Is the Nambu-Goto action defined only for the torus? For simplicity, I will use the Nambu-Goto action, but the following question would probably be the same for the Polyakov action.
According to David Tong's lecture notes on string theory, the Nambu-Goto action of a bosonic string is given by
$$S[X] = - T \int_{D} \sqrt{- \det(X^\ast \eta)} \mathrm{d} x^1 \wedge \mathrm{d} x^2$$
where $X: D \to \mathbb{R}^{d-1,1}$ describes the path of the string and $D = I \times [0,2\pi]$ is a rectangle. For closed strings one assumes that $X(\cdot,0) = X(\cdot,2\pi)$.
My question is the following: since the map $X$ needs to be differentiable, isn't the Nambu-Goto action only describing string theory on a torus? If this is true, then the map $X$ should actually take the form $X : \Sigma_g \to \mathbb{R}^{d-1,1}$ where $\Sigma_g$ is a Riemann surface. This would then be a sigma-model.
One could argue that the action is only a local description of a more complicated Riemann surface, but then we would never need the boundary condition $X(\cdot,0) = X(\cdot,2\pi)$. In this case, it would also be impossible to compute the global minimum of the action.
Maybe there is a more fundamental question lurking behind: what is the interpretation of the Nambu-Goto action? Is it describing the path of a one-dimensional object in Minkowski space, or should we interpret it as a field theory on a rectangle/Riemann surface with values in Minkowski space?
 A: I did a bit of research and I'm now able to give a partial answer to my original question. The following is based on the notes on string theory by D'Hoker. See https://www.math.ias.edu/QFT/spring/index.html.
First of all, it is possible to generalize the definition to general smooth surfaces $\Sigma$, but one has to choose a Riemannian metric $g$ on $\Sigma$, so that the action is
$$S[X,g] = -T \int_\Sigma \sqrt{-\det(X^\ast \eta)} \mathrm{vol}_g$$
where $X\colon \Sigma \to \mathbb{R}^{d-1,1}$ is a smooth map, and $\eta$ is the Minkowski metric. (Note that, for the Polyakov action we always define the action with respect to a metric.)
D'Hoker provides the following argument for a string in Minkowski space. For two different Lorentz frames $(t,x)$ and $(t',x')$, the space-slices of equal time can look similar to the slices in the following picture. For an observer in the first frame, the strings join in the space-time point $P$, while for an observer in the second frame the strings join in the space-time point $P'$. Therefore, no point on the surface can be singled out as the point of interaction. He then argues that the free string, i.e. that with the topology of a torus, determines the nature of the interactions completely.

So this explains why one only discusses string theory on the torus. However, this assumes that the string moves faster than light (because the tangent plane is space like at some points). This violates physical principles, but I posted a new question regarding this problem here: physics.stackexchange.com/questions/712146/
