Strings at finite temperature: Wilson lines, free energies, and the thermal landscape

According to the standard prescriptions, zero-temperature string theories can be extended to finite temperature by compactifying their time directions on a so-called ``thermal circle'' and implementing certain orbifold twists. However, the existence of a topologically nontrivial thermal circle leaves open the possibility that a gauge flux can pierce this circle---i.e., that a nontrivial Wilson line (or equivalently a nonzero chemical potential) might be involved in the finite-temperature extension. In this paper, we concentrate on the zero-temperature heterotic and Type I strings in ten dimensions, and survey the possible Wilson lines which might be introduced in their finite-temperature extensions. We find a rich structure of possible thermal string theories, some of which even have nontraditional Hagedorn temperatures, and we demonstrate that these new thermal string theories can be interpreted as extrema of a continuous thermal free-energy ``landscape.'' Our analysis also uncovers a unique finite-temperature extension of the heterotic $SO(32)$ and E₈×E₈ strings which involves a nontrivial Wilson line, but which---like the traditional finite-temperature extension without Wilson lines---is metastable in this thermal landscape.