Nambu–Goto and Polyakov Actions

This page defines the Nambu–Goto and Polyakov actions for a bosonic string, proves their classical equivalence, and explains why one introduces an independent worldsheet metric $\gamma_{ab}$ .

Notation and set‑up

Worldsheet coordinates $\sigma^a=(\tau,\sigma)$ , $a=0,1$ .
Target‑space coordinates $X^\mu(\sigma)$ , $\mu=0,\dots,D-1$ , with background metric $G_{\mu\nu}(X)$ (flat case: $G_{\mu\nu}=\eta_{\mu\nu}$ ).
Induced (pullback) metric on the worldsheet: $h_{ab} \equiv \partial_a X^\mu\,\partial_b X^\nu\,G_{\mu\nu}(X).$
String tension: $T \equiv \frac{1}{2\pi\alpha'}.$
Worldsheet signature $(-+)$ (Euclideanization is straightforward).

The two actions

Nambu–Goto (area) action

“Area = tension × area of the worldsheet”:

S_{\text{NG}}[X] \;=\; -\,T \int d^2\sigma\, \sqrt{-\det h_{ab}} \;=\; -\,T \int d^2\sigma\, \sqrt{-h}.

Symmetry: worldsheet diffeomorphisms (reparametrizations).

Polyakov (linearized) action

Introduce an independent worldsheet metric $\gamma_{ab}$ :

S_{\text{P}}[X,\gamma] \;=\; -\,\frac{T}{2}\int d^2\sigma\,\sqrt{-\gamma}\,\gamma^{ab}\,\partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X) \;=\; -\,\frac{T}{2}\int d^2\sigma\,\sqrt{-\gamma}\,\gamma^{ab}h_{ab}.

Symmetries: worldsheet diffeomorphisms and local Weyl (scale) transformations $\gamma_{ab}\to e^{2\omega(\sigma)}\gamma_{ab}$ . In 2D, $\sqrt{-\gamma}\,\gamma^{ab}$ is Weyl invariant.

Classical equivalence: eliminating $\gamma_{ab}$

Vary $S_{\text{P}}$ with respect to $\gamma^{ab}$ . Using $\delta\sqrt{-\gamma} = -\tfrac12\sqrt{-\gamma}\,\gamma_{cd}\,\delta\gamma^{cd}$ , one finds the (symmetric, conserved) worldsheet stress tensor:

T_{ab} \;\equiv\; -\frac{2}{\sqrt{-\gamma}}\frac{\delta S_{\text{P}}}{\delta\gamma^{ab}} \;=\; -T\left(h_{ab}-\tfrac12\gamma_{ab}\,\gamma^{cd}h_{cd}\right) \;=\; 0.

Define

\Lambda(\sigma) \;\equiv\; \tfrac12\,\gamma^{cd}h_{cd}.

Then the constraint becomes

h_{ab} = \Lambda\,\gamma_{ab}. \qquad (\star)

Thus, on shell the auxiliary metric is conformally equivalent to the induced metric.

Now evaluate $S_{\text{P}}$ on the $\gamma$ -equation of motion:

From $h_{ab}=\Lambda\gamma_{ab}$ , determinants obey
$h \equiv \det h_{ab} = \Lambda^2 \det\gamma_{ab} = \Lambda^2 \gamma \quad\Rightarrow\quad \sqrt{-h} = \sqrt{-\gamma}\,\Lambda.$
Then
$S_{\text{P}}\big|_{\gamma\text{-eom}} = -\frac{T}{2}\int d^2\sigma\,\sqrt{-\gamma}\,\gamma^{ab}(\Lambda\gamma_{ab}) = -T\int d^2\sigma\,\sqrt{-\gamma}\,\Lambda = -T\int d^2\sigma\,\sqrt{-h} = S_{\text{NG}}.$

Hence $S_{\text{P}}$ and $S_{\text{NG}}$ are classically equivalent once the auxiliary metric is eliminated.

Equations of motion for $X^\mu$ (consistency check)

From Polyakov:

\frac{1}{\sqrt{-\gamma}}\partial_a\!\left(\sqrt{-\gamma}\,\gamma^{ab}\partial_b X^\mu\right) + \Gamma^\mu_{\;\rho\sigma}(X)\,\gamma^{ab}\partial_a X^\rho\partial_b X^\sigma = 0.

From Nambu–Goto (minimal surface equation):

\frac{1}{\sqrt{-h}}\partial_a\!\left(\sqrt{-h}\,h^{ab}\partial_b X^\mu\right) + \Gamma^\mu_{\;\rho\sigma}(X)\,h^{ab}\partial_a X^\rho\partial_b X^\sigma = 0.

Using $h_{ab}=\Lambda\gamma_{ab} \Rightarrow h^{ab}=\Lambda^{-1}\gamma^{ab}$ and $\sqrt{-h}=\sqrt{-\gamma}\Lambda$ , we get the 2D identity

\sqrt{-\gamma}\,\gamma^{ab} = \sqrt{-h}\,h^{ab},

so the $X^\mu$ equations coincide exactly.

Why introduce the auxiliary metric $\gamma_{ab}$ ?

Linearization of the square root.
$S_{\text{NG}}$ has a nonpolynomial $\sqrt{\det h}$ . The Polyakov action is quadratic in $X^\mu$ , making variation and quantization tractable.
Analogy (point particle):
$S = -m\!\int\! d\tau\,\sqrt{-\dot X^2} \;\Longleftrightarrow\; S = \tfrac12\!\int\! d\tau\left(\frac{\dot X^2}{e} - e\,m^2\right),$
with the einbein $e(\tau)$ playing the role of $\gamma_{ab}$ .
Gauge structure: 2D gravity on the worldsheet.
$S_{\text{P}}$ has diffeomorphism × Weyl invariance. In 2D, a symmetric metric has 3 components and these 3 gauge freedoms let you fix to conformal gauge
$\gamma_{ab} = e^{2\phi(\sigma)}\eta_{ab},$
exposing a free 2D CFT for the $X^\mu$ fields plus ghosts.
Transparent constraints (Virasoro).
Varying $\gamma_{ab}$ imposes $T_{ab}=0$ . In conformal gauge and light‑cone coordinates $\sigma^\pm=\tau\pm\sigma$ ,
$T_{++} = \partial_+ X \cdot \partial_+ X = 0, \qquad T_{--} = \partial_- X \cdot \partial_- X = 0.$
These are the Virasoro constraints; in Nambu–Goto they are present but less manifest.
Clean path integral over geometries.
The Polyakov formulation separates the sum over embeddings and intrinsic geometries:
$\int \frac{\mathcal D\gamma\,\mathcal DX}{\text{Diff}\times\text{Weyl}}\,e^{-S_{\text{P}}}$
which, after gauge‑fixing, becomes an integral over moduli (complex structures) of the Riemann surface. The Nambu–Goto form hides this structure inside the nonpolynomial measure.
Intrinsic curvature couplings.
Terms such as the Euler characteristic $\int\!\sqrt{-\gamma}\,R(\gamma)$ or the dilaton coupling
$\frac{1}{4\pi}\int d^2\sigma\,\sqrt{-\gamma}\,\Phi(X)\,R(\gamma)$
require an intrinsic metric $\gamma_{ab}$ . Superstrings also need a zweibein/spin connection on the worldsheet.
Same physical degrees of freedom.
$D$ scalars $X^\mu$ + 3 components of $\gamma_{ab}$ − (2 diffeos + 1 Weyl) gauge = $D$ fields with 2 first‑class constraints ⇒ $D-2$ transverse modes—exactly as in Nambu–Goto.

Specialization under the conformal gauge

In conformal gauge $\gamma_{ab}=e^{2\phi}\eta_{ab}$ , the action simplifies to

S_{\text{P}} = -\frac{T}{2}\int d^2\sigma\,\eta^{ab}\,\partial_a X\cdot\partial_b X,

while the dynamics is supplemented by the Virasoro constraints above. Boundary terms from $\delta X$ give the usual Neumann/Dirichlet conditions for open strings in either formulation.

Quantum remark

Classically, $S_{\text{NG}}$ and $S_{\text{P}}$ are equivalent. Quantum‑mechanically, maintaining Weyl invariance (no conformal anomaly) picks out the critical dimension: bosonic string $D=26$ (because free bosons contribute $c=D$ and ghosts contribute $c=-26$ ), and superstring $D=10$ . In these critical dimensions, the two formulations describe the same quantum theory.