How does spacetime metric become dynamical (gravity quantized) in string theory?

Question

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?

Answer 1

In any string theory, the graviton arises in the spectrum of the string, that is, in the ordinary way in canonical quantisation. For the bosonic string, $\tilde\alpha^i_{-1}\alpha^j_{-1}|0\rangle$ transforms in a representation reducible to a symmetric traceless part along with others, which we identify as the graviton, $G_{\mu\nu}(X)$.

$G_{\mu\nu}(X)$ is a field which carries target manifold indices, and is a function of the embedding functions which can be viewed as coordinates, $X^\mu(\tau,\sigma)$.

Now, we can consider a non-linear sigma model, of the form,

$$S \sim \int d^2\sigma \sqrt{h}h^{\alpha\beta}\partial_\alpha X^\mu \partial_\beta X^\nu G_{\mu\nu}(X).$$

Now comes the question: if string theory has a graviton in the spectrum, then shouldn't $G_{\mu\nu}$ be built up from these gravitons, in the same way light is comprised of photons?

The answer is yes. If we expand $G_{\mu\nu} = \eta_{\mu\nu}+f_{\mu\nu}$ then the partition function for the theory is related to the partition function of a string in flat space by,

$$Z=\int DX Dg \, e^{-S_{\mathrm{poly}}-V}$$

where

$$V\sim \int d^2\sigma \sqrt{h}h^{\alpha\beta}\partial_\alpha X^\mu \partial_\beta X^\nu f_{\mu\nu}.$$

If we now choose $f_{\mu\nu} \sim c_{\mu\nu}e^{ip\cdot X}$ then the expression agrees with the vertex operator of a graviton in string theory, and adding $e^V$ to the path integral shifts the metric by $f_{\mu\nu}$.

Thus, to recap: a massless spin 2 gauge boson arises in the spectrum of the string upon quantisation, and the background curved metric is comprised of such gravitons, in the sense described above.

Caveat: We are not directly trying to quantise the metric in string theory, or in the description above. That approach is canonically quantised gravity. However, we can from string theory derive effective actions and these include the graviton as the metric. To obtain amplitudes from these effective actions we would quantise in the usual way.