I am reading "String Theory and M-Theory" by Becker, Becker and Schwarz. In Chapter 2, the authors try to gauge fix the auxiliary field. They start from the general expression $$h_{\alpha\beta}=\begin{pmatrix} h_{00} & h_{01}\\ h_{01} & h_{11} \end{pmatrix}\tag{2.22}$$ where it was used that $h_{10}=h_{01}$, and they say that reparametrization invariance allows us to choose two of the metric components. Then, only one component will remain and this component, in turn, can be gauged away by using the invariance of the action under Weyl transformations. What I understand from the latter claim is that there exist transformations that are taking us from $(\sigma'^0,\sigma'^1)$ to $(\sigma^0,\sigma^1)$, such that $$h_{\alpha\beta}(\sigma'^1,\sigma'^2)\rightarrow h_{\alpha\beta}(\sigma^1,\sigma^2)=e^{-\phi(\tau,\sigma)} \begin{pmatrix} -1 & 0\\ 0 & 1 \end{pmatrix}$$ So that, after the application of the abovementioned transformation, we can perform a transofrmation of the form $h_{\alpha\beta}\rightarrow e^{\phi(\tau,\sigma)}h_{\alpha\beta}$ and reduce the latter metric to the Minkowski metric $\eta_{\alpha\beta}=\text{diag}\{-1,1\}$.
My questions are the following:
Why does parametrization invariance allows us to choose two of the components of the metric? Since the action is invariant under parametrizations $\sigma'^{\alpha}\rightarrow\sigma^{\alpha}=f^{\alpha}(\sigma')$, that means we can choose $f^{\alpha}(\sigma')$ to be anything, righta? Hence, since $a=0,1$ we are allowed to choose $f^{0}(\sigma')$ and $f^{1}(\sigma')$, correct? In other words, does the number of components of the metric that I am allowed to choose equal to the number of dimensions of the world volume?
I am trying to complete an exercise I thought: I would like, given a generic metric $$h_{\alpha\beta}(\sigma'^0,\sigma'^1)=\begin{pmatrix} h_{00} & h_{01}\\ h_{01} & h_{11} \end{pmatrix}$$ to deduce which are the most generic transformations to get me to $h_{\alpha\beta}(\sigma^0,\sigma^1)=e^{-\phi(\tau,\sigma)}\text{diag}\{-1,1\}$. The transformation of the metric implies $$h_{00}(\sigma^0,\sigma^1)=-e^{-\phi(\sigma^0,\sigma^1)}= f^2(\sigma^0,\sigma^1)h_{00}+2f(\sigma^0,\sigma^1)\chi(\sigma^0,\sigma^1)h_{01}+\chi^2(\sigma^0,\sigma^1)h_{11}\\ h_{01}(\sigma^0,\sigma^1)=0= f(\sigma^0,\sigma^1)g(\sigma^0,\sigma^1)h_{00}+[f(\sigma^0,\sigma^1)\psi(\sigma^0,\sigma^1)+g(\sigma^0,\sigma^1)\chi(\sigma^0,\sigma^1)]h_{01}+\chi(\sigma^0,\sigma^1)\psi(\sigma^0,\sigma^1)h_{11}\\ h_{11}(\sigma^0,\sigma^1)=e^{-\phi(\sigma^0,\sigma^1)}= g^2(\sigma^0,\sigma^1)h_{00}+2g(\sigma^0,\sigma^1)\psi(\sigma^0,\sigma^1)h_{01}+\psi^2(\sigma^0,\sigma^1)h_{11}$$ where $$f(\sigma^0,\sigma^1)=\frac{\partial\sigma'^0}{\partial\sigma^0},\ \ g(\sigma^0,\sigma^1)=\frac{\partial\sigma'^0}{\partial\sigma^1},\ \ \chi(\sigma^0,\sigma^1)=\frac{\partial\sigma'^1}{\partial\sigma^0},\ \ \psi(\sigma^0,\sigma^1)=\frac{\partial\sigma'^1}{\partial\sigma^1}$$ How to combine the restrictions above in order to find restrictions for the functions $f(\sigma^0,\sigma^1),\ g(\sigma^0,\sigma^1),\ \chi(\sigma^0,\sigma^1)$ and $ \psi(\sigma^0,\sigma^1)$?
If the second question is too hard, I could also be satisfied with a specific example of reparametrizations that are not the most generic ones, but would do the job of reducing an arbitrary metric $h_{\alpha\beta}(\sigma'^0,\sigma'^1)$ to the metric $h_{\alpha\beta}(\sigma^0,\sigma^1)$.
Any help will be appreciated.
