Duality in nontrivially compactified heterotic strings.

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 → α′/(4R) and getting the zero temperature theory whose partition function corresponds to the Helmholtz potential, we show that the self-dual point R0 = √ α′/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 6= R0. We see that this behavior at R0 actually implies a loss of degrees of freedom below that point.