We study the implications of duality symmetry on the analyticity properties of the partition function as it depends upon the compactification length. In order to obtain non-trivial compactifications, we give a physical prescription to get the Helmholtz free energy for any heterotic string supersymmetric or not. After proving that the free energy is always invariant under the duality transformation $R\rightarrow \alpha^{ '}/(4R)$ and getting the zero temperature theory whose partition function corresponds to the Helmholtz potential, we show that the self-dual point $R_{ 0}=\sqrt{ \alpha^{ '}}/2$ is a generic singularity as the Hagedorn one. The main difference between these two critical compactification radii is that the term producing the singularity at the self-dual point is finite for any $R \neq R_{ 0}$. We see that this behavior at $R_{ 0}$ actually implies a loss of degrees of freedom below that point.
Categories: hep-th