Target space supergeometry of $\eta$ and $\lambda$-deformed strings

Abstract

We study the integrable $\eta$ and $\lambda$-deformations of supercoset string sigma models, the basic example being the deformation of the $Ad{S}_5\times S^5$ 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 $\lambda$-deformation gives rise to a standard (generically type II*) supergravity background; for the $\eta$-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 so(2,4). We argue that the corresponding backgrounds are equivalent to sequences of non-commuting TsT-transformations, and verify this explicitly for some of the examples.

32 pages. Remarks of referee addressed: references and comments added. Published version

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• $I$ in generalized supergravity
• General phase spaces: from discrete variables to rotor and continuum limits
• Generalized Dualities and Higher Derivatives
• Yang-Baxter deformations of the Principal Chiral Model plus Wess-Zumino term
• Unimodular jordanian deformations of integrable superstrings
• Marginal and non-commutative deformations via non-abelian T-duality
• Integrable sigma models and 2-loop RG flow
• Analytic integrability for strings on $\eta$ and $\lambda$ deformed backgrounds
• Poisson-Lie duals of the eta deformed symmetric space sigma model
• Integrable asymmetric $\lambda$-deformations
• An exact symmetry in $\lambda$-deformed CFTs
• Generalized Supergravity Equations and Generalized Fradkin-Tseytlin Counterterm
• Exploring The Lambda Model Of The Hybrid Superstring
• On quantum deformations of $Ad{S}_3\times S^3\times T^4$ and mirror duality
• Finite-size effect of η-deformed AdS_5 x S^5 at strong coupling
• Tri-vector deformations on compact isometries
• Novel integrable interpolations
• Asymptotics in an Asymptotic CFT
• Integrable $\lambda$-deformations of the Euclidean black string
• Extended Drinfel'd algebras and non-Abelian duality
• An invitation to Poisson-Lie T-duality in Double Field Theory and its applications