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

We study the integrable $\eta$ and $\lambda$-deformations of supercoset string sigma models, the basic example being the deformation of the $AdS_5 \times S^5$ 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.