Modular invariance of the thermo-partition function and global phase structure of the heterotic string

The global phase structure for heterotic string theory is studied. Three different types of representations for thermo-partition functions are provided. We show how modular invariance arises and analyze finite-temperature divergences. A three-phase structure, involving a new nonperturbative high-temperature phase, for the heterotic string is found. The transition into this new phase takes place at β=π/${\beta }_h$T, where ${\beta }_h$ is the conventional ‘‘Hagedorn temperature’’ for the heterotic string.