Abstract
It is interesting to ask: how does the radial space direction emerge from the CFT in gauge-gravity duality? In this context we resolve a long-standing puzzle with the gravity duals of two classes of states in the D1D5 CFT. For each class the CFT states are in the untwisted sector, suggesting that the energy gap should be 1/Ry where Ry is the radius of the circle on which the D1D5 CFT is compactified. For one class of states, the gravity dual indeed has exactly this gap, while for the other class, the gravity dual has a very deep throat, leading to an energy gap much smaller than 1/Ry. We resolve this puzzle by showing that for the latter class of states, perturbing the CFT off its free point leads to the formation of a band structure in the CFT. We also explain why such a band structure does not arise for the first class of states. Thus for the case where a deep throat emerges in the gravity description, the dynamics of falling down this throat is described in the CFT as a sequential ‘hopping’ between states all of which have the same energy at the free point; this hopping amplitude converts an integer spaced spectrum into a closely spaced band of energy levels.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [INSPIRE].
R. Dijkgraaf, Instanton strings and hyper-Kähler geometry, Nucl. Phys. B 543 (1999) 545 [hep-th/9810210] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Virasoro amplitude from the SNR24 orbifold sigma model, Theor. Math. Phys. 114 (1998) 43 [hep-th/9708129] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Four graviton scattering amplitude from SNR8 supersymmetric orbifold sigma model, Nucl. Phys. B 524 (1998) 159 [hep-th/9712061] [INSPIRE].
A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on SN(X): symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].
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].
A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].
S.R. Das and S.D. Mathur, Comparing decay rates for black holes and D-branes, Nucl. Phys. B 478 (1996) 561 [hep-th/9606185] [INSPIRE].
J.M. Maldacena and A. Strominger, Black hole grey body factors and D-brane spectroscopy, Phys. Rev. D 55 (1997) 861 [hep-th/9609026] [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].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
B.D. Chowdhury and A. Virmani, Modave lectures on fuzzballs and emission from the D1-D5 system, in 5th Modave summer school in mathematical physics, (2010) [arXiv:1001.1444] [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].
P. Heidmann and N.P. Warner, Superstratum symbiosis, JHEP 09 (2019) 059 [arXiv:1903.07631] [INSPIRE].
I. Bena, N. Ceplak, S. Hampton, Y. Li, D. Toulikas and N.P. Warner, Resolving black-hole microstructure with new momentum carriers, JHEP 10 (2022) 033 [arXiv:2202.08844] [INSPIRE].
B. Ganchev, A. Houppe and N.P. Warner, New superstrata from three-dimensional supergravity, JHEP 04 (2022) 065 [arXiv:2110.02961] [INSPIRE].
V.S. Rychkov, D1-D5 black hole microstate counting from supergravity, JHEP 01 (2006) 063 [hep-th/0512053] [INSPIRE].
E. Gava and K.S. Narain, Proving the PP wave/CFT2 duality, JHEP 12 (2002) 023 [hep-th/0208081] [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].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Excitations in the deformed D1D5 CFT, JHEP 06 (2010) 032 [arXiv:1003.2746] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for symmetric product orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [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. Pakman, L. Rastelli and S.S. Razamat, A spin chain for the symmetric product CFT2, JHEP 05 (2010) 099 [arXiv:0912.0959] [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].
B.A. Burrington, I.T. Jardine and A.W. Peet, Operator mixing in deformed D1D5 CFT and the OPE on the cover, JHEP 06 (2017) 149 [arXiv:1703.04744] [INSPIRE].
Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the twist operator in the D1D5 CFT, JHEP 08 (2014) 064 [arXiv:1405.0259] [INSPIRE].
Z. Carson, S.D. Mathur and D. Turton, Bogoliubov coefficients for the twist operator in the D1D5 CFT, Nucl. Phys. B 889 (2014) 443 [arXiv:1406.6977] [INSPIRE].
Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the deformation operator in the D1D5 CFT, JHEP 01 (2015) 071 [arXiv:1410.4543] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Second order effect of twist deformations in the D1D5 CFT, JHEP 04 (2016) 115 [arXiv:1511.04046] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, One-loop transition amplitudes in the D1D5 CFT, JHEP 01 (2017) 006 [arXiv:1606.06212] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Full action of two deformation operators in the D1D5 CFT, JHEP 11 (2017) 096 [arXiv:1612.03886] [INSPIRE].
S. Hampton and S.D. Mathur, Thermalization in the D1D5 CFT, JHEP 06 (2020) 004 [arXiv:1910.01690] [INSPIRE].
M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].
S. Hampton, S.D. Mathur and I.G. Zadeh, Lifting of D1-D5-P states, JHEP 01 (2019) 075 [arXiv:1804.10097] [INSPIRE].
C.A. Keller and I.G. Zadeh, Lifting \( \frac{1}{4} \)-BPS states on K3 and Mathieu moonshine, Commun. Math. Phys. 377 (2020) 225 [arXiv:1905.00035] [INSPIRE].
C.A. Keller and I.G. Zadeh, Conformal perturbation theory for twisted fields, J. Phys. A 53 (2020) 095401 [arXiv:1907.08207] [INSPIRE].
N. Benjamin, C.A. Keller and I.G. Zadeh, Lifting 1/4-BPS states in AdS3 × S3 × T4, JHEP 10 (2021) 089 [arXiv:2107.00655] [INSPIRE].
B. Guo and S.D. Mathur, Lifting of states in 2-dimensional N = 4 supersymmetric CFTs, JHEP 10 (2019) 155 [arXiv:1905.11923] [INSPIRE].
B. Guo and S.D. Mathur, Lifting of level-1 states in the D1D5 CFT, JHEP 03 (2020) 028 [arXiv:1912.05567] [INSPIRE].
B. Guo and S.D. Mathur, Lifting at higher levels in the D1D5 CFT, JHEP 11 (2020) 145 [arXiv:2008.01274] [INSPIRE].
B. Guo and S. Hampton, A freely falling graviton in the D1D5 CFT, arXiv:2107.11883 [INSPIRE].
B. Guo and S. Hampton, The dual of a tidal force in the D1D5 CFT, arXiv:2108.00068 [INSPIRE].
B. Guo and S. Hampton, Partial spectral flow in the D1D5 CFT, arXiv:2112.10573 [INSPIRE].
A.A. Lima, G.M. Sotkov and M. Stanishkov, Microstate renormalization in deformed D1-D5 SCFT, Phys. Lett. B 808 (2020) 135630 [arXiv:2005.06702] [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].
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, Dynamics of R-neutral Ramond fields in the D1-D5 SCFT, JHEP 07 (2021) 211 [arXiv:2012.08021] [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. Alves Lima, G.M. Sotkov and M. Stanishkov, Four-point functions with multi-cycle fields in symmetric orbifolds and the D1-D5 CFT, JHEP 05 (2022) 106 [arXiv:2202.12424] [INSPIRE].
A. Dei and L. Eberhardt, Correlators of the symmetric product orbifold, JHEP 01 (2020) 108 [arXiv:1911.08485] [INSPIRE].
L. Apolo, A. Belin, S. Bintanja, A. Castro and C.A. Keller, Deforming symmetric product orbifolds: a tale of moduli and higher spin currents, JHEP 08 (2022) 159 [arXiv:2204.07590] [INSPIRE].
L.P. Kadanoff, Multicritical behavior at the Kosterlitz-Thouless critical point, Ann. Phys. 120 (1979) 39.
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, On moduli spaces of conformal field theories with c 1, in Copenhagen 1987, Proceedings, Perspectives in string theory, World Scientific, Singapore (1988), p. 117.
J.L. Cardy, Continuously varying exponents and the value of the central charge, J. Phys. A 20 (1987) L891 [INSPIRE].
D. Kutasov, Geometry on the space of conformal field theories and contact terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
H. Eberle, Twistfield perturbations of vertex operators in the Z2 orbifold model, JHEP 06 (2002) 022 [hep-th/0103059] [INSPIRE].
M.R. Gaberdiel, A. Konechny and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402 [arXiv:0811.3149] [INSPIRE].
D. Berenstein and A. Miller, Conformal perturbation theory, dimensional regularization, and AdS/CFT correspondence, Phys. Rev. D 90 (2014) 086011 [arXiv:1406.4142] [INSPIRE].
D. Berenstein and A. Miller, Logarithmic enhancements in conformal perturbation theory and their real time interpretation, Int. J. Mod. Phys. A 35 (2020) 2050184 [arXiv:1607.01922] [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].
L. Eberhardt, AdS3/CFT2 at higher genus, JHEP 05 (2020) 150 [arXiv:2002.11729] [INSPIRE].
A. Dei, M.R. Gaberdiel, R. Gopakumar and B. Knighton, Free field world-sheet correlators for AdS3, JHEP 02 (2021) 081 [arXiv:2009.11306] [INSPIRE].
B. Knighton, Higher genus correlators for tensionless AdS3 strings, JHEP 04 (2021) 211 [arXiv:2012.01445] [INSPIRE].
M.R. Gaberdiel, B. Knighton and J. Vošmera, D-branes in AdS3 × S3 × T4 at k = 1 and their holographic duals, JHEP 12 (2021) 149 [arXiv:2110.05509] [INSPIRE].
L. Eberhardt, A perturbative CFT dual for pure NS-NS AdS3 strings, J. Phys. A 55 (2022) 064001 [arXiv:2110.07535] [INSPIRE].
M.R. Gaberdiel and B. Nairz, BPS correlators for AdS3/CFT2, JHEP 09 (2022) 244 [arXiv:2207.03956] [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].
A. Sevrin, W. Troost and A. Van Proeyen, Superconformal algebras in two-dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].
B. Guo and S.D. Hampton, Bootstrapping the effect of the twist operator in symmetric orbifold CFTs, arXiv:2206.01623 [INSPIRE].
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Guo, B., Mathur, S.D. Dynamical evolution in the D1D5 CFT. J. High Energ. Phys. 2022, 107 (2022). https://doi.org/10.1007/JHEP12(2022)107
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DOI: https://doi.org/10.1007/JHEP12(2022)107