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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 Z_4 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 x S^n for n=2,3,5. In addition we find that the backgrounds AdS_n x S^1 for n=2,3,5, AdS_4 x S^2 and AdS_2 x S^4 are self-dual at the level of the classical action, but have a non-trivial transformation of the dilaton.