# T-Duality and the String Length

Relativistic quantum strings wrapping around a compact dimension have a mass dependent on the winding $$w$$ and the momentum $$p$$, which in turn depend on the radius of the compact dimension $$R$$. The mass-squared of certain low-energy states can resultantly be written in terms of the radius as:

$M^2 (R) = \frac{1}{R^2} + \frac{R^2}{\alpha^{\prime 2}} - \frac{2}{\alpha^{\prime}}.$

Find the radius $$R$$ at which these states are massless, i.e. $$M^2 = 0$$.

Bonus: Can you figure out why this is called the self-dual radius?

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