How to Extrapolate A String Model to Finite Temperature: Interpolations and Implications for the Hagedorn Transition

In this paper, we discuss the important question of how to extrapolate a given zero-temperature string model to finite temperature. It turns out that this issue is surprisingly subtle, and we show that many of the standard results require modification. For concreteness, we focus on the case of the ten-dimensional SO(32) heterotic string, and show that the usual finite-temperature extrapolation for this string is inconsistent at the level of a proper worldsheet theory. We then derive the proper extrapolation, and in the process uncover a universal Hagedorn temperature for all tachyon-free closed string theories in ten dimensions --- both Type II and heterotic. As we discuss, these results are not in conflict with the well-known exponential growth in the degeneracies of string states in such models. This writeup is a concise summary of our recent paper hep-th/0505233, here presented using a ``bottom-up'' approach based on determining self-consistent finite-temperature extrapolations of zero-temperature string models. Some new results and observations are also added