Fine structure of Hagedorn transitions

We study non-perturbative aspects of the Hagedorn transition for IIB string theory in an anti-de Sitter spacetime in the limit that the string length goes to infinity. The theory has a holographic dual in terms of free $\text{NN}=4$ super-Yang-Mills theory on a three-dimensional sphere. We define a double scaling limit in which the width of the transition region around the Hagedorn temperature scales with the effective string coupling with a critical exponent. We show that in this limit the transition is smoothed out by quantum effects. In particular, the Hagedorn singularity of perturbative string theory is removed by summing over two different string geometries: one from the thermal AdS background, the other from a noncritical string background. The associated noncritical string has the scaling of the unconventional branch of super-Liouville theory or a branched polymer.