nLab
Nambu-Goto action

Definition

The Nambu-Goto action is an action functional for sigma-models with target space a (pseudo) Riemannian manifold $(X,g)$: it is the induced volume functional

$$S_{\mathrm{NG}}:(\Sigma \stackrel{\gamma }{\to }X)\mapsto {\int }_{\Sigma }\mathrm{dvol}({\gamma }^{*}g),$$S_{NG} : (\Sigma \stackrel{\gamma}{\to} X) \mapsto \int_\Sigma dvol(\gamma^* g) \,,

where $\mathrm{dvol}({\gamma }^{*}g)$ is the volume form of the pullback ${\gamma }^{*}g$ of the metric tensor from $X$ to $\Sigma$.

This is classically equivalent (…) to the Polyakov action.

Applications

The NG-action serves as the kinetic action functional of the sigma-model that described a fundamental brane propagating on $X$. For $\dim \Sigma =1$ this is the relativistic particle, for $\dim \Sigma =2$ the string, for $\dim \Sigma =3$ the membrane.

References

One string theory textbook that deals with the Nambu-Goto action in a bit more detail than usual is

• Barton Zwiebach, A first course in string theory , Cambridge (2009)