Self-duality of Green-Schwarz sigma-models

We study fermionic T-duality symmetries of integrable Green-Schwarz sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes, constructed as ℤ₄ supercosets of superconformal algebras. We find three algebraic conditions that guarantee self-duality of the backgrounds under fermionic T-duality, we classify those that satisfy them and construct the map of the monodromy matrix. We introduce new T-duality directions, where some of them contain no bosonic directions, along which the backgrounds are self-dual. We find that the only self-dual backgrounds are AdS_n ×Sⁿ for n = 2,3,5. In addition we find that the backgrounds AdS_n ×S¹ for n = 2,3,5, AdS₄ ×S² and AdS₂ ×S⁴ are self-dual at the level of the classical action, but have a non-trivial transformation of the dilaton.