NambuGoto

The Nambu-Goto action is a classical action describing the dynamics of a relativistic string. It assigns to each two-dimensional worldsheet the area swept by the string as it moves through spacetime, with the action proportional to that area. The action is written as S_NG = -T ∫ d^2ξ sqrt(-det γ), where ξ^a (a=0,1) are coordinates on the worldsheet, X^μ(ξ) are the embedding functions into D-dimensional spacetime, and γ_ab = ∂_a X^μ ∂_b X^ν g_μν is the induced metric on the worldsheet.

Classical dynamics follow from the variation of S_NG, yielding equations of motion that describe a minimal surface of the worldsheet. The theory is invariant under reparametrizations of the worldsheet coordinates. In convenient gauges, such as the conformal gauge, the equations reduce to wave equations for X^μ with additional constraints known as the Virasoro constraints.

Quantum aspects: The Nambu-Goto action is non-linear in the embedding fields, making direct quantization challenging. The Polyakov action, which introduces an independent worldsheet metric as an auxiliary field, is classically equivalent and more tractable for quantization. In bosonic string theory, quantum consistency requires a critical spacetime dimension of 26 (supersymmetric strings require 10).

Generalizations: The action generalizes to p-branes by replacing the worldvolume area with higher-volume measures, giving the Dirac-Nambu-Goto action for p-branes. While often superseded by Polyakov or Green–Schwarz formulations in practical use, the Nambu-Goto action remains a foundational geometric description of relativistic extended objects.