Abstract
We consider the D1D5 CFT near the orbifold point and develop methods for computing the mixing of untwisted operators to first order by using the OPE on the covering surface. We argue that the OPE on the cover encodes both the structure constants for the orbifold CFT and the explicit form of the mixing operators. We show this explicitly for some example operators. We start by considering a family of operators dual to supergravity modes, and show that the OPE implies that there is no shift in the anomalous dimension to first order, as expected. We specialize to the operator dual to the dilaton, and show that the leading order singularity in the OPE reproduces the correct structure constant. Finally, we consider an unprotected operator of conformal dimension (2,2), and show that the leading order singularity and one of the subleading singularities both reproduce the correct structure constant. We check that the operator produced at subleading order using the OPE method is correct by calculating a number of three point functions using a Mathematica package we developed. Further development of this OPE technique should lead to more efficient calculations for the D1D5 CFT perturbed away from the orbifold point.
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S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
K. Papadodimas and S. Raju, State-dependent bulk-boundary maps and black hole complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].
E. Silverstein, Backdraft: string creation in an old Schwarzschild black hole, arXiv:1402.1486 [INSPIRE].
M. Dodelson and E. Silverstein, String-theoretic breakdown of effective field theory near black hole horizons, arXiv:1504.05536 [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Emission from the D1D5 CFT, JHEP 10 (2009) 065 [arXiv:0906.2015] [INSPIRE].
S.D. Mathur and D. Turton, Comments on black holes I: the possibility of complementarity, JHEP 01 (2014) 034 [arXiv:1208.2005] [INSPIRE].
S.D. Mathur and D. Turton, The flaw in the firewall argument, Nucl. Phys. B 884 (2014) 566 [arXiv:1306.5488] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N , Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [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].
C.A. Keller and A. Maloney, Poincaré series, 3D gravity and CFT spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742] [INSPIRE].
A. Belin, J. de Boer, J. Kruthoff, B. Michel, E. Shaghoulian and M. Shyani, Universality of sparse d > 2 conformal field theory at large-N, JHEP 03 (2017) 067 [arXiv:1610.06186] [INSPIRE].
T. Anous, T. Hartman, A. Rovai and J. Sonner, Black hole collapse in the 1/c expansion, JHEP 07 (2016) 123 [arXiv:1603.04856] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3 /CFT 2, JHEP 05 (2016) 109 [arXiv:1603.08925] [INSPIRE].
H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate operators and the 1/c expansion: Lorentzian resummations, high order computations and super-Virasoro blocks, JHEP 03 (2017) 167 [arXiv:1606.02659] [INSPIRE].
S.D. Mathur and D. Turton, Oscillating supertubes and neutral rotating black hole microstates, JHEP 04 (2014) 072 [arXiv:1310.1354] [INSPIRE].
I. Bena and N.P. Warner, Resolving the structure of black holes: philosophizing with a hammer, arXiv:1311.4538 [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].
I. Bena, E. Martinec, D. Turton and N.P. Warner, Momentum fractionation on superstrata, JHEP 05 (2016) 064 [arXiv:1601.05805] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, D1/D5 moduli in SCFT and gauge theory and Hawking radiation, Nucl. Phys. B 564 (2000) 103 [hep-th/9907075] [INSPIRE].
E. Gava and K.S. Narain, Proving the PP wave/CFT 2 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].
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].
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].
B.A. Burrington, S.D. Mathur, A.W. Peet and I.G. Zadeh, Analyzing the squeezed state generated by a twist deformation, Phys. Rev. D 91 (2015) 124072 [arXiv:1410.5790] [INSPIRE].
M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [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, arXiv:1612.03886 [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
O. Lunin and S.D. Mathur, Three point functions for M N /S N orbifolds with N = 4 supersymmetry, Commun. Math. Phys. 227 (2002) 385 [hep-th/0103169] [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].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
J. Lin, A toy model of entwinement, arXiv:1608.02040 [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].
B.A. Burrington, A.W. Peet and I.G. Zadeh, Twist-nontwist correlators in M N /S N orbifold CFTs, Phys. Rev. D 87 (2013) 106008 [arXiv:1211.6689] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [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].
J. de Boer, Six-dimensional supergravity on S 3×AdS 3 and 2D conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].
R. Dijkgraaf, Instanton strings and hyper-Kähler geometry, Nucl. Phys. B 543 (1999) 545 [hep-th/9810210] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Virasoro amplitude from the S N R 24 orbifold σ-model, Theor. Math. Phys. 114 (1998) 43 [hep-th/9708129] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Four graviton scattering amplitude from S N R 8 supersymmetric orbifold σ-model, Nucl. Phys. B 524 (1998) 159 [hep-th/9712061] [INSPIRE].
A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on S N (X): symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].
S.G. Avery, Using the D1D5 CFT to understand black holes, arXiv:1012.0072 [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].
C. Loken et al., SciNet: lessons learned from building a power-efficient top-20 system and data centre, J. Phys. Conf. Ser. 256 (2010) 012026.
M.R. Gaberdiel and R. Gopakumar, Higher spins & strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Stringy symmetries and the higher spin square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, String theory as a higher spin theory, JHEP 09 (2016) 085 [arXiv:1512.07237] [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.D. Mathur and D. Turton, Microstates at the boundary of AdS, JHEP 05 (2012) 014 [arXiv:1112.6413] [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [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].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral geometry and holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
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Burrington, B.A., Jardine, I.T. & Peet, A.W. Operator mixing in deformed D1D5 CFT and the OPE on the cover. J. High Energ. Phys. 2017, 149 (2017). https://doi.org/10.1007/JHEP06(2017)149
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DOI: https://doi.org/10.1007/JHEP06(2017)149