AdS, JHEP11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
[25] Paulos, MF, The S-matrix bootstrap II: two dimensional amplitudes, JHEP, 11, 143 (2017) · Zbl 1383.81331 · doi:10.1007/JHEP11(2017)143
[26] M.F. Paulos et al., The S-matrix bootstrap. Part III: higher dimensional amplitudes, JHEP12 (2019) 040 [arXiv:1708.06765] [INSPIRE]. · Zbl 1431.81162
[27] Homrich, A., The S-matrix Bootstrap IV: Multiple Amplitudes, JHEP, 11, 076 (2019) · Zbl 1429.81072 · doi:10.1007/JHEP11(2019)076
[28] Henkel, M., Schrodinger invariance in strongly anisotropic critical systems, J. Statist. Phys., 75, 1023 (1994) · Zbl 0828.60095 · doi:10.1007/BF02186756
[29] Nishida, Y.; Son, DT, Nonrelativistic conformal field theories, Phys. Rev. D, 76 (2007) · doi:10.1103/PhysRevD.76.086004
[30] Goldberger, WD; Khandker, ZU; Prabhu, S., OPE convergence in non-relativistic conformal field theories, JHEP, 12, 048 (2015) · Zbl 1388.81042
[31] Golkar, S.; Son, DT, Operator Product Expansion and Conservation Laws in Non-Relativistic Conformal Field Theories, JHEP, 12, 063 (2014) · doi:10.1007/JHEP12(2014)063
[32] Kravec, SM; Pal, S., Nonrelativistic Conformal Field Theories in the Large Charge Sector, JHEP, 02, 008 (2019) · Zbl 1411.81183 · doi:10.1007/JHEP02(2019)008
[33] Karananas, GK; Monin, A., More on the operator-state map in nonrelativistic CFTs, Phys. Rev. D, 105 (2022) · doi:10.1103/PhysRevD.105.065008
[34] Shimada, H.; Shimada, H., Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance, JHEP, 10, 030 (2021) · Zbl 1476.81062 · doi:10.1007/JHEP10(2021)030
[35] Bagchi, A.; Chakrabortty, J.; Mehra, A., Galilean Field Theories and Conformal Structure, JHEP, 04, 144 (2018) · Zbl 1390.81480 · doi:10.1007/JHEP04(2018)144
[36] Bagchi, A.; Mehra, A.; Nandi, P., Field Theories with Conformal Carrollian Symmetry, JHEP, 05, 108 (2019) · Zbl 1416.81134 · doi:10.1007/JHEP05(2019)108
[37] Bagchi, A.; Basu, R.; Mehra, A.; Nandi, P., Field Theories on Null Manifolds, JHEP, 02, 141 (2020) · Zbl 1435.81114 · doi:10.1007/JHEP02(2020)141
[38] Banerjee, K., Interacting Conformal Carrollian Theories: Cues from Electrodynamics, Phys. Rev. D, 103 (2021) · doi:10.1103/PhysRevD.103.105001
[39] Chen, B.; Liu, R.; Zheng, Y-F, On Higher-dimensional Carrollian and Galilean Conformal Field Theories, SciPost Phys., 14, 088 (2023) · Zbl 07902431 · doi:10.21468/SciPostPhys.14.5.088
[40] Hofman, DM; Strominger, A., Chiral Scale and Conformal Invariance in 2D Quantum Field Theory, Phys. Rev. Lett., 107 (2011) · doi:10.1103/PhysRevLett.107.161601
[41] Detournay, S.; Hartman, T.; Hofman, DM, Warped Conformal Field Theory, Phys. Rev. D, 86 (2012) · doi:10.1103/PhysRevD.86.124018
[42] Chen, B.; Hao, P-X; Liu, Y-J, Supersymmetric Warped Conformal Field Theory, Phys. Rev. D, 102 (2020) · doi:10.1103/PhysRevD.102.065016
[43] Chen, B.; Hao, P-X; Yu, Z-F, 2d Galilean Field Theories with Anisotropic Scaling, Phys. Rev. D, 101 (2020) · doi:10.1103/PhysRevD.101.066029
[44] Bagchi, A.; Gopakumar, R., Galilean Conformal Algebras and AdS/CFT, JHEP, 07, 037 (2009) · doi:10.1088/1126-6708/2009/07/037
[45] Bagchi, A.; Fareghbal, R., BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP, 10, 092 (2012) · doi:10.1007/JHEP10(2012)092
[46] Barnich, G., Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP, 10, 095 (2012) · Zbl 1397.83210 · doi:10.1007/JHEP10(2012)095
[47] Bagchi, A.; Detournay, S.; Fareghbal, R.; Simón, J., Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.141302
[48] Bagchi, A.; Basu, R.; Grumiller, D.; Riegler, M., Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett., 114 (2015) · doi:10.1103/PhysRevLett.114.111602
[49] Jiang, H.; Song, W.; Wen, Q., Entanglement Entropy in Flat Holography, JHEP, 07, 142 (2017) · Zbl 1380.83148 · doi:10.1007/JHEP07(2017)142
[50] Hijano, E., Semi-classical BMS_3blocks and flat holography, JHEP, 10, 044 (2018) · Zbl 1402.81225 · doi:10.1007/JHEP10(2018)044
[51] Hijano, E., Flat space physics from AdS/CFT, JHEP, 07, 132 (2019) · Zbl 1418.81093 · doi:10.1007/JHEP07(2019)132
[52] Apolo, L.; Jiang, H.; Song, W.; Zhong, Y., Swing surfaces and holographic entanglement beyond AdS/CFT, JHEP, 12, 064 (2020) · doi:10.1007/JHEP12(2020)064
[53] Apolo, L.; Jiang, H.; Song, W.; Zhong, Y., Modular Hamiltonians in flat holography and (W)AdS/WCFT, JHEP, 09, 033 (2020) · Zbl 1454.83110 · doi:10.1007/JHEP09(2020)033
[54] Chen, B.; Hao, P-X; Liu, R.; Yu, Z-F, On Galilean conformal bootstrap, JHEP, 06, 112 (2021) · Zbl 1466.81095 · doi:10.1007/JHEP06(2021)112
[55] Chen, B.; Hao, P-X; Liu, R.; Yu, Z-F, On Galilean conformal bootstrap. Part II. ξ = 0 sector, JHEP, 12, 019 (2022) · Zbl 1536.81181 · doi:10.1007/JHEP12(2022)019
[56] Hogervorst, M.; Paulos, M.; Vichi, A., The ABC (in any D) of Logarithmic CFT, JHEP, 10, 201 (2017) · Zbl 1383.81230 · doi:10.1007/JHEP10(2017)201
[57] Maldacena, J.; Stanford, D., Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D, 94 (2016) · doi:10.1103/PhysRevD.94.106002
[58] Polchinski, J.; Rosenhaus, V., The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP, 04, 001 (2016) · Zbl 1388.81067 · doi:10.1007/JHEP04(2016)001
[59] Murugan, J.; Stanford, D.; Witten, E., More on Supersymmetric and 2d Analogs of the SYK Model, JHEP, 08, 146 (2017) · Zbl 1381.81121 · doi:10.1007/JHEP08(2017)146
[60] S. Ferrara, A.F. Grillo and G. Parisi, Nonequivalence between conformal covariant wilson expansion in euclidean and minkowski space, Lett. Nuovo Cim.5S2 (1972) 147 [INSPIRE].
[61] S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim.4S2 (1972) 115 [INSPIRE].
[62] S. Ferrara and G. Parisi, Conformal covariant correlation functions, Nucl. Phys. B42 (1972) 281 [INSPIRE].
[63] S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B49 (1972) 77 [INSPIRE].
[64] G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys.55 (1977) 1 [INSPIRE]. · Zbl 0352.22012
[65] G. Mack, Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory, Commun. Math. Phys.53 (1977) 155 [INSPIRE].
[66] V.K. Dobrev et al., On the Clebsch-Gordan Expansion for the Lorentz Group in n Dimensions, Rept. Math. Phys.9 (1976) 219 [INSPIRE]. · Zbl 0344.22013
[67] V.K. Dobrev et al., Harmonic Analysis. On the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, vol. 63, Springer (1977) [doi:10.1007/BFb0009678] [INSPIRE]. · Zbl 0407.43010
[68] Osborn, H., Conformal Blocks for Arbitrary Spins in Two Dimensions, Phys. Lett. B, 718, 169 (2012) · doi:10.1016/j.physletb.2012.09.045
[69] Simmons-Duffin, D., Projectors, Shadows, and Conformal Blocks, JHEP, 04, 146 (2014) · Zbl 1333.83125 · doi:10.1007/JHEP04(2014)146
[70] Karateev, D.; Kravchuk, P.; Simmons-Duffin, D., Weight Shifting Operators and Conformal Blocks, JHEP, 02, 081 (2018) · Zbl 1387.81323 · doi:10.1007/JHEP02(2018)081
[71] Karateev, D.; Kravchuk, P.; Simmons-Duffin, D., Harmonic Analysis and Mean Field Theory, JHEP, 10, 217 (2019) · Zbl 1427.81137 · doi:10.1007/JHEP10(2019)217
[72] Kravchuk, P.; Simmons-Duffin, D., Light-ray operators in conformal field theory, JHEP, 11, 102 (2018) · Zbl 1404.81234 · doi:10.1007/JHEP11(2018)102
[73] Liu, J.; Perlmutter, E.; Rosenhaus, V.; Simmons-Duffin, D., d-dimensional SYK, AdS Loops, and 6j Symbols, JHEP, 03, 052 (2019) · Zbl 1414.81211 · doi:10.1007/JHEP03(2019)052
[74] Duval, C.; Gibbons, GW; Horvathy, PA, Conformal Carroll groups, J. Phys. A, 47 (2014) · Zbl 1297.83028 · doi:10.1088/1751-8113/47/33/335204
[75] Duval, C.; Gibbons, GW; Horvathy, PA, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav., 31 (2014) · Zbl 1291.83084 · doi:10.1088/0264-9381/31/9/092001
[76] J. Figueroa-O’Farrill, R. Grassie and S. Prohazka, Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes, JHEP08 (2019) 119 [arXiv:1905.00034] [INSPIRE]. · Zbl 1421.83031
[77] J. Figueroa-O’Farrill, On the intrinsic torsion of spacetime structures, arXiv:2009.01948 [INSPIRE]. · Zbl 0864.17007
[78] J. Figueroa-O’Farrill, E. Have, S. Prohazka and J. Salzer, Carrollian and celestial spaces at infinity, JHEP09 (2022) 007 [arXiv:2112.03319] [INSPIRE]. · Zbl 1531.83127
[79] J. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, American Mathematical Society (2021).
[80] Penedones, J.; Trevisani, E.; Yamazaki, M., Recursion Relations for Conformal Blocks, JHEP, 09, 070 (2016) · Zbl 1390.81533 · doi:10.1007/JHEP09(2016)070
[81] Yamazaki, M., Comments on Determinant Formulas for General CFTs, JHEP, 10, 035 (2016) · Zbl 1390.81546 · doi:10.1007/JHEP10(2016)035
[82] Pasterski, S.; Puhm, A.; Trevisani, E., Celestial diamonds: conformal multiplets in celestial CFT, JHEP, 11, 072 (2021) · Zbl 1521.81324 · doi:10.1007/JHEP11(2021)072
[83] Donnay, L.; Fiorucci, A.; Herfray, Y.; Ruzziconi, R., Carrollian Perspective on Celestial Holography, Phys. Rev. Lett., 129 (2022) · doi:10.1103/PhysRevLett.129.071602
[84] Bagchi, A.; Banerjee, S.; Basu, R.; Dutta, S., Scattering Amplitudes: Celestial and Carrollian, Phys. Rev. Lett., 128 (2022) · doi:10.1103/PhysRevLett.128.241601
[85] E.P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals Math.40 (1939) 149 [INSPIRE]. · JFM 65.1129.01
[86] V. Bargmann and E.P. Wigner, Group theoretical discussion of relativistic wave equations, in Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics, pp. 82-94, Springer (1997) [doi:10.1007/978-3-662-09203-3_6]. · Zbl 0030.42306
[87] G.W. Mackey, Unitary representations of group extensions. I, Acta Math.99 (1958) 265. · Zbl 0082.11301
[88] Bekaert, X.; Boulanger, N., The unitary representations of the Poincaré group in any spacetime dimension, SciPost Phys. Lect. Notes, 30, 1 (2021)
[89] Campoleoni, A.; Gonzalez, HA; Oblak, B.; Riegler, M., BMS Modules in Three Dimensions, Int. J. Mod. Phys. A, 31, 1650068 (2016) · Zbl 1339.81063 · doi:10.1142/S0217751X16500688
[90] G. Folland, A Course in Abstract Harmonic Analysis, Textbooks in Mathematics, CRC Press (2016). · Zbl 1342.43001
[91] A.W. Knapp and E.M. Stein, Interwining operators for semisimple groups, Annals Math.93 (1971) 489. · Zbl 0257.22015
[92] A. Kleppner and R. Lipsman, The plancherel formula for group extensions, Annales Sci. Ecole Norm. Sup.5 (1972) 459. · Zbl 0239.43003
[93] A. Kleppner and R. Lipsman, The plancherel formula for group extensions II, Annales Sci. Ecole Norm. Sup.6 (1973) 103. · Zbl 0258.43001
[94] R.F. Streater and A.S. Wightman, PCT, spin and statistics, and all that, Princeton University Press (2016). · Zbl 0704.53058
[95] D. Simmons-Duffin, Conformal Field Theory in Lorentzian Signature, https://gitlab.com/davidsd/lorentzian-cft-notes.
[96] Gillioz, M., Momentum-space conformal blocks on the light cone, JHEP, 10, 125 (2018) · Zbl 1402.83089 · doi:10.1007/JHEP10(2018)125
[97] Czech, B., A Stereoscopic Look into the Bulk, JHEP, 07, 129 (2016) · Zbl 1390.83101 · doi:10.1007/JHEP07(2016)129
[98] Gadde, A., Conformal constraints on defects, JHEP, 01, 038 (2020) · Zbl 1434.81101 · doi:10.1007/JHEP01(2020)038
[99] A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS Bootstrap, Class. Quant. Grav.34 (2017) 174002 [arXiv:1705.05890] [INSPIRE]. · Zbl 1372.81134
[100] A. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].
[101] M. D’Eramo, G. Parisi and L. Peliti, Theoretical predictions for critical exponents at the λ-point of Bose liquids, Lett. Nuovo Cim.2 (1971) 878 [INSPIRE].
[102] Merbis, W.; Riegler, M., Geometric actions and flat space holography, JHEP, 02, 125 (2020) · Zbl 1435.83113 · doi:10.1007/JHEP02(2020)125
[103] Ammon, M., Semi-classical BMS-blocks from the oscillator construction, JHEP, 04, 155 (2021) · Zbl 1462.83047 · doi:10.1007/JHEP04(2021)155
[104] A. Bagchi, P. Nandi, A. Saha and Zodinmawia, BMS Modular Diaries: Torus one-point function, JHEP11 (2020) 065 [arXiv:2007.11713] [INSPIRE]. · Zbl 1456.81350
[105] Lodato, I.; W., Merbis and Zodinmawia, Supersymmetric Galilean conformal blocks, JHEP, 09, 086 (2018) · Zbl 1398.81246 · doi:10.1007/JHEP09(2018)086
[106] Hao, P-X; Song, W.; Xie, X.; Zhong, Y., BMS-invariant free scalar model, Phys. Rev. D, 105 (2022) · doi:10.1103/PhysRevD.105.125005
[107] Hogervorst, M.; van Rees, BC, Crossing symmetry in alpha space, JHEP, 11, 193 (2017) · Zbl 1383.81231 · doi:10.1007/JHEP11(2017)193
[108] M. Hogervorst, Crossing Kernels for Boundary and Crosscap CFTs, arXiv:1703.08159 [INSPIRE].
[109] Rutter, D.; Van Rees, BC, Applications of Alpha Space, JHEP, 12, 048 (2020) · Zbl 1457.81107 · doi:10.1007/JHEP12(2020)048
[110] Rajabpour, MA, Conformal symmetry in non-local field theories, JHEP, 06, 076 (2011) · Zbl 1298.81149 · doi:10.1007/JHEP06(2011)076
[111] Paulos, MF; Rychkov, S.; van Rees, BC; Zan, B., Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B, 902, 246 (2016) · Zbl 1332.82017 · doi:10.1016/j.nuclphysb.2015.10.018
[112] Bagchi, A.; Banerjee, A.; Chakrabortty, S.; Parekh, P., Inhomogeneous Tensionless Superstrings, JHEP, 02, 065 (2018) · Zbl 1387.83076 · doi:10.1007/JHEP02(2018)065
[113] A. Schild, Classical Null Strings, Phys. Rev. D16 (1977) 1722 [INSPIRE].
[114] Isberg, J.; Lindstrom, U.; Sundborg, B.; Theodoridis, G., Classical and quantized tensionless strings, Nucl. Phys. B, 411, 122 (1994) · Zbl 1049.81596 · doi:10.1016/0550-3213(94)90056-6
[115] Bagchi, A.; Chakrabortty, S.; Parekh, P., Tensionless Strings from Worldsheet Symmetries, JHEP, 01, 158 (2016) · Zbl 1388.81476 · doi:10.1007/JHEP01(2016)158
[116] Mason, L.; Skinner, D., Ambitwistor strings and the scattering equations, JHEP, 07, 048 (2014) · doi:10.1007/JHEP07(2014)048
[117] Casali, E.; Tourkine, P., On the null origin of the ambitwistor string, JHEP, 11, 036 (2016) · Zbl 1390.81493 · doi:10.1007/JHEP11(2016)036
[118] E. Casali, Y. Herfray and P. Tourkine, The complex null string, Galilean conformal algebra and scattering equations, JHEP10 (2017) 164 [arXiv:1707.09900] [INSPIRE]. · Zbl 1383.83163
[119] P. Rodríguez, D. Tempo and R. Troncoso, Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite \(\sqrt{T\overline{T}}\) deformations, JHEP11 (2021) 133 [arXiv:2106.09750] [INSPIRE]. · Zbl 1521.81330
[120] Witten, E., Anti-de Sitter space and holography, Adv. Theor. Math. Phys., 2, 253 (1998) · Zbl 0914.53048 · doi:10.4310/ATMP.1998.v2.n2.a2
[121] F. Bruhat, Sur les representations induites des groupes de lie, Bull. Soc. Math. France84 (1956) 97. · Zbl 0074.10303
[122] I. Gelfand, M. Graev and N. Vilenkin, Generalized Functions, Volume 5, AMS Chelsea Publishing, American Mathematical Society (2016). · Zbl 1339.01005
[123] Setare, MR; Kamali, V., Galilean Conformal Algebra in Semi-Infinite Space, Int. J. Mod. Phys. A, 27, 1250044 (2012) · Zbl 1247.81435 · doi:10.1142/S0217751X12500443
[124] I. Gelfand and G. Shilov, Generalized Functions, Volume 1, Elsevier Science (2014).
[125] R. Howe and E. Tan, Non-abelian Harmonic Analysis: Applications of SL(2, R), Universitext, Springer-Verlag (1992) [doi:10.1007/978-1-4613-9200-2]. · Zbl 0768.43001
[126] Bulycheva, K., A note on the SYK model with complex fermions, JHEP, 12, 069 (2017) · Zbl 1383.81188 · doi:10.1007/JHEP12(2017)069
[127] A. Kitaev, Notes on \(\overset{\sim }{\text{SL}}\left(2,\mathbb{R}\right)\) representations, arXiv:1711.08169 [INSPIRE].
[128] Z. Sun, A note on the representations of SO(1, d + 1), arXiv:2111.04591 [INSPIRE].
[129] J.-F. Fortin, W.-J. Ma and W. Skiba, All Global One- and Two-Dimensional Higher-Point Conformal Blocks, arXiv:2009.07674 [INSPIRE].
[130] F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE]. · Zbl 1097.81735
[131] J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B240 (1984) 514 [INSPIRE].
[132] D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B455 (1995) 522 [cond-mat/9505127] [INSPIRE]. · Zbl 0925.81295
[133] Liendo, P.; Rastelli, L.; van Rees, BC, The Bootstrap Program for Boundary CFT_d, JHEP, 07, 113 (2013) · Zbl 1342.81504 · doi:10.1007/JHEP07(2013)113
[134] Gaiotto, D.; Mazáč, D.; Paulos, MF, Bootstrapping the 3d Ising twist defect, JHEP, 03, 100 (2014) · doi:10.1007/JHEP03(2014)100
[135] Gliozzi, F.; Liendo, P.; Meineri, M.; Rago, A., Boundary and Interface CFTs from the Conformal Bootstrap, JHEP, 05, 036 (2015) · Zbl 1388.81150 · doi:10.1007/JHEP05(2015)036
[136] Billò, M.; Gonçalves, V.; Lauria, E.; Meineri, M., Defects in conformal field theory, JHEP, 04, 091 (2016) · Zbl 1388.81029
[137] Rastelli, L.; Zhou, X., The Mellin Formalism for Boundary CFT_d, JHEP, 10, 146 (2017) · Zbl 1383.81254 · doi:10.1007/JHEP10(2017)146
[138] Mazáč, D.; Rastelli, L.; Zhou, X., An analytic approach to BCFT_d, JHEP, 12, 004 (2019) · Zbl 1431.81139 · doi:10.1007/JHEP12(2019)004
[139] Kaviraj, A.; Paulos, MF, The Functional Bootstrap for Boundary CFT, JHEP, 04, 135 (2020) · Zbl 1436.81114 · doi:10.1007/JHEP04(2020)135
[140] J.L. Cardy, Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B275 (1986) 200 [INSPIRE]. · Zbl 0689.17016
[141] N. Ishibashi, The Boundary and Crosscap States in Conformal Field Theories, Mod. Phys. Lett. A4 (1989) 251 [INSPIRE].
[142] J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B324 (1989) 581 [INSPIRE].
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