Abstract
We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h,\( \overline{h} \)) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h,\( \overline{h} \)) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current \( Ag\le \frac{\pi \left(c-1\right){r}^2}{24} \) is satisfied, where g is the twist gap over vacuum and A is the minimal “areal gap”, generalizing the minimal gap in dimension to (h′,\( \overline{h}^{\prime } \)) plane and \( r=\frac{4\sqrt{3}}{\pi}\simeq 2.21 \). We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.
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References
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys.B 270 (1986) 186 [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
S. Ganguly and S. Pal, Bounds on density of states and spectral gap in CFT2, arXiv:1905.12636 [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP01 (2019) 025 [arXiv:1810.01335] [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, JHEP08 (2019) 063 [arXiv:1905.02191] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
H. Maxfield, Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap, JHEP12 (2019) 003 [arXiv:1906.04416] [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].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, JHEP10 (2019) 038 [arXiv:1705.05825] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
G. Cuomo, A. de la Fuente, A. Monin, D. Pirtskhalava and R. Rattazzi, Rotating superfluids and spinning charged operators in conformal field theory, Phys. Rev.D 97 (2018) 045012 [arXiv:1711.02108] [INSPIRE].
G. Cuomo, Superfluids, vortices and spinning charged operators in 4d CFT, arXiv:1906.07283 [INSPIRE].
D. Banerjee, S. Chandrasekharan, D. Orlando and S. Reffert, Conformal dimensions in the large charge sectors at the O(4) Wilson-Fisher fixed point, Phys. Rev. Lett.123 (2019) 051603 [arXiv:1902.09542] [INSPIRE].
D. Orlando, S. Reffert and F. Sannino, A safe CFT at large charge, JHEP08 (2019) 164 [arXiv:1905.00026] [INSPIRE].
S. Favrod, D. Orlando and S. Reffert, The large-charge expansion for Schr¨odinger systems, JHEP12 (2018) 052 [arXiv:1809.06371] [INSPIRE].
S.M. Kravec and S. Pal, Nonrelativistic Conformal Field Theories in the Large Charge Sector, JHEP02 (2019) 008 [arXiv:1809.08188] [INSPIRE].
S.M. Kravec and S. Pal, The Spinful Large Charge Sector of Non-Relativistic CFTs: From Phonons to Vortex Crystals, JHEP05 (2019) 194 [arXiv:1904.05462] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP11 (2007) 019 [arXiv:0708.0672] [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, JHEP12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP11 (2017) 183 [arXiv:1706.04612] [INSPIRE].
A. Ingham, A tauberian theorem for partitions, Ann. Math.42 (1941) 1075.
M.A. Subhankulov, Tauberian theorems with remainder term, American Mathematical Society Translations: Series 2 , volume 26, Providence RI U.S.A. (1963), pp. 311–338.
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, JHEP10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
S. Pal, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, arXiv:1906.11223 [INSPIRE].
B. Michel, Universality in the OPE Coefficients of Holographic 2d CFTs, arXiv:1908.02873 [INSPIRE].
E. Dyer, A.L. Fitzpatrick and Y. Xin, Constraints on Flavored 2d CFT Partition Functions, JHEP02 (2018) 148 [arXiv:1709.01533] [INSPIRE].
S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal Dynamics of Heavy Operators in CFT2, arXiv:1912.00222 [INSPIRE].
J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP10 (2017) 136 [arXiv:1705.05855] [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].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev.D 98 (2018) 126015 [arXiv:1804.07924] [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, JHEP09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys.5 (2018) 022 [arXiv:1803.04938] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast Conformal Bootstrap and Constraints on 3d Gravity, JHEP05 (2019) 087 [arXiv:1903.06272] [INSPIRE].
N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT2 , JHEP05 (2018) 092 [arXiv:1707.07717] [INSPIRE].
P. Kraus and A. Sivaramakrishnan, Light-state Dominance from the Conformal Bootstrap, JHEP08 (2019) 013 [arXiv:1812.02226] [INSPIRE].
D. Mazáč, Analytic bounds and emergence of AdS2physics from the conformal bootstrap, JHEP04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazáč and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazáč and M.F. Paulos, The analytic functional bootstrap. Part I. 1D CFTs and 2D S-matrices, JHEP02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, A Basis of Analytic Functionals for CFTs in General Dimension, arXiv:1910.12855 [INSPIRE].
M.F. Paulos, Analytic Functional Bootstrap for CFTs in d > 1, arXiv:1910.08563 [INSPIRE].
D. Carmi and S. Caron-Huot, A Conformal Dispersion Relation: Correlations from Absorption, arXiv:1910.12123 [INSPIRE].
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Pal, S., Sun, Z. Tauberian-Cardy formula with spin. J. High Energ. Phys. 2020, 135 (2020). https://doi.org/10.1007/JHEP01(2020)135
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DOI: https://doi.org/10.1007/JHEP01(2020)135