Conformal transformations that preserves the string boundary condition I am currently reading 't Hooft's lecture notes on string theory [1]. I am trying to show that given the world sheet metric $h_{ab}(\tau,\sigma)$ and the boundary condition for closed string
$$
X^\mu(\tau, \sigma) = X^\mu(\tau, \sigma + 2\pi)
$$
we can always find some coordinate transformation $\tau, \sigma \to \tilde{\tau}, \tilde{\sigma}$ so that the boundary condition still has the form
$$
X^\mu(\tilde{\tau}, \tilde{\sigma}) = X^\mu(\tilde{\tau}, \tilde{\sigma} + 2\pi)
$$
while the world sheet metric becomes conformal, i.e.,
$$
\tilde{h}_{ab}=\eta_{ab}e^{\omega(\tilde{\tau},\tilde{\sigma})}
$$
In the notes (on the bottom page 11), it was suggested that we can always first go to some conformal gauge, and then use transformation of the form
$$
\sigma^\pm\to f^\pm(\sigma^\pm), \quad \sigma^\pm = \frac{1}{\sqrt{2}}(\sigma \pm \tau)
$$
to transform the boundary conditions to the desired form while keeping the conformal gauge. But I am having a hard time to find such transformations. Any help or suggestion is appreciated.
 A: The reason why we can always find a diffeomorphism $\sigma^\alpha\rightarrow\tilde{\sigma}^\alpha(\sigma^0,\sigma^1)$ with $\sigma^\alpha\in\{\tau,\sigma\}$, that preserves the form of the boundary conditions for the closed string is due to the fact that the Nambu-Goto action is invariant under diffeomorphisms on the worldsheet. Consider in general the map$$X:\Sigma\rightarrow\mathbb{M}\quad\text{with}\quad(\sigma^0,\sigma^1)\mapsto X^\mu(\sigma^0,\sigma^1).$$Here $\Sigma$ is your worldsheet and $\mathbb{M}$ is your spacetime manifold. Then the Nambu-Goto action is given by$$S[X]=-T\int_\Sigma d^2\sigma\sqrt{|\det{(\partial_\alpha X^\mu\partial_\beta X^\nu)}|}.$$Here we are taking $\sigma^0\in(-\infty,\infty)$ and $\sigma^1\in[0,2\pi]$. Now we show that $S[X]$ is invariant under orientation preserving reparameterizations of the worldsheet $\Sigma$, of the form$$\sigma^\alpha\rightarrow\tilde{\sigma}^\alpha(\sigma^0,\sigma^1)\quad\text{where}\quad\det{\left(\frac{\partial\tilde{\sigma}^\alpha}{\partial\sigma^\beta}\right)}>0.$$Let$$J=\left(\frac{\partial\tilde{\sigma}^\alpha}{\partial\sigma^\beta}\right).$$We then have that $d^2\tilde{\sigma}=|\det{(J)}|d^2\sigma=(\det{(J)})d^2\sigma$ since $J>0$ by assumption. We can now write$$\frac{\partial X^\mu}{\partial\tilde{\sigma}^\alpha}=\frac{\partial\sigma^\gamma}{\partial\tilde{\sigma}^\alpha}\frac{\partial X^\mu}{\partial\sigma^\gamma}=J^{-1}\frac{\partial X^\mu}{\partial\sigma^\gamma}.$$Now if we write$$g:=g_{\alpha\beta}=\frac{\partial X^\mu}{\partial\sigma^\alpha}\frac{\partial X^\mu}{\partial\sigma^\beta}\quad\text{and}\quad \tilde{g}:=\tilde{g}_{\alpha\beta}=\frac{\partial X^\mu}{\partial\tilde{\sigma}^\alpha}\frac{\partial X^\mu}{\partial\tilde{\sigma}^\beta},$$we see that$$\tilde{g}=(J^{-1})^\text{T}gJ^{-1}\implies|\det{(\tilde{g})}|=|\det{\left((J^{-1})^\text{T}gJ^{-1}\right)}|=|\det{(g)}|(\det{(J)})^{-2}$$and thus$$\sqrt{|\det{(g)}|}=(\det{(J)})^{-1}\sqrt{|\det{(g)}|}.$$Putting everything together, we see that$$\tilde{S}[X]=-T\int_{\Sigma}d^2\tilde{\sigma}\sqrt{|\det{(\tilde{g}_{\alpha\beta})}|}=-T\int_\Sigma d^2\sigma(\det{(J)})(\det{(J)})^{-1}\sqrt{|\det{(g_{\alpha\beta})}|}=S[X].$$So as we can see, $S[X]$ is indeed invariant under a reparameterization of $\Sigma$. Now the boundary conditions for the closed string (as well as all the others), come about by considering the variations $X\rightarrow X+\delta X$ of the worldsheet. Since the Nambu-Goto action is invariant under reparameterizations $\sigma^\alpha\rightarrow\tilde{\sigma}^\alpha(\sigma^0,\sigma^1)$, so are the boundary conditions (since we are free to reparametrize and then compute the boundary conditions). So if we have$$X^\mu(\sigma^0,\sigma^1+2\pi)=X^\mu(\sigma^0,\sigma^1)\quad\text{then we also have}\quad X^\mu(\tilde{\sigma}^0,\tilde{\sigma}^1+2\pi)=X^\mu(\tilde{\sigma}^0,\tilde{\sigma}^1).$$
