Thermal equilibrium in string theory in the Hagedorn phase

In string theory, a thermal state is described by compactifying Euclidean time on circle $S^{1}_{\beta}$, of fixed circumference. However, this circumference dynamical field which could vary in space, therefore equilibrium not guaranteed. We discuss type II theory near and above the Hagedorn temperature show that can indeed be stabilized presence uniform isotropic flux. solve equations motion derived from an action reproduces tree-level S-matrix. find solutions with topologies $S^{1}_{\beta}\times S^2 \times {\cal M}^{d-2}$ at temperature, include space-filling winding-mode condensate Neveu-Schwarz flux supported $S^1_{\beta}\times S^2$. The we have either linear dilaton or constant dilaton, case, cosmological Ramond-Ramond then compare our to cigar cylinder backgrounds associated $SL(2,R)/U(1)$ coset but without also contrast non-uniform Horowitz-Polchinski solution, possesses characterized approximate temperature.