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

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 operatorInserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting worldsheet ghost fields giveswhere $b_{\alpha \beta }$ is symmetric and traceless and $c^{\alpha }$ is a vector. The gauge-fixed integral is consequentlyOn 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.