Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 4 / 2 / Solution

Along the common D-brane worldvolume, the Neumann solution is

X^{μ} (σ, τ) = x^{μ} + 2 α^{'} p^{μ} τ + i 2 α^{'} \sum_{n \neq = 0} \frac{α _{n}^{μ}}{n} e^{- in τ} cos nσ, μ = 0, \dots, p .

In each transverse direction, the endpoint conditions

X^{I} (0, τ) = x_{1}^{I}

and

X^{I} (π, τ) = x_{2}^{I}

give

X^{I} (σ, τ) = x_{1}^{I} + \frac{x _{2}^{I} - x _{1}^{I}}{π} σ + i 2 α^{'} \sum_{n \neq = 0} \frac{α _{n}^{I}}{n} e^{- in τ} sin nσ, I = p + 1, \dots, D - 1.

The linear term is the classical stretch between the two parallel D-branes.

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