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

The momentum integral is replaced by the sum over momentum and winding modes. Combining the zero modes with the oscillator trace gives

Z_{R} (τ, \overset{τ}{ˉ}) = \frac{1}{∣ η ( τ ) ∣ ^{2}} n, w \in Z \sum exp [- π τ_{2} (\frac{n ^{2}}{R ^{2}} + w^{2} R^{2}) + 2 πi τ_{1} n w] .

Equivalently,

Z_{R} = \frac{1}{∣ η ∣ ^{2}} \sum_{n, w} q^{p_{L}^{2} /4} \overset{q}{ˉ}^{p_{R}^{2} /4}

with the left/right convention of part (iv). The answer is invariant under the T-duality

R \leftrightarrow 1/ R

accompanied by

n \leftrightarrow w

in these units.

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