\section{Discussion}
\label{sec:discussion}

\subsection{What has been shown}

The quartic coefficient of the Polyakov-loop effective action of $\cN=4$ super Yang--Mills theory on $S^3$ is negative at order $\lambda^2$. It follows that at small $\lambda$ and large $N$ the theory has a single first-order deconfinement transition, at a temperature $T_c$ below the Hagedorn temperature by a relative amount $9.4\times10^{-5}\lambda^2$. In a window of temperatures around $T_c$ there are three saddle points: a confined one with free energy of order one, a deconfined one with free energy of order $N^2$ and a gapped eigenvalue distribution, and an unstable one between them.

This is the structure of the gravitational problem. There the three saddles are thermal anti-de Sitter space, the large black hole and the small black hole, and they exist together above a temperature $T_{\min}=\sqrt2/\pi R$, with the first two exchanging dominance at $T_{\rm HP}=3/2\pi R$. The identification of the saddles across the two regimes was proposed in \cite{AMMPV1} and sharpened in \cite{AGLW,BasuWadia,ABMW}, and the present result adds only that it is consistent at weak coupling. The simplest phase diagram in the plane of $\lambda$ and $T$, a single line of first-order transitions from $\lambda=0$ to $\lambda=\infty$, is allowed. The alternative, in which the line splits into two at some finite coupling, was the only possibility had $b$ been positive. It is now the less economical of the two.

We cannot say more than allowed. Whether the line in fact continues unbroken through intermediate coupling is not something that a weak-coupling calculation can test.

\subsection{The unstable saddle and the Hagedorn temperature}

The unstable saddle merits a comment. At weak coupling it sits at $|u_1|=u_s\propto\sqrt{T_H-T}$, it merges with the confined saddle at $T_H$, and it carries the information that the confined phase, although locally stable all the way to $T_H$, is only metastable above $T_c$. This is the scenario that Atick and Witten proposed for string theory in flat space \cite{AtickWitten}: the Hagedorn temperature is never reached in equilibrium, because a first-order transition driven by the winding mode intervenes. They argued for it from the sign of a quartic coupling of the winding mode, induced by its coupling to the dilaton and the graviton. The argument here has the same structure. The winding mode is $u_1$, and the exchange of $u_2$, which makes a negative contribution to $b$ whatever its coupling, plays the part of the massless exchange.

On the gravity side, a small black hole in anti-de Sitter space becomes as small as the string length as its temperature approaches the Hagedorn temperature of the string, and it is expected to turn into a highly excited string \cite{HorowitzPolchinski,ABMW,CMW,Urbach}. At weak coupling the corresponding statement is that the unstable saddle merges with the confined phase at $T_H$. The three temperatures \eqref{eq:three-T} thus have direct counterparts. The interpolation of $T_H(\lambda)$ is known from integrability \cite{HW1,HW2,HW3}: it rises from $0.380$ at $\lambda=0$ and grows like $\lambda^{1/4}/(2\sqrt2\,\pi)$ at large $\lambda$, where it is the Hagedorn temperature of the string in a nearly flat region. The first correction to this behaviour has been derived on the string side by Urbach, who records that Maldacena found it independently (unpublished) \cite{Urbach}, and it agrees with the numerical integrability result. Formulas for the next orders have been conjectured or proposed, partly with that numerical input, and they agree with it as well \cite{EMT,BCC,HarmarkScalar}. The transition temperature stays of order one, reaching $3/2\pi=0.477$. At weak coupling the two are separated by a relative $10^{-4}\lambda^2$, and at strong coupling by a factor that grows like $\lambda^{1/4}$. It would be interesting to follow $T_c(\lambda)$, but no analytic method is known. A Monte Carlo route through the plane-wave matrix model has been proposed \cite{IKNT}. The free energy of the deconfined phase is not a spectral problem of the planar dilatation operator.

\subsection{Open problems}

Several extensions are within reach of the same methods.

\emph{Derive the $\tfrac43$.} The finite renormalisation \eqref{eq:43} should follow from a one-loop computation of the Coulomb propagator and the fermion kinetic term with a mode cut-off. We expect it to be the statement that the cut-off renormalises the speed of light.

\emph{Resolve the pure Yang--Mills class.} This needs a comparison with the expressions and integrals that \cite{AMMPV2} give for each diagram, or a third calculation. A Lagrangian calculation of diagrams 3c, 3f and 3n alone would do.

\emph{Chemical potentials.} With chemical potentials for the R-charges and the spins, $b$ becomes a function on a five-dimensional space, which interpolates between \eqref{eq:b} and the entries of table~\ref{tab:nearbps}. The codes take the chemical potentials as input. The phase diagram with R-symmetry chemical potentials was mapped at zero coupling, and at weak coupling and high temperature, in \cite{YamadaYaffe}, and near the critical chemical potentials in \cite{HKNW}. The Hagedorn temperature with chemical potentials has been studied with integrability \cite{HW3,EMT2}. The gravitational side has charged and rotating black holes with a phase structure of their own, and a comparison at the level of the order of the transition should be possible.

\emph{Other theories.} For a gauge theory on $S^3$ with adjoint matter, the first term in \eqref{eq:master2} is a function of the letter content and the one-loop operator only, and the second is not. The one example known to us of a weakly coupled large-$N$ gauge theory with a single-trace Lagrangian on a sphere of two or more dimensions that was reported to have $b>0$ is pure Yang--Mills theory on a small $S^2$ \cite{PSV}. When scalar fields with arbitrary couplings are added, $b$ can be given either sign, in more than two space-time dimensions by a double-trace term \cite{AMMPV2}. It would be useful to know whether a single-trace example exists on $S^3$, and what decides the sign. The Yang--Mills coupling to matter seems to favour $b<0$: the plane-wave matrix model in perturbation theory \cite{HRSY}, the massive matrix quantum mechanics of \cite{AMMPVW}, pure glue on $S^3$ and now $\cN=4$ all have it.

\emph{The next order.} The $\lambda^3$ term of $b$ needs the three-loop dilatation operator off shell, and four-loop vacuum diagrams. No qualitative question that we know of needs it.

\subsection{A closing remark}

Witten's identification of the Hawking--Page transition with deconfinement is one of the oldest entries in the AdS/CFT dictionary. It has been tested at strong coupling and at zero coupling. At zero coupling it holds in a weak sense only, since the free transition is of a degenerate kind that fits no gravitational picture. The computation reported here removes the degeneracy. The first effect of the interaction is to make the transition into what it is at strong coupling: first order, with a small unstable black hole in between. This outcome was not guaranteed. The one-loop operator alone makes it plausible, and the two-loop operator, which might have opposed it, as it does in pure Yang--Mills theory, turns out to help.