Abstract
We perform non-abelian T-duality for a generic Green-Schwarz string with respect to an isometry (super)group G, and we derive the transformation rules for the supergravity background fields. Specializing to G bosonic, or G fermionic but abelian, our results reproduce those available in the literature. We discuss also continuous deformations of the T-dual models, obtained by adding a closed B-field before the dualization. This idea can also be used to generate deformations of the original (un-dualized) model, when the 2-cocycle identified from the closed B is invertible. The latter construction is the natural generalization of the so-called Yang-Baxter deformations, based on solutions of the classical Yang-Baxter equation on the Lie algebra of G and originally constructed for group manifolds and (super)coset sigma models. We find that the deformed metric and B-field are obtained through a generalization of the map between open and closed strings that was used also in the discussion by Seiberg and Witten of non-commutative field theories. When applied to integrable sigma models these deformations preserve the integrability.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
X.C. de la Ossa and F. Quevedo, Duality symmetries from nonAbelian isometries in string theory, Nucl. Phys. B 403 (1993) 377 [hep-th/9210021] [INSPIRE].
A. Giveon and M. Roček, On nonAbelian duality, Nucl. Phys. B 421 (1994) 173 [hep-th/9308154] [INSPIRE].
E. Alvarez, L. Álvarez-Gaumé, J.L.F. Barbon and Y. Lozano, Some global aspects of duality in string theory, Nucl. Phys. B 415 (1994) 71 [hep-th/9309039] [INSPIRE].
K. Sfetsos and D.C. Thompson, On non-abelian T-dual geometries with Ramond fluxes, Nucl. Phys. B 846 (2011) 21 [arXiv:1012.1320] [INSPIRE].
Y. Lozano, E. Ó Colgáin, D. Rodríguez-Gómez and K. Sfetsos, Supersymmetric AdS 6 via T Duality, Phys. Rev. Lett. 110 (2013) 231601 [arXiv:1212.1043] [INSPIRE].
G. Itsios, C. Núñez, K. Sfetsos and D.C. Thompson, On Non-Abelian T-duality and new N = 1 backgrounds, Phys. Lett. B 721 (2013) 342 [arXiv:1212.4840] [INSPIRE].
G. Itsios, C. Núñez, K. Sfetsos and D.C. Thompson, Non-Abelian T-duality and the AdS/CFT correspondence:new N = 1 backgrounds, Nucl. Phys. B 873 (2013) 1 [arXiv:1301.6755] [INSPIRE].
Y. Lozano and C. Núñez, Field theory aspects of non-Abelian T-duality and \( \mathcal{N} \) = 2 linear quivers, JHEP 05 (2016) 107 [arXiv:1603.04440] [INSPIRE].
Y. Lozano, N.T. Macpherson, J. Montero and C. Núñez, Three-dimensional \( \mathcal{N} \) = 4 linear quivers and non-Abelian T-duals, JHEP 11 (2016) 133 [arXiv:1609.09061] [INSPIRE].
Y. Lozano, C. Núñez and S. Zacarias, BMN Vacua, Superstars and Non-Abelian T-duality, JHEP 09 (2017) 000 [arXiv:1703.00417] [INSPIRE].
Y. Lozano, NonAbelian duality and canonical transformations, Phys. Lett. B 355 (1995) 165 [hep-th/9503045] [INSPIRE].
K. Sfetsos, NonAbelian duality, parafermions and supersymmetry, Phys. Rev. D 54 (1996) 1682 [hep-th/9602179] [INSPIRE].
K. Sfetsos, Canonical equivalence of nonisometric σ-models and Poisson-Lie T duality, Nucl. Phys. B 517 (1998) 549 [hep-th/9710163] [INSPIRE].
B. Hoare and A.A. Tseytlin, Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS 5 σ-model, J. Phys. A 49 (2016) 494001 [arXiv:1609.02550] [INSPIRE].
R. Borsato and L. Wulff, Integrable Deformations of T -Dual σ Models, Phys. Rev. Lett. 117 (2016) 251602 [arXiv:1609.09834] [INSPIRE].
R. Borsato and L. Wulff, On non-abelian T-duality and deformations of supercoset string σ-models, JHEP 10 (2017) 024 [arXiv:1706.10169] [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS 5 × S 5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
Y. Lozano, E. Ó Colgáin, K. Sfetsos and D.C. Thompson, Non-abelian T-duality, Ramond Fields and Coset Geometries, JHEP 06 (2011) 106 [arXiv:1104.5196] [INSPIRE].
S.F. Hassan, T duality, space-time spinors and RR fields in curved backgrounds, Nucl. Phys. B 568 (2000) 145 [hep-th/9907152] [INSPIRE].
N. Berkovits and J. Maldacena, Fermionic T-duality, Dual Superconformal Symmetry and the Amplitude/Wilson Loop Connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [INSPIRE].
L. Wulff and A.A. Tseytlin, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS 5 × S 5 superstring, T-duality and modified type-II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].
E. Alvarez, L. Álvarez-Gaumé and Y. Lozano, On nonAbelian duality, Nucl. Phys. B 424 (1994) 155 [hep-th/9403155] [INSPIRE].
S. Elitzur, A. Giveon, E. Rabinovici, A. Schwimmer and G. Veneziano, Remarks on nonAbelian duality, Nucl. Phys. B 435 (1995) 147 [hep-th/9409011] [INSPIRE].
L. Wulff, Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly, Phys. Lett. B 781 (2018) 417 [arXiv:1803.07391] [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (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, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 × S 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
T. Matsumoto and K. Yoshida, Yang-Baxter σ-models based on the CYBE, Nucl. Phys. B 893 (2015) 287 [arXiv:1501.03665] [INSPIRE].
S.J. van Tongeren, On classical Yang-Baxter based deformations of the AdS 5 × S 5 superstring, JHEP 06 (2015) 048 [arXiv:1504.05516] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
B. Hoare and D.C. Thompson, Marginal and non-commutative deformations via non-abelian T-duality, JHEP 02 (2017) 059 [arXiv:1611.08020] [INSPIRE].
D. Osten and S.J. van Tongeren, Abelian Yang-Baxter deformations and TsT transformations, Nucl. Phys. B 915 (2017) 184 [arXiv:1608.08504] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
T. Araujo, I. Bakhmatov, E. Ó Colgáin, J. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, Yang-Baxter σ-models, conformal twists and noncommutative Yang-Mills theory, Phys. Rev. D 95 (2017) 105006 [arXiv:1702.02861] [INSPIRE].
I. Bakhmatov, Ö. Kelekci, E. Ó Colgáin and M.M. Sheikh-Jabbari, Classical Yang-Baxter Equation from Supergravity, Phys. Rev. D 98 (2018) 021901 [arXiv:1710.06784] [INSPIRE].
I. Bakhmatov, E. Ó Colgáin, M.M. Sheikh-Jabbari and H. Yavartanoo, Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT), JHEP 06 (2018) 161 [arXiv:1803.07498] [INSPIRE].
J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Homogeneous Yang-Baxter deformations as generalized diffeomorphisms, J. Phys. A 50 (2017) 415401 [arXiv:1705.07116] [INSPIRE].
D. Lüst and D. Osten, Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T-duality, JHEP 05 (2018) 165 [arXiv:1803.03971] [INSPIRE].
J.-I. Sakamoto and Y. Sakatani, Local β-deformations and Yang-Baxter σ-model, JHEP 06 (2018) 147 [arXiv:1803.05903] [INSPIRE].
L. Wulff, The type-II superstring to order θ 4, JHEP 07 (2013) 123 [arXiv:1304.6422] [INSPIRE].
M. Hong, Y. Kim and E. Ó Colgáin, On non-Abelian T-duality for non-semisimple groups, arXiv:1801.09567 [INSPIRE].
K. Sfetsos, Duality invariant class of two-dimensional field theories, Nucl. Phys. B 561 (1999) 316 [hep-th/9904188] [INSPIRE].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
C. Klimčík and P. Ševera, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
S.F. Hassan, SO(d, d) transformations of Ramond-Ramond fields and space-time spinors, Nucl. Phys. B 583 (2000) 431 [hep-th/9912236] [INSPIRE].
B. Hoare and S.J. van Tongeren, On Jordanian deformations of AdS 5 and supergravity, J. Phys. A 49 (2016) 434006 [arXiv:1605.03554] [INSPIRE].
S.J. van Tongeren, Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory, Nucl. Phys. B 904 (2016) 148 [arXiv:1506.01023] [INSPIRE].
S.J. van Tongeren, Almost abelian twists and AdS/CFT, Phys. Lett. B 765 (2017) 344 [arXiv:1610.05677] [INSPIRE].
N. Seiberg, L. Susskind and N. Toumbas, Strings in background electric field, space/time noncommutativity and a new noncritical string theory, JHEP 06 (2000) 021 [hep-th/0005040] [INSPIRE].
T. Matsumoto and K. Yoshida, Integrability of classical strings dual for noncommutative gauge theories, JHEP 06 (2014) 163 [arXiv:1404.3657] [INSPIRE].
E. Tyurin and R. von Unge, Poisson-lie T duality: The Path integral derivation, Phys. Lett. B 382 (1996) 233 [hep-th/9512025] [INSPIRE].
R. Von Unge, Poisson Lie T plurality, JHEP 07 (2002) 014 [hep-th/0205245] [INSPIRE].
F. Cordonier-Tello, D. Lüst and E. Plauschinn, Open-string T-duality and applications to non-geometric backgrounds, arXiv:1806.01308 [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, Puzzles of η-deformed AdS 5 × S 5, JHEP 12 (2015) 049 [arXiv:1507.04239] [INSPIRE].
B. Hoare and A.A. Tseytlin, Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring, JHEP 10 (2015) 060 [arXiv:1508.01150] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
D. Bombardelli et al., An integrability primer for the gauge-gravity correspondence: An introduction, J. Phys. A 49 (2016) 320301 [arXiv:1606.02945] [INSPIRE].
N. Beisert and R. Roiban, Beauty and the twist: The Bethe ansatz for twisted N = 4 SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, The spectral problem for strings on twisted AdS 5 × S 5, Nucl. Phys. B 860 (2012) 339 [arXiv:1201.1451] [INSPIRE].
M. Guica, F. Levkovich-Maslyuk and K. Zarembo, Integrability in dipole-deformed \( \mathcal{N} \) = 4 super Yang-Mills, J. Phys. A 50 (2017) 394001 [arXiv:1706.07957] [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: 1806.04083
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
Borsato, R., Wulff, L. Non-abelian T-duality and Yang-Baxter deformations of Green-Schwarz strings. J. High Energ. Phys. 2018, 27 (2018). https://doi.org/10.1007/JHEP08(2018)027
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)027