I asked a similar question in What do we mean by worldsheet metric fluctuating in string theory, when we have a "target manifold"?, but the question had my misunderstanding that in Polyakov action, worldsheet metric is an independent variable, when it should just be understood as being redundant. This should have been obvious, but I was confused for the reasons below.
It is said that string theories allow metric to be "dynamic" instead of "fixed." Or said differently, correspondence like AdS/CFT allows one to study conformal field theories (on a fixed background manifold/metric) to get to a string theory indirectly, but without correspondence we study string theories that allow metric tensor to be dynamical. Or that has been my elementary understanding.
I can think of intuitive picture that "gravity" is supposed to be quantized, so I guess this metric tensor being dynamical should be the right understanding.
But again, thinking of Polyakov action as an example, if equation of motion removes degree of freedom (worldsheet metric comes out of target manifold metric tensor by equation of motion), I cannot see how one can make spacetime metric tensor dynamical, and by this one means that spacetime manifold is changing for sure. Target spacetime manifold/metric is already fixed, so how can "gravity" ever be quantized in Polyakov interpretation of string theories?