Abstract
We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can ‘lift’. The lifting can be computed by a path integral containing two twist deformations; however, the relevant 4-point amplitude cannot be computed explicitly in many cases. We analyze an older proposal by Gava and Narain where the lift can be computed in terms of a finite number of 3-point functions. A direct Hamiltonian decomposition of the path integral involves an infinite number of 3-point functions, as well the first order correction to the starting state. We note that these corrections to the state account for the infinite number of 3-point functions arising from higher energy states, and one can indeed express the path-integral result in terms of a finite number of 3-point functions involving only the leading order states that are degenerate. The first order correction to the super-charge \( {\overline{G}}^{(1)} \) gets replaced by a projection \( {\overline{G}}^{(P)} \); this projected operator can also be used to group the states into multiplets whose members have the same lifting.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
C. Vafa, Instantons on D-branes, Nucl. Phys.B 463 (1996) 435 [hep-th/9512078] [INSPIRE].
R. Dijkgraaf, Instanton strings and hyperKähler geometry, Nucl. Phys.B 543 (1999) 545 [hep-th/9810210] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP04 (1999) 017 [hep-th/9903224] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP06 (1999) 019 [hep-th/9905064] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Virasoro amplitude from the SNR24orbifold σ-model, Theor. Math. Phys.114 (1998) 43 [hep-th/9708129] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, Four graviton scattering amplitude from SNR8supersymmetric orbifold σ-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].
J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [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].
S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys.53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP06 (2007) 056 [arXiv:0704.0690] [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, Modave, Belgium, 17–21 August 2009 (2010) [arXiv:1001.1444] [INSPIRE].
M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP10 (2015) 101 [arXiv:1506.02045] [INSPIRE].
C.A. Keller and I.G. Zadeh, Lifting1-BPS States on K 3 and Mathieu Moonshine, arXiv:1905.00035 [INSPIRE].
S. Hampton, S.D. Mathur and I.G. Zadeh, Lifting of D1-D5-P states, JHEP01 (2019) 075 [arXiv:1804.10097] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for MN/SNorbifolds, Commun. Math. Phys.219 (2001) 399 [hep-th/0006196] [INSPIRE].
O. Lunin and S.D. Mathur, Three point functions for MN/SNorbifolds with N = 4 supersymmetry, Commun. Math. Phys.227 (2002) 385 [hep-th/0103169] [INSPIRE].
E. Gava and K.S. Narain, Proving the PP-wave/CFT2 duality, JHEP12 (2002) 023 [hep-th/0208081] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP10 (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 CFT(2), JHEP05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP06 (2010) 031 [arXiv:1002.3132] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Excitations in the deformed D1D5 CFT, JHEP06 (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].
B.A. Burrington, A.W. Peet and I.G. Zadeh, Twist-nontwist correlators in MN/SNorbifold CFTs, Phys. Rev.D 87 (2013) 106008 [arXiv:1211.6689] [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].
B.A. Burrington, I.T. Jardine and A.W. Peet, Operator mixing in deformed D1D5 CFT and the OPE on the cover, JHEP06 (2017) 149 [arXiv:1703.04744] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Full action of two deformation operators in the D1D5 CFT, JHEP11 (2017) 096 [arXiv:1612.03886] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, One-Loop Transition Amplitudes in the D1D5 CFT, JHEP01 (2017) 006 [arXiv:1606.06212] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Second order effect of twist deformations in the D1D5 CFT, JHEP04 (2016) 115 [arXiv:1511.04046] [INSPIRE].
Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the deformation operator in the D1D5 CFT, JHEP01 (2015) 071 [arXiv:1410.4543] [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 twist operator in the D1D5 CFT, JHEP08 (2014) 064 [arXiv:1405.0259] [INSPIRE].
L.P. Kadanoff, Multicritical behavior at the kosterlitz-thouless critical point Annals 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, pp. 117–137 [INSPIRE].
J.L. Cardy, Continuously Varying Exponents and the Value of the Central Charge, J. Phys.A 20 (1987) L891 [INSPIRE].
H. Eberle, Twistfield perturbations of vertex operators in the Z(2) orbifold model, JHEP06 (2002) 022 [hep-th/0103059] [INSPIRE].
H. Eberle, Twistfield perturbations of vertex operators in the Z(2) orbifold model, Ph.D. Thesis, University of Bonn, Bonn (2006).
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, arXiv:1607.01922 [INSPIRE].
J. de Boer, J. Manschot, K. Papadodimas and E. Verlinde, The Chiral ring of AdS3/C F T2 and the attractor mechanism, JHEP03 (2009) 030 [arXiv:0809.0507] [INSPIRE].
A. Sen, On the Background Independence of String Field Theory, Nucl. Phys.B 345 (1990) 551 [INSPIRE].
A. Sen, Background Independence of Closed Superstring Field Theory, JHEP02 (2018) 155 [arXiv:1711.08468] [INSPIRE].
M. Campbell, P.C. Nelson and E. Wong, Stress tensor perturbations in conformal field theory, Int. J. Mod. Phys.A 6 (1991) 4909 [INSPIRE].
M. Evans and B.A. Ovrut, Deformations of Conformal Field Theories and Symmetries of the String, Phys. Rev.D 41 (1990) 3149 [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: 1905.11923
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guo, B., Mathur, S.D. Lifting of states in 2-dimensional N = 4 supersymmetric CFTs. J. High Energ. Phys. 2019, 155 (2019). https://doi.org/10.1007/JHEP10(2019)155
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)155