The Polyakov action is invariant under Weyl transformations, that is local rescaling of the metric tensor on the world sheet. However, I don't really understand the physical meaning of this. What would it mean for the action to not have this symmetry? I also have another question concerning reparametrization invariance. Because of this gauge invariance, it means that we have a redundancy in the system and we actually have fewer degrees of freedom than what it appears to be in the action. But why does the presence of gauge invariance reduce the number of degrees of freedom?