On the Hagedorn Temperature in Holographic Confining Gauge Theories

The divergence of the string partition function due to the exponential growth of states is a well-understood issue in flat spacetime. It can be interpreted as the appearance of tachyon modes above a certain temperature, known as the Hagedorn temperature $T_H$. In the literature, one can find some intuitions about its generalization to curved spacetimes, where computations are extremely hard and explicit results cannot be provided in general. In this paper, we present a genus-zero estimate of $T_H$, at leading order in $\alpha'$, for string theories on curved backgrounds holographically dual to confining gauge theories. This is a particularly interesting case, since the holographic correspondence equates $T_H$ with the Hagedorn temperature of the dual gauge theories. For concreteness we focus on Type IIA string theory on a well known background dual to an $SU(N)$ Yang-Mills theory. The resulting Hagedorn temperature turns out to be proportional to the square root of the Yang-Mills confining string tension. The related coefficient, which at leading order is analytically determined, is the same as the one for Type II theories in flat space. While the calculation is performed in a specific model, the result applies in full generality to confining gauge theories with a top-down holographic dual.