Abstract
We study the integrable η and λ-deformations of supercoset string sigma models, the basic example being the deformation of the AdS 5 × S 5 superstring. We prove that the kappa symmetry variations for these models are of the standard Green-Schwarz form, and we determine the target space supergeometry by computing the superspace torsion. We check that the λ-deformation gives rise to a standard (generically type II*) supergravity background; for the η-model the requirement that the target space is a supergravity solution translates into a simple condition on the R-matrix which enters the definition of the deformation. We further construct all such non-abelian R-matrices of rank four which solve the homogeneous classical Yang-Baxter equation for the algebra \( \mathfrak{so} \)(2, 4). We argue that most of the corresponding backgrounds are equivalent to sequences of non-commuting TsT-transformations, and verify this explicitly for some of the examples.
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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].
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. Wulff, Superisometries and integrability of superstrings, JHEP 05 (2014) 115 [arXiv:1402.3122] [INSPIRE].
L. Wulff, On integrability of strings on symmetric spaces, JHEP 09 (2015) 115 [arXiv:1505.03525] [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].
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].
I.V. Cherednik, Relativistically invariant quasiclassical limits of integrable two-dimensional quantum models, Theor. Math. Phys. 47 (1981) 422 [Teor. Mat. Fiz. 47 (1981) 225] [INSPIRE].
C. Klimč´ık, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimč´ık, 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].
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].
A.A. Tseytlin, On a ‘universal’ class of WZW type conformal models, Nucl. Phys. B 418 (1994) 173 [hep-th/9311062] [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].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, S-matrices and quantum group symmetry of k-deformed σ-models, arXiv:1506.06601 [INSPIRE].
C. Klimčík and P. Ševera, Dual non-Abelian duality and the Drinfel’d double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
C. Klimčík and P. Ševera, Poisson-Lie T duality and loop groups of Drinfel’d doubles, Phys. Lett. B 372 (1996) 65 [hep-th/9512040] [INSPIRE].
B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys. A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring σ-models related to AdS n × S n supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
C. Klimčík, η and λ deformations as ε-models, Nucl. Phys. B 900 (2015) 259 [arXiv:1508.05832] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS 5 × S 5, JHEP 04 (2014) 002 [arXiv:1312.3542] [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, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
R. Borsato, Integrable strings for AdS/CFT, arXiv:1605.03173 [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].
R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS 2 × S 2 supercoset, Nucl. Phys. B 905 (2016) 264 [arXiv:1601.08192] [INSPIRE].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS 3 × S 3 supercoset, Nucl. Phys. B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: squashing coset CFTs and AdS 5 × S 5, JHEP 07 (2015) 019 [arXiv:1504.02781] [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].
A. Mikhailov, Cornering the unphysical vertex, JHEP 11 (2012) 082 [arXiv:1203.0677] [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].
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, Integrable deformations of the AdS 5 × S 5 superstring and the classical Yang-Baxter equation — towards the gravity/CYBE correspondence, J. Phys. Conf. Ser. 563 (2014) 012020 [arXiv:1410.0575] [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].
L. Wulff, The type-II superstring to order θ 4, JHEP 07 (2013) 123 [arXiv:1304.6422] [INSPIRE].
H. Kyono and K. Yoshida, Supercoset construction of Yang-Baxter deformed AdS 5 × S 5 backgrounds, arXiv:1605.02519 [INSPIRE].
O.A. Bedoya, L.I. Bevilaqua, A. Mikhailov and V.O. Rivelles, Notes on β-deformations of the pure spinor superstring in AdS 5 × S 5, Nucl. Phys. B 848 (2011) 155 [arXiv:1005.0049] [INSPIRE].
B. Hoare and S.J. van Tongeren, Non-split and split deformations of AdS 5, arXiv:1605.03552 [INSPIRE].
B. Hoare and S.J. van Tongeren, On Jordanian deformations of AdS 5 and supergravity, arXiv:1605.03554 [INSPIRE].
D. Orlando, S. Reffert, J.-I. Sakamoto and K. Yoshida, Generalized type IIB supergravity equations and non-Abelian classical r-matrices, arXiv:1607.00795 [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.A. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for superconformal deformations of N = 4 super Yang-Mills theory, JHEP 07 (2005) 045 [hep-th/0503192] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
U. Gürsoy and C. Núñez, Dipole deformations of N = 1 SYM and supergravity backgrounds with U(1) × U(1) global symmetry, Nucl. Phys. B 725 (2005) 45 [hep-th/0505100] [INSPIRE].
R.R. Metsaev, Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background, Nucl. Phys. B 625 (2002) 70 [hep-th/0112044] [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].
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. Stefanski, jr, Green-Schwarz action for type IIA strings on AdS 4 × CP 3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [INSPIRE].
J. Gomis, D. Sorokin and L. Wulff, The complete AdS 4 × CP 3 superspace for the type IIA superstring and D-branes, JHEP 03 (2009) 015 [arXiv:0811.1566] [INSPIRE].
A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3 /CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
D. Sorokin, A. Tseytlin, L. Wulff and K. Zarembo, Superstrings in AdS 2 × S 2 × T 6, J. Phys. A 44 (2011) 275401 [arXiv:1104.1793] [INSPIRE].
V.N. Tolstoy, Chains of extended Jordanian twists for Lie superalgebras, math/0402433.
A. Stolin, On rational solutions of Yang-Baxter equation for \( \mathfrak{s}\mathfrak{l} \)(n), Math. Scand. 69 (1991) 57.
A. Stolin, Rational solutions of the classical Yang-Baxter equation and quasi Frobenius Lie algebras, J. Pure Appl. Alg. 137 (1999) 285.
M. Gerstenhaber and A. Giaquinto, Boundary solutions of the classical Yang-Baxter equation, Lett. Math. Phys. 40 (1997) 337.
A. Stolin, Constant solutions of Yang-Baxter equation for \( \mathfrak{s}\mathfrak{l} \)(2) and \( \mathfrak{s}\mathfrak{l} \)(3), Math. Scand. 69 (1991) 81.
A. Lichnerowicz and A. Medina, On Lie groups with left-invariant symplectic or Kählerian structures, Lett. Math. Phys. 16 (1988) 225.
G. Ovando, Four dimensional symplectic Lie algebras, Beiträge Alg. Geom. 47 (2006) 419.
J. Patera, P. Winternitz and H. Zassenhaus, The maximal solvable subgroups of the SU(p,q) groups and all subgroups of SU(2, 1), J. Math. Phys. 15 (1974) 1378 [INSPIRE].
J. Patera, P. Winternitz and H. Zassenhaus, The maximal solvable subgroups of SO(p,q) groups, J. Math. Phys. 15 (1974) 1932.
L.F. Alday, G. Arutyunov and S. Frolov, Green-Schwarz strings in TsT-transformed backgrounds, JHEP 06 (2006) 018 [hep-th/0512253] [INSPIRE].
S. Elitzur, A. Giveon, E. Rabinovici, A. Schwimmer and G. Veneziano, Remarks on non-Abelian duality, Nucl. Phys. B 435 (1995) 147 [hep-th/9409011] [INSPIRE].
E. Bergshoeff, C.M. Hull and T. Ortín, Duality in the type-II superstring effective action, Nucl. Phys. B 451 (1995) 547 [hep-th/9504081] [INSPIRE].
M.B. Green, C.M. Hull and P.K. Townsend, D-brane Wess-Zumino actions, t duality and the cosmological constant, Phys. Lett. B 382 (1996) 65 [hep-th/9604119] [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].
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].
T. Matsumoto and K. Yoshida, Integrability of classical strings dual for noncommutative gauge theories, JHEP 06 (2014) 163 [arXiv:1404.3657] [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].
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Borsato, R., Wulff, L. Target space supergeometry of η and λ-deformed strings. J. High Energ. Phys. 2016, 45 (2016). https://doi.org/10.1007/JHEP10(2016)045
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DOI: https://doi.org/10.1007/JHEP10(2016)045