The Nambu-Goto action is an action functional for sigma-models with target space a (pseudo) Riemannian manifold : it is the induced volume functional
where is the volume form of the pullback of the metric tensor from to , and where (the brane-“tension”, e.g. the string tension for ) is an inverse unit of length to the power the dimension .
Let
(for p-brane dynamics);
a compact smooth manifold of dimension (worldvolume).
the diffeological space of smooth functions .
For the induced “proper volume” or Nambu-Goto action of is the integral over of the volume form of the pullback of the target space metric to .
Notice that the rank-2 tensor is in general not non-degenerate (unless is an embedding), hence is in general not, strictly speaking a pseudo-Riemannian metric on , but nevertheless it induces a volume form by the standard formula, only that this allowed to vanish pointwise (and even globally, for instance if is constant on a single point). In the literature is usually written as .
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 . For this is the relativistic particle, for the string, for 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:
Relation to D=3 gravity:
Last revised on August 20, 2024 at 05:57:09. See the history of this page for a list of all contributions to it.