Protected states in AdS3 backgrounds from integrability.

We write down the algebraic Bethe ansatz for string theory on AdS3 × S3 × T4 and AdS3 × S3 × K3 in its orbifold limits. We use it to determine the wave-functions of protected closed strings in these backgrounds and prove that their energies are protected to all orders in α′. We further apply the ABA to find the wave functions of protected states of AdS3 × S3 × S3 × S1 and its Z 2 orbifold. Our findings match with protected spectrum calculations from supergravity, SymN orbifolds and apply to the complete moduli space of these theories, excluding orbifold blow-up modes for which further analysis is necessary. [ABSTRACT FROM AUTHOR]