Dienes and Lennek argue that this widely believed argument is wrong.
- In the UV picture, the divergence is wrong in string theory, they say, because it effectively leads to you to treat string theory as field theory with many fields, which means that the one-loop integrals over "tau" are made over "Im(tau)" being positive and "Re(tau)" smaller than one-half. The correct stringy integral involves the fundamental domain of the modular group "SL(2,Z)", and if this domain is used, the divergence at the (old) Hagedorn point disappears, they claim.
- In the IR picture based on the existence of tachyon, they note that the tachyon is actually projected out by the GSO projections, and consequently, it has no physical consequences. The usual Hagedorn phase transition is therefore absent in heterotic string theories and other tachyon-free superstring theories. They affirm that the Hagedorn behavior in type 0 theories, for example, still holds.
The authors also say that one of the conventional Hagedorn transitions does occur but it is a 10th order phase transition where "10" is indeed the spacetime dimension "D" (and it would be replaced by "D-1" if the dimension were odd).
I am curious what others think about this paper.