Dispelling the Hagedorn Myth: Canonical and Microcanonical Strings

We clarify why the well-known exponential growth in the ultraviolet asymptotics of the mass level expansion in any superstring theory does not imply a divergence in the one-loop string free energy, even when restricted to target spacetime bosonic modes alone: perturbative string theory is not merely an infinite tower of point particle field theories of increasing mass; it is both finite, and renormalizable, and equivalent to the 2d gauge theory of diffeomorphism and Weyl invariances. We calculate the free energy of the type IIA and type IIB superstring canonical ensembles at finite temperature, demonstrating the absence of thermal tachyons in the physical state spectrum. We derive the exact temperature dependence of the free energy in the form of a power series expansion by explicit evaluation of the one-loop modular integrals of closed string theory. Similar expressions are derived for the free energy of both heterotic closed string ensembles, as well as for the unoriented type I open and closed superstring ensembles, consistent with the thermal (Euclidean T-duality) transformations of the String/M Duality Web. Our string theory analysis leads to an expression for the pair correlator of Wilson loops in the low energy finite temperature Yang Mills gauge theory, evidence for a first order thermal phase transition at the self-dual temperature. Finally, we derive an explicit expression for the microcanonical density of states of the type IB open string ensemble, demonstrating that the growth in the density of string states as a function of ensemble energy is slower than that of an exponential. Our elucidation of both the canonical, and the microcanonical, type IB open string ensembles is irrefutable evidence that the Hagedorn "phase transition" in string theory is a myth