Nambu Goto Polyakov Equivalence

Let $\mathrm{\Sigma }$ be an oriented smooth Lorentzian two-manifold, possibly with smooth boundary, and let $X:\mathrm{\Sigma }\to {\mathbb{R}}^{1,D-1}$ be a $C^2$ embedding whose differential has rank two and whose induced metric ${\gamma }_{ab}={\mathrm{\partial }}_{a}X\cdot {\mathrm{\partial }}_{b}X$ is nondegenerate Lorentzian. Let $h$ be a $C^1$ Lorentzian auxiliary metric and define $\lambda =\frac{1}{2}{h}^{ab}{\gamma }_{ab}$. Use the positive branch $\lambda >0$, the adopted free Polyakov action $S_P=-\frac{T}{2}{\int }_{\mathrm{\Sigma }}\sqrt{-\det h}{h}^{ab}{\mathrm{\partial }}_{a}X\cdot {\mathrm{\partial }}_{b}X{d}^2\sigma$, and $T=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}$. Define the Nambu–Goto action by $S_{\mathrm{N}\mathrm{G}}[X]=-T{\int }_{\mathrm{\Sigma }}\sqrt{-\det \gamma }{d}^2\sigma$, using the positive square root. For either action, stationarity requires vanishing first variation for all admissible compactly supported $X$-variations $V=\delta X$ that are $C^2$ up to the boundary. Polyakov stationarity additionally requires vanishing first variation for all compactly supported $C^1$ symmetric variations of the auxiliary metric; the Nambu–Goto action has no independent metric variable. On each boundary piece choose the same fixed target components in both actions, imposing $V^{\mu }=0$ there; the other components are free and their boundary flux coefficients must vanish (natural data). Variations vanish on any initial/final time slices. On noncompact worldsheets, use these local first variations even when total actions diverge. Write $d{\mathrm{\ell }}_a$ for the oriented coordinate boundary element characterized by ${\int }_{\mathrm{\Sigma }}{\mathrm{\partial }}_a{J}^a{d}^2\sigma ={\int }_{\mathrm{\partial }\mathrm{\Sigma }}{J}^{a}d{\mathrm{\ell }}_a$ for a compactly supported vector density $J$; this convention also applies at null boundaries. The auxiliary-metric Euler–Lagrange equation is $T_{ab}={\gamma }_{ab}-\frac{1}{2}{h}_{ab}{h}^{cd}{\gamma }_{cd}=0$, which gives ${\gamma }_{ab}=\lambda h_{ab}$. When this metric equation holds on the positive branch, the Polyakov and Nambu–Goto action densities agree pointwise, so their action integrals agree whenever defined; the $X$-equations and their boundary fluxes agree when the same variations and fixed or natural boundary data are used in both formulations. Conversely, any stationary Nambu–Goto embedding with nondegenerate Lorentzian $\gamma$ lifts to a Polyakov stationary pair by taking $h=\gamma$. This is a classical equivalence only.

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