Abstract
We consider so-called Yang-Baxter deformations of bosonic string sigma- models, based on an R-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on R is sufficient for Weyl invariance at least to two loops (first order in α′). Here we ask what the necessary condition is. We find that in cases where the matrix (G + B)mn, constructed from the metric and B-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in α′). The calculations are simplified by working in an O(D, D)-covariant doubled formulation.
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References
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].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS5 × S5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS5 × S5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
B. Hoare and A.A. Tseytlin, Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS5 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].
G. Arutyunov, R. Borsato and S. Frolov, Puzzles of η-deformed AdS5 × S5 , JHEP 12 (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 × S5 superstring, T-duality and modified type-II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [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].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
R. Borsato and L. Wulff, Non-abelian T-duality and Yang-Baxter deformations of Green-Schwarz strings, JHEP 08 (2018) 027 [arXiv:1806.04083] [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].
J.-I. Sakamoto and Y. Sakatani, Local β-deformations and Yang-Baxter σ-model, JHEP 06 (2018) 147 [arXiv:1803.05903] [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].
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].
D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring double field theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].
R. Borsato, A. Vilar López and L. Wulff, The first α′ -correction to homogeneous Yang-Baxter deformations using O(d, d), JHEP 07 (2020) 103 [arXiv:2003.05867] [INSPIRE].
M. Graña and D. Marques, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].
R. Borsato and L. Wulff, Two-loop conformal invariance for Yang-Baxter deformed strings, JHEP 03 (2020) 126 [arXiv:1910.02011] [INSPIRE].
G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [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].
D.S. Berman and D.C. Thompson, Duality symmetric string and M-theory, Phys. Rept. 566 (2014) 1 [arXiv:1306.2643] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
O. Hohm and S.K. Kwak, Frame-like geometry of double field theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].
D. Marques and C.A. Núñez, T-duality and α′ -corrections, JHEP 10 (2015) 084 [arXiv:1507.00652] [INSPIRE].
W.H. Baron, J.J. Fernandez-Melgarejo, D. Marques and C. Núñez, The odd story of α′ -corrections, JHEP 04 (2017) 078 [arXiv:1702.05489] [INSPIRE].
T. Araujo et al., Yang-Baxter σ-models, conformal twists, and noncommutative Yang-Mills theory, Phys. Rev. D 95 (2017) 105006 [arXiv:1702.02861] [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].
I. Bakhmatov, E. Ó Colgáain, 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].
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].
L. Wulff, Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly, Phys. Lett. B 781 (2018) 417 [arXiv:1803.07391] [INSPIRE].
A. Medina and P. Revoy, Lattices in symplectic Lie groups, J. Lie Theor. 17 (2007) 27.
G.P. Ovando, Four dimensional symplectic Lie algebras, Beit. Alg. Geom. 47 (2006) 419 [math/0407501].
R. Campoamor-Stursberg, Symplectic forms on six-dimensional real solvable Lie algebras I, Alg. Colloq. 16 (2009) 253 [math/0507499].
R. Borsato and L. Wulff, Quantum correction to generalized T-dualities, arXiv:2007.07902 [INSPIRE].
F. Hassler and T. Rochais, α′ -corrected Poisson-Lie T-duality, arXiv:2007.07897 [INSPIRE].
T. Codina and D. Marques, Generalized dualities and higher derivatives, arXiv:2007.09494 [INSPIRE].
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Hronek, S., Wulff, L. Relaxing unimodularity for Yang-Baxter deformed strings. J. High Energ. Phys. 2020, 65 (2020). https://doi.org/10.1007/JHEP10(2020)065
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DOI: https://doi.org/10.1007/JHEP10(2020)065