Abstract
We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-point function in any dimension in the limit of large scaling dimensions Δ0 of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite Δ0 as well as for large Δ0. We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.
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References
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].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
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].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
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].
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].
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].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal Limit for Conformal Blocks in d Dimensions, JHEP 08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
E.B. Saff and V. Totik, Lond. Math. Soc. 39 (1989) 487.
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press (2004).
M.G. Krein and A.A. Nudel’man, The Markov Moment Problem and Extremal Problems, Translations of Mathematical Monographs, Vol. 50 American Mathematical Society (1977).
T.J. Rivlin, The Chebyshev Polynomials, Wiley-Interscience publication (1974).
C.-M. Chang and Y.-H. Lin, Bootstrapping 2D CFTs in the Semiclassical Limit, arXiv:1510.02464 [INSPIRE].
S. Rychkov and P. Yvernay, Remarks on the Convergence Properties of the Conformal Block Expansion, Phys. Lett. B 753 (2016) 682 [arXiv:1510.08486] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and D. Poland, Conformal Blocks in the Large D Limit, JHEP 08 (2013) 107 [arXiv:1305.0004] [INSPIRE].
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ArXiv ePrint: 1510.08772
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Kim, H., Kravchuk, P. & Ooguri, H. Reflections on conformal spectra. J. High Energ. Phys. 2016, 184 (2016). https://doi.org/10.1007/JHEP04(2016)184
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DOI: https://doi.org/10.1007/JHEP04(2016)184