Abstract
In this work, we continue our work on two dimensional Galilean conformal field theory (GCFT2). Our previous work (2011.11092) focused on the ξ ≠ 0 sector, here we investigate the more subtle ξ = 0 sector to complete the discussion. The case ξ = 0 is degenerate since there emerge interesting null states in a general ξ = 0 boost multiplet. We specify these null states and work out the resulting selection rules. Then, we compute the ξ = 0 global GCA blocks and find that they can be written as a linear combination of several building blocks, each of which can be obtained from a sl(2, ℝ) Casimir equation. These building blocks allow us to give an Euclidean inversion formula as well. As a consistency check, we study 4-point functions of certain vertex operators in the BMS free scalar theory. In this case, the ξ = 0 sector is the only allowable sector in the propagating channel. We find that the direct expansion of the 4-point function reproduces the global GCA block and is consistent with the inversion formula.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
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].
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].
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].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
D. Mazáč, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazáč and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
D. Mazáč and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, A basis of analytic functionals for CFTs in general dimension, JHEP 08 (2021) 140 [arXiv:1910.12855] [INSPIRE].
M.F. Paulos, Analytic functional bootstrap for CFTs in d > 1, JHEP 04 (2020) 093 [arXiv:1910.08563] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
L.F. Alday and A. Bissi, Unitarity and positivity constraints for CFT at large central charge, JHEP 07 (2017) 044 [arXiv:1606.09593] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part III: higher dimensional amplitudes, JHEP 12 (2019) 040 [arXiv:1708.06765] [INSPIRE].
A. Homrich, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix Bootstrap IV: Multiple Amplitudes, JHEP 11 (2019) 076 [arXiv:1905.06905] [INSPIRE].
B. Chen, P.-X. Hao, R. Liu and Z.-F. Yu, On Galilean conformal bootstrap, JHEP 06 (2021) 112 [arXiv:2011.11092] [INSPIRE].
A. Bagchi, M. Gary and Zodinmawia, Bondi-Metzner-Sachs bootstrap, Phys. Rev. D 96 (2017) 025007 [arXiv:1612.01730] [INSPIRE].
A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS Bootstrap, Class. Quant. Grav. 34 (2017) 174002 [arXiv:1705.05890] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
P.-x. Hao, W. Song, X. Xie and Y. Zhong, BMS-invariant free scalar model, Phys. Rev. D 105 (2022) 125005 [arXiv:2111.04701] [INSPIRE].
A. Bagchi and I. Mandal, On Representations and Correlation Functions of Galilean Conformal Algebras, Phys. Lett. B 675 (2009) 393 [arXiv:0903.4524] [INSPIRE].
J. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, American Mathematical Society (2021).
J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
M. Yamazaki, Comments on Determinant Formulas for General CFTs, JHEP 10 (2016) 035 [arXiv:1601.04072] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Celestial diamonds: conformal multiplets in celestial CFT, JHEP 11 (2021) 072 [arXiv:2105.03516] [INSPIRE].
V.G. Kac, Contravariant Form for Infinite Dimensional Lie Algebras and Superalgebras, in 7th International Group Theory Colloquium: The Integrative Conference on Group Theory and Mathematical Physics, (1978), pp 441–445.
B.L. Feigin and D.B. Fuks, Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra, Funct. Anal. Appl. 16 (1982) 114 [INSPIRE].
M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP 10 (2017) 201 [arXiv:1605.03959] [INSPIRE].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic Analysis and Mean Field Theory, JHEP 10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
A. Bagchi and R. Fareghbal, BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].
B. Chen and R. Liu, The Shadow Formalism of Galilean CFT2, arXiv:2203.10490 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2207.01474
Rights and permissions
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.
About this article
Cite this article
Chen, B., Hao, Px., Liu, R. et al. On Galilean conformal bootstrap. Part II. ξ = 0 sector. J. High Energ. Phys. 2022, 19 (2022). https://doi.org/10.1007/JHEP12(2022)019
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2022)019