$AdS_3/CFT_2$, finite-gap equations and massless modes

It is known that string theory on $AdS_3\times M_7$ backgrounds preserving 16 supercharges is classically integrable. This integrability has been previously used to write down a set of integral equations, known as the finite-gap equations. These equations can be solved for the closed string spectrum of the theory. However, it has been known for some time that the $AdS_3\times M_7$ finite-gap equations do not capture the dynamics of the massless modes of the closed string theory. In this paper we re-examine the derivation of the $AdS_3\times M_7$ finite-gap system. We find that the conditions that had previously been imposed on these integral equations in order to implement the Virasoro constraints are too strict, and are in fact not required. We identify the correct implementation of the Virasoro constraints on finite-gap equations and show that this new, less restrictive condition captures the complete closed string spectrum on $AdS_3\times M_7$.