T-Duality and the String Length

Classical Mechanics Level 3

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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