= Thermal winding instability of a closed bosonic string
{title2=$m_{\rm th}^2=\beta^2/(4\pi^2\alpha'^2)-4/\alpha'$}
Euclidean finite <temperature> compactifies time to a circle of radius $R_\beta=\beta/(2\pi)$. The <string oscillator> ground states with zero <momentum> number and winding $w=\pm1$ have the displayed mass from the <bosonic string mass spectrum>. They become massless at $\beta_H=4\pi\sqrt{\alpha'}$ and tachyonic at higher <temperature>. This thermal winding mode diagnoses breakdown of the perturbative low-temperature string gas at the <Hagedorn temperature>. The endpoint and order of a transition require interactions and a suitable stable vacuum; the ordinary zero-winding bosonic <tachyon> is already unstable independently of this thermal threshold.
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