Hi. Suppose that I want to look at the BDH action for a bosonic string:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]S_{BDH} = -\frac{T}{2}\int d^2\sigma \sqrt{\gamma}\gamma^{ij}\partial_iX^a\partial_jX^b[/tex]

The string is in flat spacetime and [tex]\gamma_{ij}[/tex] is an independent world-sheet metric. I know that going to conformal gauge allows me to write the equation of motion for this string as

[tex]\left(\frac{\partial}{\partial\tau} - \frac{\partial}{\partial\sigma}\right)X^a(\tau,\sigma)[/tex].

This is just the massless wave equation and needs to be supplemented with boundary conditions. However, the case for an open string with Neumann boundary conditions is straightforward. I can express the solution as a combination of left and right-moving solutions, apply the boundary conditions, and obtain a mode expansion for the solution.

What I'm having trouble with is the mode expansion for the closed string. I've had a browse through chapter 12 of Zweibach and, while he mentions the mode expansion for the closed string, he doesn't go through the derivation. Can anyone give me a couple of pointers as to how I can produce the mode expansion for the closed string?

Thanks.

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# Mode expansion for a closed string?

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