Abstract
We study asymptotics of three point coefficients (light-light-heavy) and two point correlators in heavy states in unitary, compact 2D CFTs. We prove an upper and lower bound on such quantities using numerically assisted Tauberian techniques. We obtain an optimal upper bound on the spectrum of operators appearing with fixed spin from the OPE of two identical scalars. While all the CFTs obey this bound, rational CFTs come close to saturating it. This mimics the scenario of bounds on asymptotic density of states and thereby pronounces an universal feature in asymptotics of 2D CFTs. Next, we clarify the role of smearing in interpreting the asymptotic results pertaining to considerations of eigenstate thermalization in 2D CFTs. In the context of light-light-heavy three point coefficients, we find that the order one number in the bound is sensitive to how close the light operators are from the \( \frac{c}{32} \) threshold. In context of two point correlator in heavy state, we find the presence of an enigmatic regime which separates the AdS3 thermal physics and the BTZ black hole physics. Furthermore, we present some new numerical results on the behaviour of spherical conformal block.
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J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity and Random Matrices, JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [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].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
S. Ganguly and S. Pal, Bounds on the density of states and the spectral gap in CFT2, Phys. Rev. D 101 (2020) 106022 [arXiv:1905.12636] [INSPIRE].
S. Pal and Z. Sun, Tauberian-Cardy formula with spin, JHEP 01 (2020) 135 [arXiv:1910.07727] [INSPIRE].
B. Mukhametzhanov and S. Pal, Beurling-Selberg Extremization and Modular Bootstrap at High Energies, SciPost Phys. 8 (2020) 088 [arXiv:2003.14316] [INSPIRE].
S. Pal and Z. Sun, High Energy Modular Bootstrap, Global Symmetries and Defects, JHEP 08 (2020) 064 [arXiv:2004.12557] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
N. Lashkari, A. Dymarsky and H. Liu, Universality of Quantum Information in Chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].
D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE].
D. Das, S. Datta and S. Pal, Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks, Phys. Rev. D 98 (2018) 101901 [arXiv:1712.01842] [INSPIRE].
J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].
Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2, JHEP 07 (2020) 074 [arXiv:1912.00222] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Hawking from Catalan, JHEP 05 (2016) 069 [arXiv:1510.00014] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP 06 (2018) 123 [arXiv:1712.03464] [INSPIRE].
T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys. 7 (2019) 003 [arXiv:1903.03143] [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
N. Benjamin, E. Dyer, A.L. Fitzpatrick and Y. Xin, The Most Irrational Rational Theories, JHEP 04 (2019) 025 [arXiv:1812.07579] [INSPIRE].
N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Light-cone modular bootstrap and pure gravity, Phys. Rev. D 100 (2019) 066029 [arXiv:1906.04184] [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [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].
B. Michel, Universality in the OPE Coefficients of Holographic 2d CFTs, arXiv:1908.02873 [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
E. Dyer, A.L. Fitzpatrick and Y. Xin, Constraints on Flavored 2d CFT Partition Functions, JHEP 02 (2018) 148 [arXiv:1709.01533] [INSPIRE].
J.-B. Bae, S. Lee and J. Song, Modular Constraints on Superconformal Field Theories, JHEP 01 (2019) 209 [arXiv:1811.00976] [INSPIRE].
L.F. Alday and J.-B. Bae, Rademacher Expansions and the Spectrum of 2d CFT, JHEP 11 (2020) 134 [arXiv:2001.00022] [INSPIRE].
Y.-H. Lin and S.-H. Shao, Anomalies and Bounds on Charged Operators, Phys. Rev. D 100 (2019) 025013 [arXiv:1904.04833] [INSPIRE].
M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
J.-B. Bae, S. Lee and J. Song, Modular Constraints on Conformal Field Theories with Currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast Conformal Bootstrap and Constraints on 3d Gravity, JHEP 05 (2019) 087 [arXiv:1903.06272] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Twist gap and global symmetry in two dimensions, Phys. Rev. D 101 (2020) 106026 [arXiv:2003.02844] [INSPIRE].
N. Benjamin and Y.-H. Lin, Lessons from the Ramond sector, SciPost Phys. 9 (2020) 065 [arXiv:2005.02394] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, JHEP 12 (2020) 066 [arXiv:2006.02560] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
J. Kaidi and E. Perlmutter, Discreteness and integrality in Conformal Field Theory, JHEP 02 (2021) 064 [arXiv:2008.02190] [INSPIRE].
M.F. Paulos, Analytic functional bootstrap for CFTs in d > 1, JHEP 04 (2020) 093 [arXiv:1910.08563] [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, arXiv:1910.12855 [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, arXiv:2008.04931 [INSPIRE].
M. Shyani, Analyticity of replica correlators and Modular ETH, arXiv:2009.01246 [INSPIRE].
C. Cardona, Virasoro blocks at large exchange dimension, Nucl. Phys. B 963 (2021) 115284 [arXiv:2006.01237] [INSPIRE].
D. Das, S. Datta and M. Raman, Virasoro blocks and quasimodular forms, JHEP 11 (2020) 010 [arXiv:2007.10998] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP 01 (2019) 098 [arXiv:1810.11025] [INSPIRE].
S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, JHEP 07 (2019) 143 [arXiv:1904.00668] [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].
J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, JHEP 10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
S. Pal, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, JHEP 01 (2020) 023 [arXiv:1906.11223] [INSPIRE].
B. Mukhametzhanov and S. Pal, Extremal functions for high energy modular bootstrap, unpublished note.
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
Y. Kusuki, Large c Virasoro Blocks from Monodromy Method beyond Known Limits, JHEP 08 (2018) 161 [arXiv:1806.04352] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
E.E. Berdysheva, Two related extremal problems for entire functions of several variables, Math. Notes 66 (1999) 271.
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP 04 (2019) 018 [arXiv:1703.09805] [INSPIRE].
J.-B. Bae, K. Lee and S. Lee, Bootstrapping Pure Quantum Gravity in AdS3, arXiv:1610.05814 [INSPIRE].
S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ, JHEP 09 (2018) 150 [arXiv:1702.00423] [INSPIRE].
I. Esterlis, A.L. Fitzpatrick and D. Ramirez, Closure of the Operator Product Expansion in the Non-Unitary Bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].
Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N} \) = 4 superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
Y. Kusuki and M. Miyaji, Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block, JHEP 08 (2019) 063 [arXiv:1905.02191] [INSPIRE].
P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech. 1811 (2018) 113101 [arXiv:1805.05975] [INSPIRE].
C.-M. Chang, D.M. Ramirez and M. Rangamani, Spinning constraints on chaotic large c CFTs, JHEP 03 (2019) 068 [arXiv:1812.05585] [INSPIRE].
Y. Kusuki, New Properties of Large-c Conformal Blocks from Recursion Relation, JHEP 07 (2018) 010 [arXiv:1804.06171] [INSPIRE].
E.M. Brehm and D. Das, Aspects of the S transformation Bootstrap, J. Stat. Mech. 2005 (2020) 053103 [arXiv:1911.02309] [INSPIRE].
J. De Boer, R. Van Breukelen, S.F. Lokhande, K. Papadodimas and E. Verlinde, Probing typical black hole microstates, JHEP 01 (2020) 062 [arXiv:1901.08527] [INSPIRE].
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
A. Belin and J. de Boer, Random Statistics of OPE Coefficients and Euclidean Wormholes, arXiv:2006.05499 [INSPIRE].
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Das, D., Kusuki, Y. & Pal, S. Universality in asymptotic bounds and its saturation in 2D CFT. J. High Energ. Phys. 2021, 288 (2021). https://doi.org/10.1007/JHEP04(2021)288
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DOI: https://doi.org/10.1007/JHEP04(2021)288