Protected states in $\mbox{AdS}_3$ backgrounds from integrability

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.