Delayed Deconfinement and the Hawking-Page Transition

Abstract

We revisit the confinement/deconfinement transition in $\mathcal{N}=4$ super Yang-Mills (SYM) theory and its relation to the Hawking-Page transition in gravity. Recently there has been substantial progress on counting the microstates of 1/16-BPS extremal black holes. However, there is presently a mismatch between the Hawking-Page transition and its avatar in $\mathcal{N}=4$ SYM. This led to speculations about the existence of new gravitational saddles that would resolve the mismatch. Here we exhibit a phenomenon in complex matrix models which we call "delayed deconfinement". It turns out that when the action is complex, due to destructive interference, tachyonic modes do not necessarily condense. We demonstrate this phenomenon in ordinary integrals, a simple unitary matrix model, and finally in the context of $\mathcal{N}=4$ SYM. Delayed deconfinement implies a first-order transition, in contrast to the more familiar cases of higher-order transitions in unitary matrix models. We determine the deconfinement line and find remarkable agreement with the prediction of gravity. On the way, we derive some results about the Gross-Witten-Wadia model with complex couplings. Our techniques apply to a wide variety of (SUSY and non-SUSY) gauge theories though in this paper we only discuss the case of $\mathcal{N}=4$ SYM.

Figures (9)

• Figure 1: Structure of Lefschetz thimbles (blue color) $\mathcal{J}_{0,\pm}$ and upward flow lines (red color) $\mathcal{K}_{0,\pm}$ respectively for $\theta = 0.7$ (left frame), $\theta = 1.7$ (center frame) and $\theta = 2.7$ (right frame).
• Figure 2: Regions of dominance of various phases in the complex-$g$ plane. The regions are delimited by the anti-Stokes lines $\mathop{\rm Re}(A^{\textrm{s}/\textrm{w}}(g))=0$. In the blue region the one-cut gapped phase dominates, in the orange region the ungapped one-cut phase dominates and in the yellow region presumably none of them dominates. While as shown in the appendix one cut phases exist in the yellow "multi-cut" region, presumably they do not dominate since the instanton actions for eigenvalues tunneling out of the one cut phases are not suppressed.
• Figure 3: Deconfinement in the complexified model \ref{['trunc']}. In both gray and blue areas the gapped one-cut phase contributes as a saddle point. However, only in the grey region we have deconfinement ($\mathop{\rm Re}(Q)>0$) . The deconfinement curve $\mathcal{C}_D$ is drawn in red. The black dot is $a_1=1$.
• Figure 4: A plot of \ref{['Qcomplex']} in the real (left) and complex case (right, we take $\mathop{\mathrm{Arg}}\nolimits(a_1)=\pi/4$ for the sake of illustration). Different colors correspond to different values of $|a_1|$. In the second plot, the phase of $g$ is always chosen to be that of $g_*(a_1)$ in \ref{['g1c']}. Notice that in the complex case the appearance of a second maximum does not coincide with deconfinement. The latter happens only when the second maximum becomes positive. The intermediate dashed region is presumably dominated by multi-cut configurations.
• Figure 5: Left: Deconfinement for the truncated model in the $\mathcal{N}=4$ case. The grey area is the prediction coming from our analysis, while the blue area is the region where tachyonic modes exist ($\mathop{\rm Re} a_1 >1$) but the index is not yet deconfined. The black dot is the point $p_\star$ while the red point is the gravity prediction $p_{\text{HP}}$. Right: The numerical values of higher $a_n$ as a function of $y$.