# nLab Nambu-Goto action

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## 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

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

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

## Definition

Let

For $\phi \colon \Sigma \longrightarrow X$ the induced “proper volume” or Nambu-Goto actionof $\phi$ is the integral over $\phi$ of the volume form of the pullback of the target space metric $g$ to $\Sigma$.

$S_{NG}(\phi) \coloneqq \int_{\Sigma} dvol(\phi^\ast(g)) \,.$

Notice that the rank-2 tensor $\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma)$ is in general not degenerate (unless $\phi$ is an embedding), hence is in general not, strictly speaking a pseudo-Riemannian metric on $\Sigma$, but nevertheless it induces a volume form by the standard formula, only that this allowed to vanish pointwise (and even globally, for instance if $\phi$ is constant on a single point). In the literature $dvol(\phi^\ast g)$ is usually written as $\sqrt{-g}d^{p+1}\sigma$.

## Properties

### Relation to the Polyakov action

The NG is classically equivalent to the Polyakov action with “worldvolume cosmological constant”. See at Polyakov action – Relation to Nambu-Goto 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

The Nambu-Goto action functional is named after Yoichiro Nambu.

Detailed discussion of the relation to the Polyakov action and the Dirac-Born-Infeld action is in

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

Revised on November 19, 2015 09:37:46 by Urs Schreiber (31.55.6.173)