Abstract
Homogeneous Yang-Baxter (YB) deformation of AdS5 × S5 superstring is revisited. We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to β-deformations of the AdS5 ×S5 background when the classical r-matrices consist of bosonic generators. In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory. We further provide an O(10, 10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadratic order in fermions. When an AdS background contains a non-vanishing H-flux, it is not straightforward to perform homogeneous YB deformations. In order to get any hint for such YB deformations, we study β-deformations of H-fluxed AdS backgrounds and obtain various solutions of (generalized) type II supergravity.
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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].
T. Matsumoto and K. Yoshida, Yang-Baxter σ-models based on the CYBE, Nucl. Phys. B 893 (2015) 287 [arXiv:1501.03665] [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].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS 5 × S 5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [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, Lunin-Maldacena backgrounds from the classical Yang-Baxter equation - towards the gravity/CYBE correspondence, JHEP 06 (2014) 135 [arXiv:1404.1838] [INSPIRE].
T. Matsumoto and K. Yoshida, Integrability of classical strings dual for noncommutative gauge theories, JHEP 06 (2014) 163 [arXiv:1404.3657] [INSPIRE].
T. Matsumoto and K. Yoshida, Schrödinger geometries arising from Yang-Baxter deformations, JHEP 04 (2015) 180 [arXiv:1502.00740] [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].
S.J. van Tongeren, Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory, Nucl. Phys. B 904 (2016) 148 [arXiv:1506.01023] [INSPIRE].
H. Kyono and K. Yoshida, Supercoset construction of Yang-Baxter deformed AdS 5 × S 5 backgrounds, PTEP 2016 (2016) 083B03 [arXiv:1605.02519] [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].
A. Hashimoto and N. Itzhaki, Noncommutative Yang-Mills and the AdS/CFT correspondence, Phys. Lett. B 465 (1999) 142 [hep-th/9907166] [INSPIRE].
J.M. Maldacena and J.G. Russo, Large N limit of noncommutative gauge theories, JHEP 09 (1999) 025 [hep-th/9908134] [INSPIRE].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [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].
A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [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].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
D. Orlando, S. Reffert, J. Sakamoto and K. Yoshida, Generalized type IIB supergravity equations and non-Abelian classical r-matrices, J. Phys. A 49 (2016) 445403 [arXiv:1607.00795] [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].
B. Hoare and D.C. Thompson, Marginal and non-commutative deformations via non-abelian T-duality, JHEP 02 (2017) 059 [arXiv:1611.08020] [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].
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].
B.E. Fridling and A. Jevicki, Dual Representations and Ultraviolet Divergences in Nonlinear σ Models, Phys. Lett. B 134 (1984) 70 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Quantum Equivalence of Dual Field Theories, Annals Phys. 162 (1985) 31 [INSPIRE].
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].
M. Gasperini, R. Ricci and G. Veneziano, A Problem with nonAbelian duality?, Phys. Lett. B 319 (1993) 438 [hep-th/9308112] [INSPIRE].
A. Giveon and M. Roček, On nonAbelian duality, Nucl. Phys. B 421 (1994) 173 [hep-th/9308154] [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].
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, K. Sfetsos and D.C. Thompson, Non-abelian T-duality, Ramond Fields and Coset Geometries, JHEP 06 (2011) 106 [arXiv:1104.5196] [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].
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].
A.A. Tseytlin and L. Wulff, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
L. Wulff, Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly, Phys. Lett. B 781 (2018) 417 [arXiv:1803.07391] [INSPIRE].
M. Hong, Y. Kim and E. Ó Colgáin, On non-Abelian T-duality for non-semisimple groups, arXiv:1801.09567 [INSPIRE].
Y. Sakatani, S. Uehara and K. Yoshida, Generalized gravity from modified DFT, JHEP 04 (2017) 123 [arXiv:1611.05856] [INSPIRE].
A. Baguet, M. Magro and H. Samtleben, Generalized IIB supergravity from exceptional field theory, JHEP 03 (2017) 100 [arXiv:1612.07210] [INSPIRE].
J. Sakamoto, Y. Sakatani and K. Yoshida, Weyl invariance for generalized supergravity backgrounds from the doubled formalism, PTEP 2017 (2017) 053B07 [arXiv:1703.09213] [INSPIRE].
J. Sakamoto, Y. Sakatani and K. Yoshida, Homogeneous Yang-Baxter deformations as generalized diffeomorphisms, J. Phys. A 50 (2017) 415401 [arXiv:1705.07116] [INSPIRE].
J.J. Fernandez-Melgarejo, J. Sakamoto, Y. Sakatani and K. Yoshida, T -folds from Yang-Baxter deformations, JHEP 12 (2017) 108 [arXiv:1710.06849] [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
W. Siegel, Manifest duality in low-energy superstrings, in International Conference on Strings 93 Berkeley, California, May 24-29, 1993, pp. 353-363 [hep-th/9308133] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
M.B. Green and J.H. Schwarz, Covariant Description of Superstrings, Phys. Lett. B 136 (1984) 367 [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D 84 (2011) 044022 [arXiv:1105.6294] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity, Phys. Rev. D 85 (2012) 081501 [Erratum ibid. D 86 (2012) 089903] [arXiv:1112.0069] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Ramond-Ramond Cohomology and O(D, D) T-duality, JHEP 09 (2012) 079 [arXiv:1206.3478] [INSPIRE].
I. Jeon, K. Lee, J.-H. Park and Y. Suh, Stringy Unification of Type IIA and IIB Supergravities under N = 2 D = 10 Supersymmetric Double Field Theory, Phys. Lett. B 723 (2013) 245 [arXiv:1210.5078] [INSPIRE].
C.M. Hull, Doubled Geometry and T-Folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].
C.D.A. Blair, E. Malek and A.J. Routh, An O(D, D) invariant Hamiltonian action for the superstring, Class. Quant. Grav. 31 (2014) 205011 [arXiv:1308.4829] [INSPIRE].
S. Driezen, A. Sevrin and D.C. Thompson, Aspects of the Doubled Worldsheet, JHEP 12 (2016) 082 [arXiv:1609.03315] [INSPIRE].
I. Bandos, Superstring in doubled superspace, Phys. Lett. B 751 (2015) 408 [arXiv:1507.07779] [INSPIRE].
J.-H. Park, Green-Schwarz superstring on doubled-yet-gauged spacetime, JHEP 11 (2016) 005 [arXiv:1609.04265] [INSPIRE].
I. Bandos, Type II superstring in doubled superspace, Fortsch. Phys. 64 (2016) 361 [INSPIRE].
M. Hatsuda, K. Kamimura and W. Siegel, Ramond-Ramond gauge fields in superspace with manifest T-duality, JHEP 02 (2015) 134 [arXiv:1411.2206] [INSPIRE].
M. Hatsuda, K. Kamimura and W. Siegel, Type II chiral affine Lie algebras and string actions in doubled space, JHEP 09 (2015) 113 [arXiv:1507.03061] [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, A deformation of quantum affine algebra in squashed Wess-Zumino-Novikov-Witten models, J. Math. Phys. 55 (2014) 062302 [arXiv:1311.4696] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Integrable double deformation of the principal chiral model, Nucl. Phys. B 891 (2015) 312 [arXiv:1410.8066] [INSPIRE].
F. Delduc, B. Hoare, T. Kameyama and M. Magro, Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable σ-model, JHEP 10 (2017) 212 [arXiv:1707.08371] [INSPIRE].
S. Demulder, S. Driezen, A. Sevrin and D.C. Thompson, Classical and Quantum Aspects of Yang-Baxter Wess-Zumino Models, JHEP 03 (2018) 041 [arXiv:1711.00084] [INSPIRE].
O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, Puzzles of η-deformed AdS 5 × S 5, JHEP 12 (2015) 049 [arXiv:1507.04239] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Differential geometry with a projection: Application to double field theory, JHEP 04 (2011) 014 [arXiv:1011.1324] [INSPIRE].
O. Hohm and B. Zwiebach, On the Riemann Tensor in Double Field Theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].
M. Fukuma, T. Oota and H. Tanaka, Comments on T dualities of Ramond-Ramond potentials on tori, Prog. Theor. Phys. 103 (2000) 425 [hep-th/9907132] [INSPIRE].
E. Bergshoeff, R. Kallosh, T. Ortín, D. Roest and A. Van Proeyen, New formulations of D = 10 supersymmetry and D8-O8 domain walls, Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [INSPIRE].
O. Hohm, D. Lüst and B. Zwiebach, The Spacetime of Double Field Theory: Review, Remarks and Outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [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].
O. Hohm, S.K. Kwak and B. Zwiebach, Double Field Theory of Type II Strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].
M.J. Duff, Duality Rotations in String Theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].
A.A. Tseytlin, Duality Symmetric Formulation of String World Sheet Dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].
A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].
C.M. Hull, A Geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].
N.B. Copland, A Double σ-model for Double Field Theory, JHEP 04 (2012) 044 [arXiv:1111.1828] [INSPIRE].
K. Lee and J.-H. Park, Covariant action for a string in “doubled yet gauged” spacetime, Nucl. Phys. B 880 (2014) 134 [arXiv:1307.8377] [INSPIRE].
M. Cvetič, H. Lü, C.N. Pope and K.S. 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].
R.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].
S. Förste, A Truly marginal deformation of SL(2, R) in a null direction, Phys. Lett. B 338 (1994) 36 [hep-th/9407198] [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].
A. Giveon, N. Itzhaki and D. Kutasov, \( \mathrm{T}\overline{\mathrm{T}} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, A solvable irrelevant deformation of AdS 3 /CFT 2, JHEP 12 (2017) 155 [arXiv:1707.05800] [INSPIRE].
M. Asrat, A. Giveon, N. Itzhaki and D. Kutasov, Holography Beyond AdS, arXiv:1711.02690 [INSPIRE].
G. Giribet, \( T\overline{T} \) -deformations, AdS/CFT and correlation functions, JHEP 02 (2018) 114 [arXiv:1711.02716] [INSPIRE].
T. Azeyanagi, D.M. Hofman, W. Song and A. Strominger, The Spectrum of Strings on Warped AdS 3 × S 3, JHEP 04 (2013) 078 [arXiv:1207.5050] [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].
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].
T. Araujo, I. Bakhmatov, E. Ó. Colgáin, J. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, Conformal twists, Yang-Baxter σ-models & holographic noncommutativity, J. Phys. A 51 (2018) 235401 [arXiv:1705.02063] [INSPIRE].
T. Araujo, E. Ó Colgáin, J. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, I in generalized supergravity, Eur. Phys. J. C 77 (2017) 739 [arXiv:1708.03163] [INSPIRE].
O. Hohm and B. Zwiebach, Large Gauge Transformations in Double Field Theory, JHEP 02 (2013) 075 [arXiv:1207.4198] [INSPIRE].
C.M. Hull and B.J. Spence, The Gauged Nonlinear σ Model With Wess-Zumino Term, Phys. Lett. B 232 (1989) 204 [INSPIRE].
C.M. Hull and B.J. Spence, The Geometry of the gauged σ-model with Wess-Zumino term, Nucl. Phys. B 353 (1991) 379 [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), arXiv:1803.07498 [INSPIRE].
C. Klimčík and P. Ševera, Poisson-Lie T duality and loop groups of Drinfeld doubles, Phys. Lett. B 372 (1996) 65 [hep-th/9512040] [INSPIRE].
C. Klimčík and P. Ševera, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
R.A. Reid-Edwards, Bi-Algebras, Generalised Geometry and T-duality, arXiv:1001.2479 [INSPIRE].
F. Hassler, Poisson-Lie T-duality in Double Field Theory, arXiv:1707.08624 [INSPIRE].
F. Rennecke, O(d, d)-Duality in String Theory, JHEP 10 (2014) 69 [arXiv:1404.0912] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
L. Castellani, R. D’Auria and P. Fre, Supergravity and superstrings: A Geometric perspective. Vol. 1: Mathematical foundations, World Scientific, Singapore, Singapore (1991), pp. 1-603 [INSPIRE].
T. Ortin, Gravity and strings, Cambridge University Press (2004) [INSPIRE]
M.T. Grisaru, P.S. Howe, L. Mezincescu, B. Nilsson and P.K. Townsend, N = 2 Superstrings in a Supergravity Background, Phys. Lett. B 162 (1985) 116 [INSPIRE].
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Sakamoto, Ji., Sakatani, Y. Local β-deformations and Yang-Baxter sigma model. J. High Energ. Phys. 2018, 147 (2018). https://doi.org/10.1007/JHEP06(2018)147
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DOI: https://doi.org/10.1007/JHEP06(2018)147