Is there a nice derivation for the renormalization group β-functions in String Theory that leads to the Einstein Field Equations? There are many existing threads here on Physics Stack Exchange which deal with how General Relativity arises from String Theory. However, all of them simply list the renormalization group beta functions, and state that the derivation of them it outside the scope of the answer. Is there anywhere that actually covers a derivation of these beta functions in any level of detail?
 A: Dan Friedan's thesis was on this topic.  I don't know if it's easily available.
A: I will give a derivation based on Polchinski's book. So I refer to his book for notations and for references of equations. Further details (including derivations of other equations) can also be found in my notes on the book that you can access via my profile.
Recall that all these equations are valid as operator equations,  so we can use (3.4.6)
\begin{align}
\delta_w \langle \cdots \rangle = -\frac{1}{2\pi} \int d^2\sigma\,  \sqrt{g}\,  \delta\omega (\sigma) \langle T^a_a (\sigma) \cdots \rangle
\end{align}
We split the $S_\sigma$ again in the Polyakov and vertex part. The Weyl transformation of the Polyakov part is  simply the Weyl anomaly (3.4.15)
\begin{align}
(T_p)^a_a = -\frac{c}{12} R =- \frac{D-26}{12} R
\end{align}
The Weyl variation of the vertex part is given by (3.6.16)
\begin{align}
\delta_w V_1 =&\,  \frac{g_c}{2}  \int d^2\sigma\,  \sqrt{g}\,  \delta\omega (\sigma) \Big\{ (g^{ab} S_{\mu\nu} + i\epsilon^{ab} A_{\mu\nu}) \left[\partial_a X^\mu \partial_b X^\nu e^{i k\cdot X(\sigma)} \right]_{\mathrm{r}}+ \alpha' F R \left[e^{i k\cdot X(\sigma)} \right]_{\mathrm{r}} \Big\}  \nonumber\\
=&\,   -\frac{1}{2\pi} \int d^2\sigma\,  \sqrt{g}\,  \delta\omega (\sigma)  (T_\sigma)^a_a (\sigma)
\end{align}
with $S_{\mu\nu},  A_{\mu\nu}$ and $F$ given by (3.6.17). We have written it is the form of (3.4.6) in the understanding that this is valid as an operator equation.
We can thus write
\begin{align}
 (T_\sigma)^a_a  =&\,  (-2\pi) \frac{g_c}{2}  \Big\{ (g^{ab} S_{\mu\nu} + i\epsilon^{ab} A_{\mu\nu}) \left[\partial_a X^\mu \partial_b X^\nu e^{i k\cdot X(\sigma)} \right]_{\mathrm{r}}+ \alpha' F R \left[e^{i k\cdot X(\sigma)} \right]_{\mathrm{r}} \Big\}
\end{align}
We thus have,  combining both parts
\begin{align}
 T^a_a  =&\,   -g_c \pi  (g^{ab} S_{\mu\nu} + i\epsilon^{ab} A_{\mu\nu}) \partial_a X^\mu \partial_b X^\nu e^{i k\cdot X} - g_c\pi  \alpha' F R e^{i k\cdot X} - \frac{D-26}{12} R 
\end{align}
We have dropped the renormalisation symbols for convenience.
Let us write this in the form (3.7.12)
\begin{align}
T^a_a = -\frac{1}{2\alpha'} \beta^G_{\mu\nu} g^{ab}  \partial_a X^\mu \partial_b X^\nu-\frac{i}{2\alpha'} \beta^B_{\mu\nu} \epsilon^{ab}  \partial_a X^\mu \partial_b X^\nu
-\frac{1}{2} \beta^\Phi R
\end{align}
Let us break this down in parts top avoid long formula. We start with the last term
\begin{align}
-\frac{1}{2} \beta^\Phi R = - g_c\pi  \alpha' F R e^{i k\cdot X} - \frac{D-26}{12} R 
\end{align}
Using (3.6.17c) for $F$ and taking from here on $\gamma=0$ we find
\begin{align}
 \beta^\Phi = 2g_c\pi  \alpha' (-k^2 \phi)  e^{i k\cdot X} +\frac{D-26}{6}  = 2g_c \pi \alpha' \phi\,  \partial^2 e^{i k\cdot X} +\frac{D-26}{6}  
 \end{align}
Note that here $\partial^2$ and later $\partial_\mu$ denote derivatives w.r.t. the spacetime field $X^\mu$ and not the worldsheet coordinate $\sigma^a$. We now use the definition of the dilaton field $\Phi(X)$ in (3.7.11c)
\begin{align}
 \beta^\Phi =&\,  2g_c \pi \alpha' \left( -\frac{1}{4\pi g_c} \partial^2\Phi\right) +\frac{D-26}{6}   \nonumber\\
 = &\,  \frac{D-26}{6}  -\frac{\alpha'}{2} \partial^2 \Phi
 \end{align}
Consider now the second term
\begin{align}
-\frac{i}{2\alpha'} \beta^B_{\mu\nu} \epsilon^{ab}  \partial_a X^\mu \partial_b X^\nu =&\,  -g_c\pi i\epsilon^{ab} A_{\mu\nu}\partial_a X^\mu \partial_b X^\nu e^{i k\cdot X} 
\end{align}
Using $A_{\mu\nu}$ from (3.6.16b) we find
\begin{align}
 \beta^B_{\mu\nu} \epsilon^{ab}   \partial_a X^\mu \partial_b X^\nu =&\,  2 \alpha' g_c\pi \epsilon^{ab} (-k^2 a_{\mu\nu} + k_\nu k^\omega a_{\mu\omega} - k_\mu k^\omega a_{\nu\omega} )   \partial_a X^\mu \partial_b X^\nu e^{i k\cdot X} \nonumber\\
 =&\,  2 \alpha' g_c\pi \epsilon^{ab} \partial_a X^\mu \partial_b X^\nu  (a_{\mu\nu} \partial^2 -a_{\mu\omega} \partial_\nu \partial^\omega + a_{\nu\omega}\partial_\mu \partial^\omega  )  e^{i k\cdot X} 
\end{align}
We use the expression for $B_{\mu\nu}(X)$ in (3.7.11b)
\begin{align}
 \beta^B_{\mu\nu} \epsilon^{ab}   \partial_a X^\mu \partial_b X^\nu =&\,  2 \alpha' g_c\pi \epsilon^{ab} \partial_a X^\mu \partial_b X^\nu  \left(-\frac{1}{4\pi g_c} \right)
 \left( \partial^2 B_{\mu\nu} - \partial_\nu \partial^\omega B_{\mu\omega}+ \partial_\mu \partial^\omega B_{\nu\omega} \right)\nonumber\\
 =&\,  -\frac{\alpha'}{2} \epsilon^{ab} \partial_a X^\mu \partial_b X^\nu 
 \left( \partial^2 B_{\mu\nu} - \partial_\nu \partial^\omega B_{\mu\omega}+ \partial_\mu \partial^\omega B_{\nu\omega} \right)
 \end{align}
from which we get
\begin{align}
 \beta^B_{\mu\nu}  =&\,  -\frac{\alpha'}{2} 
 \left( \partial^2 B_{\mu\nu} - \partial_\nu \partial^\omega B_{\mu\omega}+ \partial_\mu \partial^\omega B_{\nu\omega} \right)\nonumber\\
 = &\,  -\frac{\alpha'}{2} \partial^\omega H_{\omega \mu\nu}
 \end{align}
where we have used the definition of the field strength (3.7.8). Finally we take the first term
\begin{align}
 -\frac{1}{2\alpha'} \beta^G_{\mu\nu} g^{ab}  \partial_a X^\mu \partial_b X^\nu= &\,  -g_c\pi g^{ab} S_{\mu\nu}\partial_a X^\mu \partial_b X^\nu e^{i k\cdot X} 
\end{align}
Using the definition of $S_{\mu\nu}$ in (3.6.17b),  setting $\gamma=0$ and ignoring the $ g^{ab}  \partial_a X^\mu \partial_b X^\nu$ we have
\begin{align}
\beta^G_{\mu\nu} = &\,  2\alpha' g_c \pi (-k^2 s_{\mu\nu} + k_\nu k^\omega s_{\mu\omega} + k_\mu k^\omega s_{\nu\omega} - k_\mu k_\nu s^\omega_\omega +4 k_\mu k_\nu \phi) e^{ik\cdot X}\nonumber\\
=&\,   2\alpha' g_c \pi (s_{\mu\nu}\partial^2  - s_{\mu\omega}\partial_\nu \partial^\omega - s_{\nu\omega} \partial_\mu \partial^\omega + s^\omega_\omega  \partial_\mu \partial_\nu -4  \phi \partial_\mu \partial_\nu) e^{ik\cdot X}\nonumber\\
=&\,  -\frac{2\alpha' g_c \pi}{4\pi g_c}(\partial^2 \chi_{\mu\nu}-\partial_\nu \partial^\omega \chi_{\mu\omega} -\partial_\mu \partial^\omega  \chi _{\nu\omega} +  \partial_\mu \partial_\nu \chi^\omega_\omega  -4  \partial_\mu \partial_\nu \Phi )\nonumber\\
=&\,  -\frac{\alpha'}{2}(\partial^2 \chi_{\mu\nu}-\partial_\nu \partial^\omega \chi_{\mu\omega} -\partial_\mu \partial^\omega  \chi _{\nu\omega} +  \partial_\mu \partial_\nu \chi^\omega_\omega)   +2 \alpha'  \partial_\mu \partial_\nu \Phi )
\end{align}
where we have used (3.7.11) and the fact that $G_{\mu\nu}= \eta_{\mu\nu} + \chi_{\mu\nu}$ to that order.
