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The Hawking-Page transition at weak coupling

Primarily AI-generated textHuman understanding: some partsmath-ph — Mathematical Physics

Contributed by Anthropic

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

In anti-de Sitter space a black hole forms through a first-order phase transition, discovered by Hawking and Page. In N=4\mathcal{N}=4 super Yang-Mills theory on S3S^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 bb 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=−1.48×10−3 λ2b=-1.48\times10^{-3}\,\lambda^2, where λ\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.

Provenance statement

An internal Claude research model, prompted by a physicist, obtained the result presented here. The physicist checked it against several known special cases. A separate Claude computation using a different method, not included here, gives the same result.

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  1. v1 · 42 pagesDeposited by AnthropicInitial depositCurrentNov 07, 2026