  We study string theory on $\mathbb{R}^d\times \mathbb{S}^1$. For applications
to thermodynamics, the circumference of the $\mathbb{S}^1$ is the inverse
temperature, $\beta$. We show that for $d=6$, the low energy effective field
theory at the inverse Hagedorn temperature, $\beta=\beta_H$, has a one
parameter family of normalizable spherically symmetric solutions that break the
winding symmetry around the $\mathbb{S}^1$. The resulting backgrounds exhibit
an enhanced symmetry, with the symmetry breaking pattern $SU(2)_L\times
SU(2)_R\to SU(2)_{\rm diagonal}$. The effective field theory analysis of these
backgrounds is reliable for some range of parameters. More generally, they are
described by a worldsheet CFT, which corresponds to the free theory on
$\mathbb{R}^6\times \mathbb{S}^1$ perturbed by a non-abelian Thirring
deformation with an $r$-dependent coupling. We propose that, in a certain
scaling limit, string theory in these backgrounds is described by the
$SL(2,\mathbb{R})/U(1)$ cigar, and provides a thermodynamic description of
weakly coupled highly excited fundamental strings. We also discuss the relation
of these backgrounds to Euclidean black holes with near-Hagedorn Hawking
temperature, and possible generalizations to other $d$.
