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

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

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

Definition

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 $\Sigma$.

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

Notice that the rank-2 tensor $\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma)$ is in general not 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 $\sqrt{-g}d^{p+1}\sigma$.

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.

Related concepts

• minimal surface

• Polyakov action

• Dirac-Born-Infeld action

• Veneziano amplitude

• particle, string, membrane

• membrane instanton

• for more on brane tension see also Worldsheet and brane instantons

References

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

• J. A. Nieto, Remarks on Weyl invariant p-branes and Dp-branes (arXiv:hep-th/0110227)

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)

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:

• S. Martinoli, Michele Schiavina, BV analysis of Polyakov and Nambu–Goto theories with boundary, Lett. Math. Phys. 112 35 (2022) [doi:10.1007/s11005-022-01526-1, arXiv:2106.02983]

Relation of the Nambu-Goto string to Liouville theory:

• Yuri Makeenko, The Nambu-Goto string as higher-derivative Liouville theory [arXiv:2407.01136]

Relation to D=3 gravity:

• Avik Banerjee, Ayan Mukhopadhyay, Giuseppe Policastro: Nambu-Goto equation from three-dimensional gravity [arXiv:2404.02149]