Abstract
We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2, \( \mathbb{Z} \)) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2, \( \mathbb{Z} \)). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the PSL(2, \( \mathbb{Z} \)) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over PSL(2, \( \mathbb{Z} \)) in CFT2 has a natural AdS3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.
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].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [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].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics (2016) [arXiv:1601.05000] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
H. Rademacher, The fourier series and the functional equation of the absolute modular invariant J (τ ), Am. J. Math. 61 (1939) 237.
T.M. Apostol, Modular functions and Dirichlet series in number theory, vol. 41, Springer Science & Business Media (2012).
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
P. Kraus, Lectures on black holes and the AdS 3 /CFT 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
J.F. Duncan and I.B. Frenkel, Rademacher sums, Moonshine and Gravity, Commun. Num. Theor. Phys. 5 (2011) 849 [arXiv:0907.4529] [INSPIRE].
A. Castro, N. Lashkari and A. Maloney, A de Sitter Farey Tail, Phys. Rev. D 83 (2011) 124027 [arXiv:1103.4620] [INSPIRE].
C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
J. Manschot and G.W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
A. Castro, M.R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The Gravity Dual of the Ising Model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
M. Cheng, T. Gannon and G. Lockhart, in preparation.
M. Cheng, A Bit More Physics from A Bit More Number Theory, in proceedings of Strings 2016, Beijing, China, 1-5 August 2016.
V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere and F. Galli, Heavy-Heavy-Light-Light correlators in Liouville theory, arXiv:1705.08004 [INSPIRE].
M. Headrick, A. Maloney, E. Perlmutter and I.G. Zadeh, Rényi entropies, the analytic bootstrap and 3D quantum gravity at higher genus, JHEP 07 (2015) 059 [arXiv:1503.07111] [INSPIRE].
P. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media (2012).
M. Hogervorst and B.C. van Rees, Crossing Symmetry in Alpha Space, arXiv:1702.08471 [INSPIRE].
L. Alvarez-Gaumé, G. Sierra and C. Gomez, Topics in conformal field theory, CERN-TH-5540-89 (1989) [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Operator Algebra of Two-Dimensional Conformal Theories with Central Charge C ≤ 1, Phys. Lett. B 154 (1985) 291 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].
D. Friedan, Z.-a. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. B 151 (1985) 37 [INSPIRE].
Z.A. Qiu, Supersymmetry, Two-dimensional Critical Phenomena and the Tricritical Ising Model, Nucl. Phys. B 270 (1986) 205 [INSPIRE].
J. Fuchs and A. Klemm, The Computation of the Operator Algebra in Nondiagonal Conformal Field Theories, Annals Phys. 194 (1989) 303 [INSPIRE].
J. Fuchs, Operator Product Coefficients in Nondiagonal Conformal Field Theories, Phys. Rev. Lett. 62 (1989) 1705 [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].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, The ADE Classification of Minimal and A1(1) Conformal Invariant Theories, Commun. Math. Phys. 113 (1987) 1 [INSPIRE].
A. Kato, Classification of Modular Invariant Partition Functions in Two-dimensions, Mod. Phys. Lett. A 2 (1987) 585 [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [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].
A. Belin, C.A. Keller and A. Maloney, String Universality for Permutation Orbifolds, Phys. Rev. D 91 (2015) 106005 [arXiv:1412.7159] [INSPIRE].
F.M. Haehl and M. Rangamani, Permutation orbifolds and holography, JHEP 03 (2015) 163 [arXiv:1412.2759] [INSPIRE].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [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 and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Bootstrapping 2D CFTs in the Semiclassical Limit, JHEP 08 (2016) 056 [arXiv:1510.02464] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra, Proc. Conf., Oxford (1967), Elsevier (1970), pg. 329-358.
J.R. Goldman and L.H. Kauffman, Rational tangles, Adv. Appl. Math. 18 (1997) 300.
O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1609.02165
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Maloney, A., Maxfield, H. & Ng, G.S. A conformal block Farey tail. J. High Energ. Phys. 2017, 117 (2017). https://doi.org/10.1007/JHEP06(2017)117
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2017)117