We write down the Algebraic Bethe Ansatz for string theory on $\mbox{ AdS}_3\times\mbox{ S}^3\times\mbox{ T}^4$ and $\mbox{ AdS}_3\times\mbox{ S}^3\times\mbox{ K3}$ 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 $\alpha'$. We further apply the ABA to find the wave functions of protected states of $\mbox{ AdS}_3\times\mbox{ S}^3\times\mbox{ S}^3\times \mbox{ S}^1$ and its $\mathbf{ Z}_2$ orbifold. Our findings match with protected spectrum calculations from supergravity, $\mbox{ Sym}^N$ orbifolds and apply to the complete moduli space of these theories, excluding orbifold blow-up modes for which further analysis is necessary.
Categories: hep-th