Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 1 / b / Solution

A classical string must also satisfy the equations obtained by varying the worldsheet metric before imposing conformal gauge. Vanishing of the worldsheet stress-energy tensor gives

T_{++} = \frac{1}{α ^{'}} \partial_{+} X \cdot \partial_{+} X = 0, T_{--} = \frac{1}{α ^{'}} \partial_{-} X \cdot \partial_{-} X = 0.

In modes these are the Virasoro constraints

L_{m} = \frac{1}{2} \sum_{r} α_{m - r} \cdot α_{r} = 0, L_{m} = \frac{1}{2} \sum_{r} α_{m - r} \cdot α_{r} = 0

for every integer

m

X^{μ}

also requires

(α_{r}^{μ})^{*} = α_{- r}^{μ}, (α_{r}^{μ})^{*} = α_{- r}^{μ} .

Together with periodicity and the wave equation, these conditions remove the unphysical longitudinal worldsheet excitations.

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