We construct supergravity backgrounds for the integrable η-deformations of the AdS2 × S2 × T6 and AdS5 × S5 superstring sigma models. The η-deformation is governed by an R-matrix that solves the non-split modified classical Yang-Baxter equation on the superisometry algebra of the model. Such R-matrices include those of Drinfel’d-Jimbo type, which are constructed from a Dynkin diagram and the associated Cartan-Weyl basis. Drinfel’d-Jimbo R-matrices associated with inequivalent bases will typically lead to different deformed backgrounds. For the two models under consideration we find that the unimodularity condition, implying that there is no Weyl anomaly, is satisfied if and only if all the simple roots are fermionic. For AdS2 × S2 × T6 we construct backgrounds corresponding to the three Dynkin diagrams. When all the simple roots are fermionic we find a supergravity background previously obtained by directly solving the supergravity equations. For AdS5 × S5 we construct a supergravity background corresponding to the Dynkin diagram with all fermionic simple roots.
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].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [Dokl. Akad. Nauk Ser. Fiz. 283 (1985) 1060] [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.
M.B. Green and J.H. Schwarz, Covariant description of superstrings, Phys. Lett. B 136 (1984) 367 [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].
M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983) 259 [Funkt. Anal. Pril. 17N4 (1983) 17] [INSPIRE].
V.A. Fateev, E. Onofri and A.B. Zamolodchikov, The sausage model (integrable deformations of O(3) σ-model), Nucl. Phys. B 406 (1993) 521 [INSPIRE].
C. Klimčík, Poisson-Lie T duality, Nucl. Phys. Proc. Suppl. 46 (1996) 116 [hep-th/9509095] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS 5 × S 5 ) η superstring, Theor. Math. Phys. 182 (2015) 23 [Teor. Mat. Fiz. 182 (2014) 28] [arXiv:1403.6104] [INSPIRE].
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ArXiv ePrint: 1811.07841
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Hoare, B., Seibold, F.K. Supergravity backgrounds of the η-deformed AdS2 × S2 × T6 and AdS5 × S5 superstrings. J. High Energ. Phys. 2019, 125 (2019). https://doi.org/10.1007/JHEP01(2019)125
- Integrable Field Theories
- Sigma Models
- Superstrings and Heterotic Strings