Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 2 / iv

Solution

X \sim X + 2 π R

, a closed spatial circuit of the worldsheet may wind around the target circle:

X (τ, σ + 2 π) = X (τ, σ) + 2 π Rw, w \in Z .

p = n / R

n \in Z

. In the convention

p_{L} = \frac{n}{R} + wR, p_{R} = \frac{n}{R} - wR,

H = \frac{1}{2} (\frac{n ^{2}}{R ^{2}} + w^{2} R^{2}) - \frac{1}{12} + k \geq 1 \sum k (N_{k} + N_{k}),

P = n w + k \geq 1 \sum k (N_{k} - N_{k}) .

The

n w

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