On Integrable Backgrounds Self-dual under Fermionic T-duality

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 AdS₂ \times S² and AdS₃ \times S³ are self-dual under fermionic T-duality, while supercosets of OSp supergroups such as non-critical AdS₂ and AdS₄ models, and the critical AdS₄ \times CP³ 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