Finite temperature bosonic closed strings: Thermal duality and the Kosterlitz-Thouless transition

We elucidate the properties of a gas of free closed bosonic strings in thermal equilibrium. Our starting point is the intensive generating functional of connected one-loop closed bosonic string vacuum graphs on which we impose invariance under a timelike T-duality transformation. Invariance of the Polyakov path integral under thermal duality transformations gives a modular invariant expression for the free energy of free closed strings at finite temperature. We show that a Z₂ twist in Euclidean time is necessary for the absence of spurious vector bosons in the massless spectrum at finite temperature, thereby resolving an ambiguity in the Euclidean time prescription for perturbative string theory. The spurious modes have no correspondence among the propagating degrees of freedom in a finite temperature gauge theory. We find that states localized at the fixed points of the Z₂ orbifold contribute a constant term to the entropy but are otherwise absent from the thermodynamic potentials including both the internal energy and specific heat. We show that a Kosterlitz-Thouless continuous phase transition at the self-dual temperature results in the formation of the long string phase in a self-dual gas of free closed strings at high temperature. There is no Hagedorn phase transition in the free energy of the free bosonic string gas.