nLab Nambu-Goto action



Riemannian geometry

String theory



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

S NG:(ΣγX)T Σdvol(γ *g), S_{NG} \;\colon\; (\Sigma \stackrel{\gamma}{\to} X) \mapsto T \int_\Sigma dvol(\gamma^* g) \,,

where dvol(γ *g)dvol(\gamma^* g) is the volume form of the pullback γ *g\gamma^* g of the metric tensor from XX to Σ\Sigma, and where TT (the brane-“tension”, e.g. the string tension for dim(Σ)=2dim(\Sigma) = 2) is an inverse unit of length to the power the dimension dim(Σ)dim(\Sigma).



For ϕ:ΣX\phi \colon \Sigma \longrightarrow X the induced “proper volume” or Nambu-Goto action of ϕ\phi is the integral over ϕ\phi of the volume form of the pullback of the target space metric gg to Σ\Sigma.

S NG(ϕ) Σdvol(ϕ *(g)). S_{NG}(\phi) \coloneqq \int_{\Sigma} dvol(\phi^\ast(g)) \,.

Notice that the rank-2 tensor ϕ *gΓ(T*ΣT*Σ)\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma) is in general not non-degenerate (unless ϕ\phi is an embedding), hence is in general not, strictly speaking a pseudo-Riemannian metric on Σ\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 ϕ\phi is constant on a single point). In the literature dvol(ϕ *g)dvol(\phi^\ast g) is usually written as gd p+1σ\sqrt{-g}d^{p+1}\sigma.


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.


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


The Nambu-Goto action functional originates as a proposal for the dynamics of strings meant to explain the “dual resonance model” for hadron bound states (quantum hadrodynamics, cf. Polyakov gauge-string duality):

Historical review:

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

Discussion of the Nambu-Goto action and Polyakov action on worldsheets with boundary (i.e. in the generality of open strings) and cast in BV-BRST formalism:

Relation of the Nambu-Goto string to Liouville theory:

Last revised on July 2, 2024 at 08:04:50. See the history of this page for a list of all contributions to it.