Abstract
String theory on AdS3 × S3 × T4 has 20 moduli. We investigate how the perturbative closed string spectrum changes as we move around this moduli space in both the RR and NSNS flux backgrounds. We find that, at weak string coupling, only four of the moduli affect the energies. In the RR background the only effect of these moduli is to change the radius of curvature of the background. On the other hand, in the NSNS background, the moduli introduce worldsheet interactions which enable the use of integrability methods to solve the spectral problem. Our results show that the worldsheet theory is integrable across the 20 dimensional moduli space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu.I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
M.R. Douglas, Gauge fields and D-branes, J. Geom. Phys. 28 (1998) 255 [hep-th/9604198] [INSPIRE].
R. Dijkgraaf, Instanton strings and hyperKähler geometry, Nucl. Phys. B 543 (1999) 545 [hep-th/9810210] [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].
E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].
C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415 [hep-th/9511088] [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].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr., A. Torrielli and O. Ohlsson Sax, On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS3 × S3 × T4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS3 /CFT2 Correspondence and Integrability, JHEP 11 (2012) 133 [Erratum ibid. 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
B. Hoare and A.A. Tseytlin, Massive S-matrix of AdS3 × S3 × T4 superstring theory with mixed 3-form flux, Nucl. Phys. B 873 (2013) 395 [arXiv:1304.4099] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete worldsheet S matrix of superstrings on AdS3 × S3 × T4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, ℝ) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) R5412 [hep-th/9508072] [INSPIRE].
S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].
S. Ferrara and R. Kallosh, Universality of supersymmetric attractors, Phys. Rev. D 54 (1996) 1525 [hep-th/9603090] [INSPIRE].
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., On the spectrum of AdS3 × S3 × T4 strings with Ramond-Ramond flux, J. Phys. A 49 (2016) 41LT03 [arXiv:1605.00518] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete AdS3 × S3 × T4 worldsheet S matrix, JHEP 10 (2014) 66 [arXiv:1406.0453] [INSPIRE].
D. Marolf, Chern-Simons terms and the three notions of charge, in Quantization, gauge theory and strings. Proceedings, International Conference dedicated to the memory of ProfeSSOR Efim Fradkin, Moscow, Russia, June 5-10, 2000, vol. 1+2, pp. 312-320 [hep-th/0006117] [INSPIRE].
M.R. Douglas, Branes within branes, in Strings, branes and dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26-June 14, 1997, pp. 267-275 [hep-th/9512077] [INSPIRE].
J.M. Maldacena and J.G. Russo, Large N limit of noncommutative gauge theories, JHEP 09 (1999) 025 [hep-th/9908134] [INSPIRE].
A. Dhar, G. Mandal, S.R. Wadia and K.P. Yogendran, D-1/D-5 system with B field, noncommutative geometry and the CFT of the Higgs branch, Nucl. Phys. B 575 (2000) 177 [hep-th/9910194] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
L.F. Alday, G. Arutyunov and S. Frolov, Green-Schwarz strings in TsT-transformed backgrounds, JHEP 06 (2006) 018 [hep-th/0512253] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic SU(1|1)2 S-matrix for AdS3 /CFT2, JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, The all-loop integrable spin-chain for strings on AdS3 × S3 × T4 : the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., Towards the All-Loop Worldsheet S Matrix for AdS3 × S3 × T4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, Dressing phases of AdS3 /CFT2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
G. Arutyunov and S. Frolov, Uniform light-cone gauge for strings in AdS5 × S5 : Solving SU(1|1) sector, JHEP 01 (2006) 055 [hep-th/0510208] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, Protected string spectrum in AdS3 /CFT2 from worldsheet integrability, JHEP 04 (2017) 091 [arXiv:1701.03501] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and C. Hull, Stringy AdS3 from the worldsheet, JHEP 07 (2017) 090 [arXiv:1704.08665] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3 at k = 1, arXiv:1803.04420 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless String Spectra on AdS3, arXiv:1803.04423 [INSPIRE].
S.F. Hassan and A. Sen, Marginal deformations of WZNW and coset models from O(d, d) transformation, Nucl. Phys. B 405 (1993) 143 [hep-th/9210121] [INSPIRE].
A. Giveon and E. Kiritsis, Axial vector duality as a gauge symmetry and topology change in string theory, Nucl. Phys. B 411 (1994) 487 [hep-th/9303016] [INSPIRE].
S. Förste, A Truly marginal deformation of SL(2, ℝ) in a null direction, Phys. Lett. B 338 (1994) 36 [hep-th/9407198] [INSPIRE].
A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS3 /CFT2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
O. Ohlsson Sax, B. Stefanski Jr. and A. Torrielli, On the massless modes of the AdS3 /CFT2 integrable systems, JHEP 03 (2013) 109 [arXiv:1211.1952] [INSPIRE].
R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
A. Babichenko, A. Dekel and O. Ohlsson Sax, Finite-gap equations for strings on AdS 3 x S 3 x T 4 with mixed 3-form flux, JHEP 11 (2014) 122 [arXiv:1405.6087] [INSPIRE].
J.R. David and B. Sahoo, Giant magnons in the D1-D5 system, JHEP 07 (2008) 033 [arXiv:0804.3267] [INSPIRE].
J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].
M.C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz, Phys. Rev. D 93 (2016) 106006 [arXiv:1512.08761] [INSPIRE].
D. Bombardelli, A. Torrielli and B. Stefanski Jr., The low-energy limit of AdS3/CFT2 and its TBA, to appear.
T. Nishioka and T. Takayanagi, On Type IIA Penrose Limit and N = 6 Chern-Simons Theories, JHEP 08 (2008) 001 [arXiv:0806.3391] [INSPIRE].
D. Gaiotto, S. Giombi and X. Yin, Spin Chains in N = 6 Superconformal Chern-Simons-Matter Theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [INSPIRE].
N. Gromov and G. Sizov, Exact Slope and Interpolating Functions in N = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].
P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].
P. Sundin and L. Wulff, One- and two-loop checks for the AdS3 × S3 × T4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
J. de Boer, Six-dimensional supergravity on S3 × AdS3 and 2d conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
R. Hernández and J.M. Nieto, Spinning strings in AdS3 × S3 with NS-NS flux, Nucl. Phys. B 888 (2014) 236 [Erratum ibid. B 895 (2015) 303] [arXiv:1407.7475] [INSPIRE].
J.R. David and A. Sadhukhan, Spinning strings and minimal surfaces in AdS3 with mixed 3-form fluxes, JHEP 10 (2014) 49 [arXiv:1405.2687] [INSPIRE].
A. Banerjee and A. Sadhukhan, Multi-spike strings in AdS3 with mixed three-form fluxes, JHEP 05 (2016) 083 [arXiv:1512.01816] [INSPIRE].
A. Banerjee, K.L. Panigrahi and M. Samal, A note on oscillating strings in AdS3 × S3 with mixed three-form fluxes, JHEP 11 (2015) 133 [arXiv:1508.03430] [INSPIRE].
R. Hernández, J.M. Nieto and R. Ruiz, Pulsating strings with mixed three-form flux, JHEP 04 (2018) 078 [arXiv:1803.03078] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].
G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, The Off-shell Symmetry Algebra of the Light-cone AdS5 × S5 Superstring, J. Phys. A 40 (2007) 3583 [hep-th/0609157] [INSPIRE].
B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS3 × S3 × T4 with mixed flux, Nucl. Phys. B 879 (2014) 318 [arXiv:1311.1794] [INSPIRE].
O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [INSPIRE].
O. Ohlsson Sax and B. Stefanski Jr., Integrability, spin-chains and the AdS3 /CFT2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations for AdS3 /CFT2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].
M.C. Abbott, The AdS3 × S3 × S3 × S1 Hernández-López phases: a semiclassical derivation, J. Phys. A 46 (2013) 445401 [arXiv:1306.5106] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The AdS3 × S3 × S3 × S1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].
S. Gukov, E. Martinec, G.W. Moore and A. Strominger, The Search for a holographic dual to AdS3 × S3 × S3 × S1, Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].
D. Tong, The holographic dual of AdS3 × S3 × S3 × S1, JHEP 04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and W. Li, A holographic dual for string theory on AdS3 × S3 × S3 × S1, JHEP 08 (2017) 111 [arXiv:1707.02705] [INSPIRE].
G. Itsios, Y. Lozano, E. Ó Colgáin and K. Sfetsos, Non-Abelian T-duality and consistent truncations in type-II supergravity, JHEP 08 (2012) 132 [arXiv:1205.2274] [INSPIRE].
M.J.D. Hamilton, The field and Killing spinor equations of M-theory and type IIA/ IIB supergravity in coordinate-free notation, arXiv:1607.00327 [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: 1804.02023
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
Sax, O.O., Stefański, B. Closed strings and moduli in AdS3/CFT2. J. High Energ. Phys. 2018, 101 (2018). https://doi.org/10.1007/JHEP05(2018)101
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2018)101