Choose a reference metric in each conformal class. An infinitesimal <worldsheet diffeomorphism> generated by $c^\alpha$ and a <Weyl transformation> decompose the metric variation into its trace and the traceless operator
$$
(P_1c)_{\alpha\beta}
=\nabla_\alpha c_\beta+\nabla_\beta c_\alpha
-g_{\alpha\beta}\nabla_\gamma c^\gamma.
$$
Inserting the gauge condition and its <Faddeev-Popov determinant> cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting <worldsheet ghost fields> gives
$$
S_{\rm Gh}[b,c,g]
=\frac1{2\pi}\int_\Sigma d^2\sigma\,\sqrt g\,
b^{\alpha\beta}(P_1c)_{\alpha\beta},
$$
where $b_{\alpha\beta}$ is symmetric and traceless and $c^\alpha$ is a vector. The gauge-fixed integral is consequently
$$
Z=\int\mathcal DX\,\mathcal Db\,\mathcal Dc\,
\exp(-S_{\rm P}-S_{\rm Gh}).
$$
On higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 4 / a / Solution

{wiki=Diffeomorphism}

A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which $X^\mu$ is a scalar and $g_{\alpha\beta}$ is a rank-two tensor.


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