Abstract
We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [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].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049 [arXiv:1504.05943] [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].
P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP 10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
K.B. Alkalaev, Many-point classical conformal blocks and geodesic networks on the hyperbolic plane, JHEP 12 (2016) 070 [arXiv:1610.06717] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles, arXiv:1609.07153 [INSPIRE].
O. Hulík, T. Procházka and J. Raeymaekers, Multi-centered AdS3 solutions from Virasoro conformal blocks, JHEP 03 (2017) 129 [arXiv:1612.03879] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [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].
P. Menotti, Riemann-Hilbert treatment of Liouville theory on the torus, J. Phys. A 44 (2011) 115403 [arXiv:1010.4946] [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].
K.B. Alkalaev and V.A. Belavin, Monodromic vs geodesic computation of Virasoro classical conformal blocks, Nucl. Phys. B 904 (2016) 367 [arXiv:1510.06685] [INSPIRE].
S. Datta, J.R. David and S.P. Kumar, Conformal perturbation theory and higher spin entanglement entropy on the torus, JHEP 04 (2015) 041 [arXiv:1412.3946] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, arXiv:1609.01287 [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Holographic interpretation of 1-point toroidal block in the semiclassical limit, JHEP 06 (2016) 183 [arXiv:1603.08440] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, From global to heavy-light: 5-point conformal blocks, JHEP 03 (2016) 184 [arXiv:1512.07627] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
A. Zamolodchikov, Conformal Symmetry in Two-dimensional Space: Recursion Representation of the Conformal Block, Teor. Mat. Fiz. 73 (1987) 103.
A. Bhatta, P. Raman and N.V. Suryanarayana, Holographic Conformal Partial Waves as Gravitational Open Wilson Networks, JHEP 06 (2016) 119 [arXiv:1602.02962] [INSPIRE].
V.A. Fateev and S. Ribault, The Large central charge limit of conformal blocks, JHEP 02 (2012) 001 [arXiv:1109.6764] [INSPIRE].
H. Poghosyan, R. Poghossian and G. Sarkissian, The light asymptotic limit of conformal blocks in Toda field theory, JHEP 05 (2016) 087 [arXiv:1602.04829] [INSPIRE].
P. Kraus and A. Maloney, A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots, arXiv:1608.03284 [INSPIRE].
E. Inonu and E.P. Wigner, On the Contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].
A.O. Barut and L. Girardello, New ‘coherent’ states associated with noncompact groups, Commun. Math. Phys. 21 (1971) 41 [INSPIRE].
A. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Zh. Eksp. Teor. Fiz. 90 (1986) 1808.
M. Piatek, Classical torus conformal block, \( \mathcal{N}={2}^{\ast } \) twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Classical irregular block, \( \mathcal{N}=2 \) pure gauge theory and Mathieu equation, JHEP 12 (2014) 032 [arXiv:1407.0305] [INSPIRE].
C. Rim and H. Zhang, Classical Virasoro irregular conformal block, JHEP 07 (2015) 163 [arXiv:1504.07910] [INSPIRE].
V.G. Kac, Contravariant form for infinite dimensional Lie algebras and superalgebras, in Group Theoretical Methods in Physics. Lecture Notes in Physics. Volume 94, W. Beiglböck, A. Böhm and E. Takasugi eds., Springer, Berlin and Heidelberg Germany (1979).
B.L. Feigin and D.B. Fuks, Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra, Funct. Anal. Appl. 16 (1982) 114 [INSPIRE].
G.E. Andrews, The Theory of Partitions, in Encyclopedia of Mathematics and its Applications. Volume 2, Addison-Wesley Publishing Co., Reading Massachusetts U.S.A., London U.K. and Amsterdam The Netherlands (1976).
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: 1612.05891
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
Alkalaev, K., Geiko, R. & Rappoport, V. Various semiclassical limits of torus conformal blocks. J. High Energ. Phys. 2017, 70 (2017). https://doi.org/10.1007/JHEP04(2017)070
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2017)070