Abstract
We study SN-invariant four-point functions with two generic multi-cycle fields and two twist-2 fields, at the free orbifold point of the D1-D5 CFT. We derive the explicit factorization of these functions following from the action of the symmetric group on the composite multi-cycle fields. Apart from non-trivial symmetry factors that we compute, the function with multi-cycle operators is reduced to a sum of connected correlators in which the composite fields have, at most, two cycles. The correlators with two double-cycle and two single-cycle fields give the leading order contribution in the large-N limit. We derive explicit formulas for these functions, encompassing a large class of choices for the single- and the double-cycle fields, including generic Ramond ground states, NS chiral fields and the marginal deformation operator. We are thus able to extract important dynamical information from the short-distance OPEs: conformal dimensions, R-charges and structure constants of families of BPS and non-BPS fields present in the corresponding light-light and heavy-light channels. We also discuss properties of generic multi-cycle Q-point functions in MN/SN orbifolds, using a technology due to Pakman, Rastelli and Razamat.
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S. Giusto, R. Russo and C. Wen, Holographic correlators in AdS3, JHEP 03 (2019) 096 [arXiv:1812.06479] [INSPIRE].
S. Giusto, R. Russo, A. Tyukov and C. Wen, Holographic correlators in AdS3 without Witten diagrams, JHEP 09 (2019) 030 [arXiv:1905.12314] [INSPIRE].
L. Rastelli, K. Roumpedakis and X. Zhou, AdS3 × S3 Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry, JHEP 10 (2019) 140 [arXiv:1905.11983] [INSPIRE].
L. Eberhardt and M.R. Gaberdiel, String theory on AdS3 and the symmetric orbifold of Liouville theory, Nucl. Phys. B 948 (2019) 114774 [arXiv:1903.00421] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, B. Knighton and P. Maity, From symmetric product CFTs to AdS3, JHEP 05 (2021) 073 [arXiv:2011.10038] [INSPIRE].
A. Galliani, S. Giusto, E. Moscato and R. Russo, Correlators at large c without information loss, JHEP 09 (2016) 065 [arXiv:1606.01119] [INSPIRE].
A. Galliani, S. Giusto and R. Russo, Holographic 4-point correlators with heavy states, JHEP 10 (2017) 040 [arXiv:1705.09250] [INSPIRE].
A. Bombini, A. Galliani, S. Giusto, E. Moscato and R. Russo, Unitary 4-point correlators from classical geometries, Eur. Phys. J. C 78 (2018) 8 [arXiv:1710.06820] [INSPIRE].
J. Tian, J. Hou and B. Chen, Holographic Correlators on Integrable Superstrata, Nucl. Phys. B 948 (2019) 114766 [arXiv:1904.04532] [INSPIRE].
A. Bombini and A. Galliani, AdS3 four-point functions from \( \frac{1}{8} \)-BPS states, JHEP 06 (2019) 044 [arXiv:1904.02656] [INSPIRE].
S. Giusto, R. Russo, A. Tyukov and C. Wen, The CFT6 origin of all tree-level 4-point correlators in AdS3 × S3 , Eur. Phys. J. C 80 (2020) 736 [arXiv:2005.08560] [INSPIRE].
N. Ceplak, S. Giusto, M.R.R. Hughes and R. Russo, Holographic correlators with multi-particle states, JHEP 09 (2021) 204 [arXiv:2105.04670] [INSPIRE].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171] [INSPIRE].
M. Taylor, Matching of correlators in AdS3/CFT2, JHEP 06 (2008) 010 [arXiv:0709.1838] [INSPIRE].
S. Giusto, E. Moscato and R. Russo, AdS3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [arXiv:1507.00945] [INSPIRE].
S. Rawash and D. Turton, Supercharged AdS3 Holography, JHEP 07 (2021) 178 [arXiv:2105.13046] [INSPIRE].
S. Giusto, S. Rawash and D. Turton, Ads3 holography at dimension two, JHEP 07 (2019) 171 [arXiv:1904.12880] [INSPIRE].
J. Garcia i Tormo and M. Taylor, Correlation functions in the D1-D5 orbifold CFT, JHEP 06 (2018) 012 [arXiv:1804.10205] [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, 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].
A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS3/CFT2, JHEP 05 (2016) 109 [arXiv:1603.08925] [INSPIRE].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049 [arXiv:1504.05943] [INSPIRE].
B. Carneiro da Cunha and M. Guica, Exploring the BTZ bulk with boundary conformal blocks, arXiv:1604.07383 [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
R. Cavalieri and E. Miles, Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, U.K. (2016), [DOI].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP 06 (2010) 031 [arXiv:1002.3132] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for MN/SN orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
O. Lunin and S.D. Mathur, Three point functions for MN/SN orbifolds with N = 4 supersymmetry, Commun. Math. Phys. 227 (2002) 385 [hep-th/0103169] [INSPIRE].
A. Dei and L. Eberhardt, Correlators of the symmetric product orbifold, JHEP 01 (2020) 108 [arXiv:1911.08485] [INSPIRE].
A.A. Lima, G.M. Sotkov and M. Stanishkov, Correlation functions of composite Ramond fields in deformed D1-D5 orbifold SCFT2, Phys. Rev. D 102 (2020) 106004 [arXiv:2006.16303] [INSPIRE].
A.A. Lima, G.M. Sotkov and M. Stanishkov, On the dynamics of protected ramond ground states in the D1-D5 CFT, JHEP 07 (2021) 120 [arXiv:2103.04459] [INSPIRE].
A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on \( {S}_X^N \): Symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].
A. Dabholkar and A. Pakman, Exact chiral ring of AdS3/CFT2, Adv. Theor. Math. Phys. 13 (2009) 409 [hep-th/0703022] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds, Phys. Rev. D 80 (2009) 086009 [arXiv:0905.3451] [INSPIRE].
A. Schwimmer and N. Seiberg, Comments on the N = 2, N = 3, N = 4 Superconformal Algebras in Two-Dimensions, Phys. Lett. B 184 (1987) 191 [INSPIRE].
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
B. Sagan, The symmetric group: representations, combinatorial algorithms, and symmetric functions, vol. 203 of Graduate Texts in Mathematics, 2 ed., Springer-Verlag, New York, U.S.A. (2001).
G.E. Arutyunov and S.A. Frolov, Virasoro amplitude from the SN R24 orbifold sigma model, Theor. Math. Phys. 114 (1998) 43 [hep-th/9708129] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT(2), JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Four graviton scattering amplitude from S**N R8 supersymmetric orbifold sigma model, Nucl. Phys. B 524 (1998) 159 [hep-th/9712061] [INSPIRE].
S.G. Avery, Using the D1D5 CFT to Understand Black Holes, other thesis, (2010) [arXiv:1012.0072] [INSPIRE].
A.A. Lima, G.M. Sotkov and M. Stanishkov, Renormalization of twisted Ramond fields in D1-D5 SCFT2, JHEP 03 (2021) 202 [arXiv:2010.00172] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Emission from the D1D5 CFT, JHEP 10 (2009) 065 [arXiv:0906.2015] [INSPIRE].
P. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media, Germany (2012).
T. De Beer, B.A. Burrington, I.T. Jardine and A.W. Peet, The large N limit of OPEs in symmetric orbifold CFTs with \( \mathcal{N} \) = (4, 4) supersymmetry, JHEP 08 (2019) 015 [arXiv:1904.07816] [INSPIRE].
B. Guo and S. Hampton, Partial Spectral Flow in the D1D5 CFT, arXiv:2112.10573 [INSPIRE].
B.A. Burrington, A.W. Peet and I.G. Zadeh, Operator mixing for string states in the D1-D5 CFT near the orbifold point, Phys. Rev. D 87 (2013) 106001 [arXiv:1211.6699] [INSPIRE].
O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nucl. Phys. B 610 (2001) 49 [hep-th/0105136] [INSPIRE].
O. Lunin and S.D. Mathur, A toy black hole S-matrix in the D1-D5 CFT, JHEP 02 (2013) 083 [arXiv:1211.5830] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates, Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, Nucl. Phys. B 710 (2005) 425 [hep-th/0406103] [INSPIRE].
B.A. Burrington, A.W. Peet and I.G. Zadeh, Bosonization, cocycles, and the D1-D5 CFT on the covering surface, Phys. Rev. D 93 (2016) 026004 [arXiv:1509.00022] [INSPIRE].
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Lima, A.A., Sotkov, G.M. & Stanishkov, M. Four-point functions with multi-cycle fields in symmetric orbifolds and the D1-D5 CFT. J. High Energ. Phys. 2022, 106 (2022). https://doi.org/10.1007/JHEP05(2022)106
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DOI: https://doi.org/10.1007/JHEP05(2022)106