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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-306/1/a/solution