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The Hawking-Page transition at weak coupling

Contributed by Anthropic

In anti-de Sitter space a black hole forms through a first-order phase transition, discovered by Hawking and Page. In $\mathcal{N}=4$ super Yang-Mills theory on $S^3$ this is deconfinement at strong coupling. Does the transition remain first order at weak coupling, where there is no classical geometry? The free theory has a transition at its Hagedorn temperature, and it was suggested in 2003 that the two could be continuously connected, either by a single first-order line or through a pair of continuous transitions at weak coupling, which have no counterpart in classical gravity. One number decides: the sign of the quartic coefficient $b$ in the effective action of the Polyakov loop. It first appears at three loops and, to our knowledge, has not been computed in the two decades since. We find $b=-1.48\times10^{-3}\,\lambda^2$ , where $\lambda$ is the 't Hooft coupling. The transition is first order. It lies slightly below the Hagedorn temperature, and its three saddle points appear, exchange dominance and disappear in the same order as thermal anti-de Sitter space and the small and large black holes. The structure of the Hawking-Page transition is thus already present in perturbation theory. The computation is Hamiltonian, and almost every number in it was obtained in two independent ways. It reproduces the two-loop Hagedorn temperature known from integrability and the exact results available in closed subsectors.

Primarily AI-generated textHuman understanding: some partsAdS/CFTBlack HolesHawking–Page transitionN=4 super Yang–Mills

Nov 07, 202642 pages

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hexagon:2611.00174v1

math-ph — Mathematical Physics

The spinless modular bootstrap at large central charge

Contributed by Anthropic

The spinless modular bootstrap bounds the dimension $\Delta_1$ of the lightest primary other than the vacuum in a two-dimensional conformal field theory, using only modular $S$ invariance of the torus partition function at zero angular potential. For unitary, compact theories with central charge $c>1$ and no conserved currents beyond the Virasoro algebra, these bounds read $\Delta_1\le c/\Lambda+o(c)$ at large $c$ . The best slope $\Lambda$ was known only to lie between $8.503$ and $12$ , where the bound would reach the BTZ black hole threshold. We show that it is exactly $\Lambda_V=12\pi^2/(\pi^2+4\,\mathrm{arcsinh}^2 1)\approx9.1266$ . No linear functional in a class that contains all finite combinations of derivatives improves on $c/\Lambda_V$ by more than $O(c^{1/3}\log^{2/3}c)$ , and an explicit functional gives $\Delta_1<c/\Lambda_V+0.483\,c^{1/3}+O(1)$ for all sufficiently large $c$ . The value has an electrostatic origin: as a function of a complexified dimension, $c^{-1}\log\lvert g\rvert$ of a functional $g$ is a two-dimensional potential with positive charges at the zeros of $g$ . Positivity forces $g$ to fall as fast as the Cardy density of states grows, and the maximum principle bounds the potential by an explicit harmonic function, which fixes $\Lambda_V$ . The field of this potential is the spectral curve of a mean-field theory of the optimal functionals, and its charge density is the distribution that the zeros of any functionals attaining $\Lambda_V$ must follow. We also conjecture the coefficient of the $c^{1/3}$ correction to the optimal bound, supported by numerics up to $c=2048$ .

Primarily AI-generated textHuman understanding: some parts3d gravityAdS/CFTconformal field theorymodular bootstrap

Nov 07, 2026131 pages

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