Abstract
In this note we study type ii superstring theory on the direct product of AdS3, the squashed three-sphere and a four-torus (AdS3 × SqS3 × T 4). We derive explicitly the Green-Schwarz action and discuss in detail the hidden integrable structure corresponding to local and non-local currents. We find that the model is classically integrable and that the currents generate a full psu(1, 1|2) superalgebra, even though the corresponding spacetime isometries are broken by the squashing.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
V. Kazakov, A. Marshakov, J. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].
N. Beisert and M. Staudacher, Long-range PSU (2, 2|4) Bethe ansatze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].
N. Beisert, V. Kazakov, K. Sakai and K. Zarembo, The Algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
G. Arutyunov and S. Frolov, Superstrings on AdS 4 × CP 3 as a coset σ-model, JHEP 09 (2008) 129 [arXiv:0806.4940] [INSPIRE].
B. Stefański Jr., Green-Schwarz action for type IIA strings on AdS 4 × CP 3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [INSPIRE].
D. Sorokin and L. Wulff, Evidence for the classical integrability of the complete AdS 4 × CP 3 superstring, JHEP 11 (2010) 143 [arXiv:1009.3498] [INSPIRE].
B. Chen, Y.-L. He, P. Zhang and X.-C. Song, Flat currents of the Green-Schwarz superstrings in AdS 5 × S 1 and AdS 3 × S 3 backgrounds, Phys. Rev. D 71 (2005) 086007 [hep-th/0503089] [INSPIRE].
I. Adam, A. Dekel, L. Mazzucato and Y. Oz, Integrability of type II superstrings on Ramond-Ramond backgrounds in various dimensions, JHEP 06 (2007) 085 [hep-th/0702083] [INSPIRE].
A. Babichenko, B. Stefański Jr. and K. Zarembo, Integrability and the AdS 3/CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
M. Rooman and P. Spindel, Godel metric as a squashed Anti-de Sitter geometry, Class. Quant. Grav. 15 (1998) 3241 [gr-qc/9804027] [INSPIRE].
S. Detournay, D. Orlando, P.M. Petropoulos and P. Spindel, Three-dimensional black holes from deformed anti-de Sitter, JHEP 07 (2005) 072 [hep-th/0504231] [INSPIRE].
D. Orlando, String theory: exact solutions, marginal deformations and hyperbolic spaces, Fortsch. Phys. 55 (2007) 161 [hep-th/0610284] [INSPIRE].
G. Compere, S. Detournay and M. Romo, Supersymmetric Godel and warped black holes in string theory, Phys. Rev. D 78 (2008) 104030 [arXiv:0808.1912] [INSPIRE].
D. Israel, C. Kounnas and M.P. Petropoulos, Superstrings on NS5 backgrounds, deformed AdS 3 and holography, JHEP 10 (2003) 028 [hep-th/0306053] [INSPIRE].
D. Israel, C. Kounnas, D. Orlando and P.M. Petropoulos, Electric/magnetic deformations of S 3 and AdS 3 and geometric cosets, Fortsch. Phys. 53 (2005) 73 [hep-th/0405213] [INSPIRE].
D. Israel, C. Kounnas, D. Orlando and P.M. Petropoulos, Heterotic strings on homogeneous spaces, Fortsch. Phys. 53 (2005) 1030 [hep-th/0412220] [INSPIRE].
D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, Warped AdS 3 black holes, JHEP 03 (2009) 130 [arXiv:0807.3040] [INSPIRE].
T. Azeyanagi, D.M. Hofman, W. Song and A. Strominger, The spectrum of strings on warped AdS 3 × S 3, arXiv:1207.5050 [INSPIRE].
D. Orlando and L.I. Uruchurtu, Warped Anti-de Sitter spaces from brane intersections in type-II string theory, JHEP 06 (2010) 049 [arXiv:1003.0712] [INSPIRE].
M.A. Melvin, Pure magnetic and electric geons, Phys. Lett. 8 (1964) 65 [INSPIRE].
A.A. Tseytlin, Melvin solution in string theory, Phys. Lett. B 346 (1995) 55 [hep-th/9411198] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, String theory of the Omega deformation, JHEP 01 (2012) 148 [arXiv:1106.0279] [INSPIRE].
S. Reffert, General Ω deformations from closed string backgrounds, JHEP 04 (2012) 059 [arXiv:1108.0644] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, The Ω deformation from string and M-theory, JHEP 07 (2012) 061 [arXiv:1204.4192] [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].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].
R. Ricci, A.A. Tseytlin and M. Wolf, On T-duality and Integrability for Strings on AdS Backgrounds, JHEP 12 (2007) 082 [arXiv:0711.0707] [INSPIRE].
N. Mohammedi, On the geometry of classically integrable two-dimensional non-linear σ-models, Nucl. Phys. B 839 (2010) 420 [arXiv:0806.0550] [INSPIRE].
T. Curtright and C.K. Zachos, Currents, charges and canonical structure of pseudodual chiral models, Phys. Rev. D 49 (1994) 5408 [hep-th/9401006] [INSPIRE].
J. Balog, P. Forgacs, Z. Horvath and L. Palla, A new family of SU(2) symmetric integrable σ-models, Phys. Lett. B 324 (1994) 403 [hep-th/9307030] [INSPIRE].
J.M. Evans and T.J. Hollowood, Integrable theories that are asymptotically CFT, Nucl. Phys. B 438 (1995) 469 [hep-th/9407113] [INSPIRE].
N. Beisert, R. Ricci, A.A. Tseytlin and M. Wolf, Dual superconformal symmetry from AdS 5 × S 5 superstring integrability, Phys. Rev. D 78 (2008) 126004 [arXiv:0807.3228] [INSPIRE].
D. Orlando, S. Reffert and L.I. Uruchurtu, Classical integrability of the squashed three-sphere, warped AdS 3 and Schrödinger spacetime via T-duality, J. Phys. A 44 (2011) 115401 [arXiv:1011.1771] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hidden yangian symmetry in σ-model on squashed sphere, JHEP 11 (2010) 032 [arXiv:1008.0776] [INSPIRE].
I. Kawaguchi, D. Orlando and K. Yoshida, Yangian symmetry in deformed WZNW models on squashed spheres, Phys. Lett. B 701 (2011) 475 [arXiv:1104.0738] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hybrid classical integrable structure of squashed σ-models: A Short summary, J. Phys. Conf. Ser. 343 (2012) 012055 [arXiv:1110.6748] [INSPIRE].
M.T. Grisaru et al., N = 2 superstrings in a supergravity background, Phys. Lett. B 162 (1985) 116 [INSPIRE].
M. Cvetič, H. Lü, C. Pope and K. Stelle, T duality in the Green-Schwarz formalism and the massless/massive IIA duality map, Nucl. Phys. B 573 (2000) 149 [hep-th/9907202] [INSPIRE].
N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Zwiebach, Superstring theory on AdS 2 × S 2 as a coset supermanifold, Nucl. Phys. B 567 (2000) 61 [hep-th/9907200] [INSPIRE].
N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].
A.A. Tseytlin, Harmonic superpositions of M-branes, Nucl. Phys. B 475 (1996) 149 [hep-th/9604035] [INSPIRE].
H.J. Boonstra, B. Peeters and K. Skenderis, Brane intersections, anti-de Sitter space-times and dual superconformal theories, Nucl. Phys. B 533 (1998) 127 [hep-th/9803231] [INSPIRE].
D. Sorokin et al., Superstrings in AdS(2) × S(2) × T(6), J. Phys. A 44 (2011) 275401 [arXiv:1104.1793] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1208.3680
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Orlando, D., Uruchurtu, L.I. Integrable superstrings on the squashed three-sphere. J. High Energ. Phys. 2012, 7 (2012). https://doi.org/10.1007/JHEP10(2012)007
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2012)007