Dilaton in Background Field How can one show that the action of dilaton in the String Background Fields must be of the form: $ S_\Phi = \frac1{4\pi} \int d^2 \sigma \sqrt{h} R(h) \Phi(X) $?
Thank you.
 A: There seems to be a basic confusion in the other answer, and maybe in the beliefs of the OP,  which prevents folks from seeing the right answer which is simple. The main thing to notice is that $h,R$ in the formula aren't spacetime metric and curvature: they are the world sheet metric and curvature!
The dilaton is normalized so that $\exp(\Phi)$ is proportional to the (closed) string coupling constant $g_s$. We know that the diagrams should scale with a power of $g_s$ that depend on the topology of the Riemann surface (it's Euclidean signature, not Minkowskian one, Prahar), namely as $g_s^{2h}$ where $h$ is the number of handles or with $g_s^{-\chi}$. That translates to $\exp(-\Phi \chi)$ for a fixed dilaton (log of the string coupling).
The Euclidean action above is simply the way to write down the Euler characteristic,
$$\chi =  \frac{1}{4\pi}\int d^2 z\,\sqrt{h} R $$
multiplied by $\Phi$ because we want $-\Phi\chi$ to be the exponent from the Euclideanized path integral, from $\exp(-S)$, giving us $S=\Phi\chi$. Note that $R=2/a^2$ for a sphere of radius $a$ and we want $\chi=2$ when the curvature is integrated over its $4\pi a^2$ surface, therefore the $1/4\pi$ factor in the formula above.
Your formula above differs from mine simply by allowing $\Phi$ to depend on $X^\mu$.
The explanation above was an explanation using some known facts about "what spacetime physics we expect to get", especially when it comes to the coupling-dependence of individual Riemann surfaces (stringy Feynman diagrams).
If you wanted a derivation from the first principles, you would have to start with the most general world sheet conformal field theory. The integral of $R\sqrt{h}$ is proportional to the Euler characteristic which is conformal because it is even topological. Adding $X^\mu(z,\bar z)$-dependent coefficients in all the terms is the standard way to construct a more general action (a non-constant, possibly curved, background for the metric, dilaton, and other fields). The conformality requirement (including the quantum loop corrections) tells us that the background – expressed in these coefficients – has to obey the spacetime equations of motion. This is true for the dilaton profile $\Phi(X)$, too.
To correctly label the coefficients of such terms in the general world sheet conformal field theory action, you need to compare the spacetime predictions of the action with the expectations because this is about the interpretation or terminology. So you rerun the beginning of this answer and realize that the coefficient of the $\chi$-like integral above is what we want to call the spacetime dilaton because for a (nearly) constant value of the dilaton, it gives us the right coupling-constant-dependence that we expect from the dilaton. You may pick your psychological starting point differently. 
The truly stringy approach will simply lead you to write the most general CFT with the most general coefficients that make the theory conformal, and predict physics out of it. The physics will turn out to be equivalent to some dynamics in a spacetime and this dynamics in the spacetime will – at least at long distances – look like coming from a field theory. So you may identify the parameters in such a way that this map will work. However, in principle, you may parameterize the CFTs in any way you want and describe the spacetime physics in terms of these parameters. In this approach, you will see that the CFTs may have the term above, with a general function $\Phi(X)$ that has to obey a Klein-Gordon-like (with a potential) equation, the spacetime field equation for the dilaton field, for the 1-loop corrections to the world sheet scale invariance to cancel and everything else in the spacetime will depend on this $\Phi(X)$ as if it were the background configuration for a dilaton field, too.
A: The way you would so this, is to take $h_{\mu\nu} = \eta_{\mu\nu} + g_{\mu\nu}$, where $g$ is small. We are then treating gravity as a perturbation over Minkowski. The action is then takes the form of the vertex operator corresponding to the Dilaton $\Phi(X)$. Can you then convince yourself that this implies that $S_\Phi$ should be of the form above? (Hint: A similar calculation is done for $G_{\mu\nu}$ and $B_{\mu\nu}$ in Polchinski somewhere and Tong's notes.)
