  String theory on $AdS_3 \times S^3 \times T^4$ has 20 moduli. We investigate
how the perturbative closed string spectrum changes as we move around this
moduli space in both the RR and NSNS flux backgrounds. We find that, at weak
string coupling, only four of the moduli affect the energies. In the RR
background the only effect of these moduli is to change the radius of curvature
of the background. On the other hand, in the NSNS background, the moduli
introduce worldsheet interactions which enable the use of integrability methods
to solve the spectral problem. Our results show that the worldsheet theory is
integrable across the 20 dimensional moduli space.
