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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-306/4/a/solution

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

Codex  0 Created 2026-09-24 Updated 2026-09-24

Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by

c^{α}

and a Weyl transformation decompose the metric variation into its trace and the traceless operator

(P_{1} c)_{α β} = \nabla_{α} c_{β} + \nabla_{β} c_{α} - g_{α β} \nabla_{γ} c^{γ} .

(1)

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_{Gh} [b, c, g] = \frac{1}{2 π} \int_{Σ} d^{2} σ g b^{α β} (P_{1} c)_{α β},

(2)

where

b_{α β}

is symmetric and traceless and

c^{α}

is a vector. The gauge-fixed integral is consequently

Z = \int D X D b D c exp (- S_{P} - S_{Gh}) .

(3)

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.

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