# 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 sigma model is where the target space $R^{d-1,1}$ changes to some arbitrary manifold. Not the base space as you have it. String theory considers changes in both manifolds. Jun 4, 2022 at 21:15

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. 