 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

Since

δ γ_{α β} = 2 \partial_{(α} X_{μ} \partial_{β)} δ X^{μ},

δ S_{NG} = - \frac{T}{2} \int d^{2} σ - γ γ^{α β} δ γ_{α β} = - T \int d^{2} σ - γ γ^{α β} \partial_{α} X_{μ} \partial_{β} δ X^{μ} .

Integrating by parts and taking

δ X^{μ}

to vanish on the boundary yields

δ S_{NG} = T \int d^{2} σ δ X_{μ} \partial_{α} (- γ γ^{α β} \partial_{β} X^{μ}) .

\partial_{α} (- det γ γ^{α β} \partial_{β} X^{μ}) = 0.

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