T-duality -- Very fast words -- ( just words no equations, don't have time)

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 Consider:


$X(\tau,\sigma + 2 \pi) = X(\tau,\sigma) + m(2\pi R)$

Set 

$X(\tau,\sigma + 2\pi) = X(\tau, \sigma) + 2\pi \alpha$

Follow through as usual.

 - Do Fourier mode expansion


-Compute the momentum, and write the Hamiltonian

-Use creation and annihilation operators to compute the spectrum



At the level of the spectrum:

"the closed string spectrum for a compactification with radius R is identical to the closed string spectrum for a compactification with radius $\bar{R} = \frac{\alpha}{R}$."

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