Abstract
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide two new closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. The other is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.
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].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
A.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.
O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012) 038 [Erratum ibid. 1210 (2012) 183] [arXiv:1207.0787] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, arXiv:1501.05315 [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
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].
A. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
P.H. Ginsparg, Applied conformal field theory, hep-th/9108028 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory. Springer, (1997).
P. Bowcock, Quasi-primary Fields and Associativity of Chiral Algebras, Nucl. Phys. B 356 (1991) 367 [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Properties of Partial Wave Amplitudes in Conformal Invariant Field Theories, Nuovo Cim. A 26 (1975) 226 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Conformal field theory and 2-D critical phenomena. 3. Conformal bootstrap and degenerate representations of conformal algebra, ITEP-90-31 (1990) [INSPIRE].
I. Runkel and G.M.T. Watts, A nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].
A. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
V.A. Fateev, A.V. Litvinov, A. Neveu and E. Onofri, Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks, J. Phys. A 42 (2009) 304011 [arXiv:0902.1331] [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
X. Yin, Partition Functions of Three-Dimensional Pure Gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [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 II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
E. Perlmutter, A universal feature of CFT Rényi entropy, JHEP 03 (2014) 117 [arXiv:1308.1083] [INSPIRE].
J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Rényi entropy, stationarity and entanglement of the conformal scalar, JHEP 03 (2015) 075 [arXiv:1407.7816] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10 (2014) 178 [arXiv:1407.6429] [INSPIRE].
B. Chen, J. Long and J.-j. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].
E. Hijano and P. Kraus, A new spin on entanglement entropy, JHEP 12 (2014) 041 [arXiv:1406.1804] [INSPIRE].
J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and W N conformal blocks, JHEP 07 (2015) 168 [arXiv:1412.7520] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum Entanglement of Localized Excited States at Finite Temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [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].
A.L. Fitzpatrick and J. Kaplan, AdS Field Theory from Conformal Field Theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [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].
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: 1502.07742
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
Perlmutter, E. Virasoro conformal blocks in closed form. J. High Energ. Phys. 2015, 88 (2015). https://doi.org/10.1007/JHEP08(2015)088
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)088