We study the fermionic T-duality symmetry of integrable Green-Schwarz sigma-models on AdS backgrounds with Ramond-Ramond fluxes in various dimensions. We show that sigma-models based on supercosets of PSU supergroups, such as AdS 2 × S 2 and AdS 3 × S 3 are self-dual under fermionic T-duality, while supercosets of OSp supergroups such as non-critical AdS 2 and AdS 4 models, and the critical AdS 4 × CP 3 background are not. We present a general algebraic argument to when a supercoset is expected to have a fermionic T-duality symmetry, and when it will fail to have one.

We consider type II superstrings on AdS backgrounds with Ramond-Ramond flux in various dimensions. We realize the backgrounds as supercosets and analyze explicitly two classes of models: non-critical superstrings on AdS 2d and critical superstrings on AdS p × S p × CY . We work both in the Green-Schwarz and in the pure spinor formalisms. We construct a one-parameter family of flat currents (a Lax connection), leading to an infinite number of conserved non-local charges, which imply the classical integrability of both sigma-models. In the pure spinor formulation, we use the BRST symmetry to prove the quantum integrability of the sigma-model. We discuss how classical κ-symmetry implies one-loop conformal invariance. We consider the addition of space-filling D-branes to the pure spinor formalism.

We examine the Weyl anomaly for a four-dimensional z = 3 Lifshitz scalar coupled to Hořava's theory of anisotropic gravity. We find a one-loop break-down of scale-invariance at second order in the gravitational background.