 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-342/1/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a worldline winding once around a noncontractible torus cycle, a large gauge transformation can satisfy

Δ Λ^{I} = 2 π ℏ n^{I}, n^{I} \in Z .

The source action changes by

Δ S_{c} = - q_{I} \oint d Λ^{I} = - 2 π ℏ q_{I} n^{I} .

n

requires

exp (\frac{i Δ S _{c}}{ℏ}) = e^{- 2 πi q \cdot n} = 1.

Therefore

q \in Z^{n} .

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