Coordinates vs. parametrization of a worldsheet In introductory string theory, the worldsheet is described (e.g. Tong, Polchinski) as a surface $X^\mu(\tau,\sigma)$ in Minkowski spacetime indexed by two parameters: $(\tau,\sigma)$. Now, I initially wrote this off as classic abuse of notation, where we're taking $X^\mu(\tau,\sigma)$ to be shorthand for $X^\mu(S(\tau,\sigma))$ where $S:\mathbb{R}^2\to M$ is the map to the "abstract" worldsheet and $X^\mu :M\to\mathbb{R}^4$ is the (homeomorphism defined by the) chart. However, then I was introduced to the induced metric on the worldsheet,$$\gamma_{\alpha\beta}=\frac{\partial X^\mu}{\partial\sigma^\alpha}\frac{\partial X^\nu}{\partial\sigma^\beta}\eta_{\mu\nu},$$ and I realized the $\sigma^\alpha\in\{\tau,\sigma\}$ are meant to be understood as genuine coordinates, despite the fact they were clearly introduced as parameters. But the parameters and coordinates need not have anything to do with each other. I could happily reparametrize the worldsheet and leave the chart alone and that obviously wouldn't be a coordinate transformation right?
The only way I can make sense of this is if really $X^\mu :\mathbb{R}^2\to\mathbb{R}^4$ and $X^\mu(\tau,\sigma)$ is shorthand for $X^\mu[\varphi(S(\tau,\sigma))]$ with the worldsheet chart $\varphi=S^{-1}$. Then whenever we perform a reparameterization we modify $S$ and $\varphi$ simultaneously so that it is a genuine coordinate transformation. As in, the reparametrization takes us from $S(\tau,\sigma)\rightarrow S^\prime(\tau^\prime,\sigma^\prime)$, and we modify the chart, $S^{-1}(p)\rightarrow (S^\prime)^{-1}(p)$ so that rather than a given point mapping to the coordinates $(\tau,\sigma)$ it maps to $(\tau^\prime,\sigma^\prime)$. So in effect, we simply choose the coordinates to numerically coincide with the parameters.
Is this correct or am I overthinking it?
 A: I believe the core issue of your question is the sentence "But the parameters and coordinates need not have anything to do with each other". This is not true. If $\cal M$ is a smooth manifold, $U\subset \cal M$ is an open subset on which we define a coordinate chart $x: U\to \mathbb{R}^D$ the inverse $x^{-1}:\mathbb{R}^D\to U$ is called the associated parameterization. Coordinates and parameters are distinct names for the same thing.
The origin of this issue is probably because you recall that a parameterized curve is a map $\gamma:\mathbb{R}\to \mathbb{R}^n$ and then you seem to think that an embedding $\phi:\Sigma\to {\cal M}$ presuposes a parameterization. That is not true. Given one embedding $\phi : \Sigma\to {\cal M}$ you will have a parameterization when you introduce coordinates on $\Sigma$. In that setting a reparameterization is just a change of coordinates in $\Sigma$.
In the parameterized curve story, the thing is that one implicitly assumes that one is using the trivial chart in $\mathbb{R}$, namely $t: \mathbb{R}\to \mathbb{R}$ given as the identity element. A reparameterization, as you know from the study of curves, is then given by a change of coordinates.
So in summary: an embedding is just one smooth injective immersion which is a homeomorphism onto its image. An embedding exists independently of any choice of parameterization. When you introduce coordinates on the domain you then have a parameterization of the embedded submanifold.
