Abstract
In this paper, we use crossing symmetry and unitarity constraints to put a lower bound on the central charge of conformal field theories in large space-time dimensions D. Specifically, we work with the four-point function of identical scalars ϕ with scaling dimension ∆ϕ, and use a certain class of analytic functionals to show that the OPE coefficient squared \( {c}_{\phi \phi {T}^{\mu \nu}}^2 \) must be exponentially small in D. For this to hold, we need to make a certain assumption about the nature of the spectrum below 2∆ϕ. Our argument is robust and can be applied to any OPE coefficient squared \( {c}_{\phi \phi O}^2 \) with ∆O < 2∆ϕ. This suggests that conformal field theories in large dimensions (if they exist) must be exponentially close to generalized free field theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, Boulder, U.S.A. (2017), pg. 1, https://doi.org/10.1142/9789813149441_0001 [arXiv:1602.07982] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics, Springer Cham (2016), https://doi.org/10.1007/978-3-319-43626-5 [arXiv:1601.05000] [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].
S.M. Chester, Weizmann Lectures on the Numerical Conformal Bootstrap, arXiv:1907.05147 [INSPIRE].
A. Bissi, A. Sinha and X. Zhou, Selected topics in analytic conformal bootstrap: A guided journey, Phys. Rept. 991 (2022) 1 [arXiv:2202.08475] [INSPIRE].
A. Gadde and T. Sharma, Constraining conformal theories in large dimensions, JHEP 02 (2022) 035 [arXiv:2002.10147] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP 06 (2017) 076 [arXiv:1705.01357] [INSPIRE].
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].
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].
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].
P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part I. Cross-ratio space, JHEP 05 (2020) 137 [arXiv:2001.08778] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and D. Poland, Conformal Blocks in the Large D Limit, JHEP 08 (2013) 107 [arXiv:1305.0004] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, JHEP 05 (2021) 243 [arXiv:2008.04931] [INSPIRE].
Acknowledgments
We would like to thank the TIFR string theory group, in particular Gautam Mandal, Shiraz Minwalla, Onkar Parrikar, Sandip Trivedi for useful discussions. We would like to thank Adwait Gaikwad for collaboration in related projects. This work is supported by the Infosys Endowment for the study of the Quantum Structure of Spacetime. The work of A.G. is also supported by the SERB Ramanujan fellowship. We acknowledge support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4002. We would also like to acknowledge our debt to the people of India for their steady support to the study of the basic sciences.
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: 2301.04980
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
Gadde, A., Jagadale, M., Jain, S. et al. Bound on the central charge of CFTs in large dimension. J. High Energ. Phys. 2023, 146 (2023). https://doi.org/10.1007/JHEP05(2023)146
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)146