Hagedorn Temperature of ${\text{AdS}}_5/{\mathrm{CFT}}_4$ via Integrability

We establish a framework for calculating the Hagedorn temperature of ${\text{AdS}}_5/{\mathrm{CFT}}_4$ via integrability. Concretely, we derive the thermodynamic Bethe ansatz equations that yield the Hagedorn temperature of planar $\mathcal{N}=4$ super Yang-Mills theory at any value of the ’t Hooft coupling. We solve these equations perturbatively at weak coupling via the associated $Y$ system, confirming the known results at tree level and one-loop order as well as deriving the previously unknown two-loop Hagedorn temperature. Finally, we comment on solving the equations at finite coupling.