Abstract
For supercosets with isometry group of the form \( \hat{\mathrm{G}} \)×\( \hat{\mathrm{G}} \), the η-deformation can be generalised to a two-parameter integrable deformation with independent q-deformations of the two copies. We study its kappa-symmetry and write down a formula for the Ramond- Ramond fluxes. We then focus on \( \hat{\mathrm{G}} \) = PSU(1, 1|2) and construct two supergravity back- grounds for the two-parameter integrable deformation of the AdS3× S3× T4 superstring, as well as explore their limits. We also construct backgrounds that are solutions of the weaker generalised supergravity equations of motion and compare them to the literature.
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 [Adv. Theor. Math. Phys.2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Babichenko, B. Stefański Jr. and K. Zarembo, Integrability and the AdS3/C F T2 correspondence, JHEP03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS3/C F T2Correspondence and Integrability, JHEP11 (2012) 133 [Erratum JHEP04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
O. Ohlsson Sax and B. Stefański Jr., Integrability, spin-chains and the AdS3/C F T2 correspondence, JHEP08 (2011) 029 [arXiv:1106.2558] [INSPIRE].
P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS3× S3× S3× S1superstring, JHEP10 (2012) 109 [arXiv:1207.5531] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic su(1|1)2S-matrix for AdS3/C F T2, JHEP04 (2013) 113 [arXiv:1211.5119] [INSPIRE].
B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS3× S3× T4with mixed flux, Nucl. Phys.B 879 (2014) 318 [arXiv:1311.1794] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, The complete AdS3× S3× T4worldsheet S matrix, JHEP10 (2014) 066 [arXiv:1406.0453] [INSPIRE].
R. Hernández and J.M. Nieto, Spinning strings in AdS3× S3with NS-NS flux, Nucl. Phys.B 888 (2014) 236 [Corrigendum ibid.B 895 (2015) 303] [arXiv:1407.7475] [INSPIRE].
M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefański and A. Torrielli, Protected string spectrum in AdS3/CFT2from worldsheet integrability, JHEP04 (2017) 091 [arXiv:1701.03501] [INSPIRE].
A. Sfondrini, Towards integrability for AdS3/C F T2, J. Phys.A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3and SL(2, ℝ) WZW model. I: The Spectrum, J. Math. Phys.42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS3and the SL(2, ℝ) WZW model. II: Euclidean black hole, J. Math. Phys.42 (2001) 2961 [hep-th/0005183] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3and the SL(2, ℝ) WZW model. III. Correlation functions, Phys. Rev.D 65 (2002) 106006 [hep-th/0111180] [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].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS5× S5superstring action, Phys. Rev. Lett.112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS5× S5superstring, JHEP10 (2014) 132 [arXiv:1406.6286] [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys.50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl.32 (1985) 254 [INSPIRE].
M. Jimbo, A q difference analog of U (g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63 [INSPIRE].
A.A. Belavin and V.G. Drinfel’d, Triangle equations and simple Lie algebras, Sov. Sci. Rev.C 4 (1984) 93.
G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS5× S5, JHEP04 (2014) 002 [arXiv:1312.3542] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, Puzzles of η-deformed AdS5× S5 , JHEP12 (2015) 049 [arXiv:1507.04239] [INSPIRE].
G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS5× S5superstring, T-duality and modified type-II equations, Nucl. Phys.B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].
A.A. Tseytlin and L. Wulff, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Weyl invariance for generalized supergravity backgrounds from the doubled formalism, Prog. Theor. Exp. Phys.2017 (2017) 053B07 [arXiv:1703.09213] [INSPIRE].
J.J. Fernández-Melgarejo, J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Weyl Invariance of String Theories in Generalized Supergravity Backgrounds, Phys. Rev. Lett.122 (2019) 111602 [arXiv:1811.10600] [INSPIRE].
W. Mück, Generalized Supergravity Equations and Generalized Fradkin-Tseytlin Counterterm, JHEP05 (2019) 063 [arXiv:1904.06126] [INSPIRE].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
B. Hoare and F.K. Seibold, Supergravity backgrounds of the η-deformed AdS2× S2× T6and AdS5× S5superstrings, JHEP01 (2019) 125 [arXiv:1811.07841] [INSPIRE].
G. Arutyunov and S. Frolov, On String S-matrix, Bound States and TBA, JHEP12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
G. Arutyunov and S.J. van Tongeren, AdS5× S5mirror model as a string σ-model, Phys. Rev. Lett.113 (2014) 261605 [arXiv:1406.2304] [INSPIRE].
G. Arutyunov and S.J. van Tongeren, Double Wick rotating Green-Schwarz strings, JHEP05 (2015) 027 [arXiv:1412.5137] [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models, J. Phys.A 49 (2016) 415402 [arXiv:1606.01712] [INSPIRE].
N. Beisert and P. Koroteev, Quantum Deformations of the One-Dimensional Hubbard Model, J. Phys.A 41 (2008) 255204 [arXiv:0802.0777] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS5× S5 )η superstring, Theor. Math. Phys.182 (2015) 23 [arXiv:1403.6104] [INSPIRE].
B. Hoare and A.A. Tseytlin, Tree-level S-matrix of Pohlmeyer reduced form of AdS5× S5superstring theory, JHEP02 (2010) 094 [arXiv:0912.2958] [INSPIRE].
N. Beisert, The Classical Trigonometric r-Matrix for the Quantum-Deformed Hubbard Chain, J. Phys.A 44 (2011) 265202 [arXiv:1002.1097] [INSPIRE].
B. Hoare and A.A. Tseytlin, Towards the quantum S-matrix of the Pohlmeyer reduced version of AdS5× S5superstring theory, Nucl. Phys.B 851 (2011) 161 [arXiv:1104.2423] [INSPIRE].
B. Hoare, Towards a two-parameter q-deformation of AdS3× S3× M4superstrings, Nucl. Phys.B 891 (2015) 259 [arXiv:1411.1266] [INSPIRE].
C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys.104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
O. Lunin, R. Roiban and A.A. Tseytlin, Supergravity backgrounds for deformations of AdSn × Snsupercoset string models, Nucl. Phys.B 891 (2015) 106 [arXiv:1411.1066] [INSPIRE].
M. Grigoriev and A.A. Tseytlin, Pohlmeyer reduction of AdS5× S5superstring σ-model, Nucl. Phys.B 800 (2008) 450 [arXiv:0711.0155] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS5× S5Superstring. Part I, J. Phys.A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl.17 (1983) 259 [INSPIRE].
L. Wulff, Trivial solutions of generalized supergravity vs. non-abelian T-duality anomaly, Phys. Lett.B 781 (2018) 417 [arXiv:1803.07391] [INSPIRE].
R. Borsato and L. Wulff, Marginal deformations of WZW models and the classical Yang-Baxter equation, J. Phys.A 52 (2019) 225401 [arXiv:1812.07287] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP04 (2002) 013 [hep-th/0202021] [INSPIRE].
M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, Penrose limits and maximal supersymmetry, Class. Quant. Grav.19 (2002) L87 [hep-th/0201081] [INSPIRE].
D. Roychowdhury, On pp wave limit for η deformed superstrings, JHEP05 (2018) 018 [arXiv:1801.07680] [INSPIRE].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdSn × Snsupercosets, JHEP06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
M. Grigoriev and A.A. Tseytlin, On reduced models for superstrings on AdSn × Sn , Int. J. Mod. Phys.A 23 (2008) 2107 [arXiv:0806.2623] [INSPIRE].
A. Pacho-l and S.J. van Tongeren, Quantum deformations of the flat space superstring, Phys. Rev.D 93 (2016) 026008 [arXiv:1510.02389] [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].
T. Araujo, E.Ó. Colgáin and H. Yavartanoo, Embedding the modified CYBE in Supergravity, Eur. Phys. J.C 78 (2018) 854 [arXiv:1806.02602] [INSPIRE].
B. Hoare and S.J. van Tongeren, On Jordanian deformations of AdS5and supergravity, J. Phys.A 49 (2016) 434006 [arXiv:1605.03554] [INSPIRE].
S.J. van Tongeren, Unimodular Jordanian deformations of integrable superstrings, Sci Post Phys.7 (2019) 011 [arXiv:1904.08892] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS5× S5superstring, JHEP04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
D. Orlando, S. Reffert and L.I. Uruchurtu, Classical Integrability of the Squashed Three-sphere, Warped AdS3and Schrödinger Spacetime via T-duality, J. Phys.A 44 (2011) 115401 [arXiv:1011.1771] [INSPIRE].
C. Klimčík and P. Ševera, Dual non-Abelian duality and the Drinfeld double, Phys. Lett.B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
C. Klimčík, Poisson-Lie T duality, Nucl. Phys. Proc. Suppl.46 (1996) 116 [hep-th/9509095] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring σ-models related to AdSn × Snsupercosets, Nucl. Phys.B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS5× S5Superstring, J. Phys.A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS3× S3supercoset, Nucl. Phys.B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
G. Georgiou and K. Sfetsos, The most general λ-deformation of CFTs and integrability, JHEP03 (2019) 094 [arXiv:1812.04033] [INSPIRE].
F. Delduc, B. Hoare, T. Kameyama, S. Lacroix and M. Magro, Three-parameter integrable deformation of ℤ4permutation supercosets, JHEP01 (2019) 109 [arXiv:1811.00453] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3 , JHEP05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3at k = 1, JHEP08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt and M.R. Gaberdiel, String theory on AdS3and the symmetric orbifold of Liouville theory, arXiv:1903.00421 [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: 1907.05430
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
Seibold, F.K. Two-parameter integrable deformations of the AdS3× S3× T4 superstring. J. High Energ. Phys. 2019, 49 (2019). https://doi.org/10.1007/JHEP10(2019)049
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)049