Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories incorporate CFT but allows massive states?
Is it because the CFT is on the worldsheet and therefore applies to the worldsheet coordinate X (X is treated as the field) - however the physical states arise from the the creation/annihilation operators that we get from X? Therefore the CFT doesn't actually act on the states (massive or massless) but instead it acts on the field X.