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
where $dvol(\gamma^* g)$ is the volume form of the pullback $\gamma^* g$ of the metric tensor from $X$ to $\Sigma$, and where $T$ (the brane-“tension”, e.g. the string tension for $dim(\Sigma) = 2$) is an inverse unit of length to the power the dimension $dim(\Sigma)$.
Let
$p \in \mathbb{N}$ (for p-brane dynamics);
$(X,g)$ a pseudo-Riemannian manifold (target spacetime);
$\Sigma$ a compact smooth manifold of dimension $(p+1)$ (worldvolume).
$[\Sigma,X]$ the diffeological space of smooth functions $\Sigma \to X$.
For $\phi \colon \Sigma \longrightarrow X$ the induced “proper volume” or Nambu-Goto action of $\phi$ is the integral over $\phi$ of the volume form of the pullback of the target space metric $g$ to $\Sigma$.
Notice that the rank-2 tensor $\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma)$ is in general not non-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 $\sqrt{-g}d^{p+1}\sigma$.
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 $X$. For $dim \Sigma = 1$ this is the relativistic particle, for $dim \Sigma = 2$ the string, for $dim \Sigma = 3$ the membrane.
for more on brane tension see also Worldsheet and brane instantons
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):
Yoichiro Nambu, Duality and Hadrodynamics, Notes prepared for the Copenhagen High Energy Symposium (1970) [doi:10.1142/9789812795823_0026, pdf]
Tetsuo Gotō, Relativistic Quantum Mechanics of One-Dimensional Mechanical Continuum and Subsidiary Condition of Dual Resonance Model, Progress of Theoretical Physics 46 5 (1971) 1560–1569 [doi:10.1143/PTP.46.1560]
Historical review:
Joël Scherk, An introduction to the theory of dual models and strings, Rev. Mod. Phys. 47 123 (1975) [doi:10.1103/RevModPhys.47.123]
Hiroshi Itoyama, Birth of String Theory, Progress of Theoretical and Experimental Physics 2016 6 (2016) 06A103 [arXiv:1604.03701, doi:10.1093/ptep/ptw063]
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.