Nambu-Goto action in nLab

Contents

Contents

Definition

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:(Σ→γX)↦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).

Definition

Let

• p∈ℕp \in \mathbb{N} (for p-brane dynamics);

• (X,g)(X,g) a pseudo-Riemannian manifold (target spacetime);

• Σ\Sigma a compact smooth manifold of dimension (p+1)(p+1) (worldvolume).

• [Σ,X][\Sigma,X] the diffeological space of smooth functions Σ→X\Sigma \to X.

For ϕ:Σ⟶X\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 *g∈Γ(T*Σ⊕T*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.

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 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.

• 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]