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∫D−det⁡(X∗η) dx1∧dx2S[X] = - T \int_{D} \sqrt{- \det(X^\ast \eta)} \mathrm{d} x^1 \wedge \mathrm{d} x^2 where X:D→Rd−1,1X: D \to \mathbb{R}^{d-1,1} describes the path of the string and D=I×[0,2π]D = I \times [0,2\pi] is a rectangle. For closed strings one assumes that X(⋅,0)=X(⋅,2π)X(\cdot,0) = X(\cdot,2\pi) . My question is the following: since the map XX needs to be differentiable, isn't the Nambu-Goto action only describing string theory on a torus? If this is true, then the map XX should actually take the form X:Σg→Rd−1,1X : \Sigma_g \to \mathbb{R}^{d-1,1} where Σg\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(⋅,0)=X(⋅,2π)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?