Touring the Hagedorn Ridge

TL;DR

This work surveys the Hagedorn regime in critical string theories, linking the perturbative picture of a limiting temperature $T_s$ to a global, holography-inspired viewpoint in which a topology change governs the high-energy transition. It introduces the thermal scalar as an effective winding-mode degree of freedom that signals criticality at $\beta=\beta_s$, and emphasizes that the transition is better understood as a geometry/topology change (via Euclidean black-hole nucleation) rather than a simple particle tachyon picture. By employing AdS regularization and toy models with T-duality and Gregory–Laflamme-type transitions on tori, the paper connects string thermodynamics to large-$N$ gauge theories and their holographic duals, illustrating how the Hagedorn plateau can be stabilized or exited through black-hole phases. The overall message is that the high-energy fate of hot string gases is governed by topology-changing processes that can be fruitfully analyzed within a string/black-hole correspondence framework, with implications for holography and the nonperturbative structure of quantum gravity.

Abstract

We review aspects of the Hagedorn regime in critical string theories, from basic facts about the ideal gas approximation to the proposal of a global picture inspired by general ideas of holography. It was suggested that the condensation of thermal winding modes triggers a first-order phase transition. We propose, by an Euclidean analogue of the string/black hole correspondence principle, that the transition is actually related to a topology change in spacetime. Similar phase transitions induced by unstable winding modes can be studied in toy models. There, using T-duality of supersymmetric cycles, one can identify a topology change of the Gregory--Laflamme type, which we associate with large-N phase transitions of Yang--Mills theories on tori. This essay is dedicated to the memory of Ian Kogan.

Figures (18)

• Figure 1: The microcanonical temperature function $T(E) = (\partial S /\partial E )^{-1}$ for a string gas in the two-component approximation. For energies $0<E<E_s$ below the string-scale threshold, with $E_s = \rho_s V$, and $\rho_s = O(1)$ in string units, the temperature grows as $T\sim E^{1/ (d+1)}$, dominated by the massless modes. It gets saturated at $T\approx T_s$ by the highly excited strings. The dotted line represents the sensitivity to small interaction effects that can perturb the Hagedorn plateau either way, into a regime of positive specific heat (with $dT(E)/dE >0$) or a negative one.
• Figure 2: The temperature function in the three-component approximation. Black hole dominance of the density of states at very high energies implies a phase of negative specific heat, starting at $E_g \sim 1/g_s^2$ (in string units), corresponding to the nucleation of small black holes of size $\ell_s$, that subsequently grow as the temperature drops. The Jeans energy $E_L \sim L^{d-2} /g_s^2$ represents the limit beyond which back-reaction effects cannot be neglected on the scale $L$ at the finite-volume box. We have also included the low-energy cutoff at energies of order $1/L$ because the standard scaling $T\sim E^{1/(d+1)}$ only applies for energies above the gap of finite-volume excitations.
• Figure 3: The phase diagram of quantum gravity in ${\rm AdS}_5 \times {\bf S}^5$ with $N$ units of Ramond--Ramond flux, according to the AdS/CFT correspondence. A ten-dimensional weakly-curved description only arises for $R/\ell_s \sim (g_s N)^{1/4} \gg 1$. In this case, the Hagedorn regime is bounded by black hole and massless graviton phases. It only exists for sufficiently weak string coupling, $g_s \ll N^{-9/17}$. At very weak couplings, $g_s < 1/N$, one must use four-dimensional descriptions based on the Yang--Mills degrees of freedom, whereas at strong coupling, $g_s >1$, the diagram mirrors itself by the action of S-duality.
• Figure 4: The complete microcanonical temperature function for thermal AdS spaces, in the four-component approximation. In addition to the gravitons, heavy strings and small black holes, we now include large AdS black holes. When small Schwarzschild black holes grow to size $R$, at energies $E_{R} \sim N^2 /R$, their specific heat becomes positive and the temperature can grow without bound with $T\sim E^{1 / d}$, corresponding to the dual CFT in $d$ dimensions. A Maxwell construction (in the thick line) shows that the Hagedorn plateau is only accessible to superheated states. A first-order phase transition at $T_c \sim 1/R$ nucleates very large black holes of mass $M \sim N^2 R^3 T_c^4$, directly out of the massless graviton phase.
• Figure 5: At very weak coupling $g_s \sim 1/N$, the system matches the perturbative regime of Yang--Mills theory and several hierarchical windows of the system shut down. The ten-dimensional description becomes strongly coupled and standard geometrical intuition breaks down. The string energy $E_s$ that signals the beginning of the Hagedorn plateau becomes of the same order as the finite-size gap of the gauge theory $E_{\rm gap} \sim 1/R$, so that the phase of ten-dimensional graviton entropy disappears. The plateau ends at $E_R \sim N^2 /R$, the energy of the phase transition into the Yang--Mills plasma. This threshold coincides with $E_g$, and the phase of ten-dimensional black holes with negative specific heat also disappears. Instead, the details of the plateau must be worked out in Yang--Mills perturbation theory in the 't Hooft coupling $g_s N < 1$, where some "precursors" of the strong-coupling behaviour described in Fig 4 can be identified (c.f. aha).