The <induced worldsheet metric> is
$$
\gamma_{\alpha\beta}
=\partial_\alpha X^\mu\partial_\beta X^\nu\eta_{\mu\nu}
=\partial_\alpha X\mathbin\cdot\partial_\beta X.
$$
The <Nambu–Goto action> is
$$
S_{\rm NG}=-T\int_\Sigma d^2\sigma\,\sqrt{-\det\gamma},
$$
which is minus the <string tension> times the invariant area of the <string worldsheet>. Varying the independent metric in the <Polyakov action> gives
$$
0=T_{\alpha\beta}
=\gamma_{\alpha\beta}
-\frac12g_{\alpha\beta}g^{\rho\sigma}\gamma_{\rho\sigma}.
$$
In two dimensions this says $g_{\alpha\beta}=e^{2\omega}\gamma_{\alpha\beta}$ for an undetermined <Weyl transformation> factor. Substitution gives $\sqrt{-g}\,g^{\alpha\beta}\gamma_{\alpha\beta}=2\sqrt{-\det\gamma}$, and hence $S_{\rm P}=S_{\rm NG}$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 1 / a / Solution

{c}
{wiki}

The Polyakov action introduces an independent worldsheet metric $g_{\alpha\beta}$:
$$
S_{\rm P}=-\frac T2\int_\Sigma d^2\sigma\,\sqrt{-g}\,
g^{\alpha\beta}\partial_\alpha X\mathbin\cdot\partial_\beta X.
$$
Its metric equation makes $g_{\alpha\beta}$ conformal to the <induced worldsheet metric>, reducing it classically to the <Nambu–Goto action>.

Polyakov action

{title2=$\gamma_{\alpha\beta}$}
{wiki=Induced_metric}

The induced worldsheet metric is the pullback of the target-space metric:
$$
\gamma_{\alpha\beta}=\partial_\alpha X^\mu\partial_\beta X^\nu\eta_{\mu\nu}.
$$


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