Delayed deconfinement and the Hawking-Page transition

Abstract

We revisit the confinement/deconfinement transition in $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 4 SYM.

Spectral form factor for free large N gauge theory and strings

“…There are several benefits of considering Y E,∆ (t) rather than Z(β + it). First, if we consider Z(β + it) for a higher dimensional AdS black hole, as we increase t, due to the cancellation among high energy states from the phases in (1.1), the temperature effectively lowers and at some t ∼ O(β), the quantity will be dominated by the thermal AdS geometry rather than the black hole [13,14]. On the other hand, the thermal AdS will not contribute in the microcanonical quantity Y E,∆ (t) if one sets the energy high enough, so one can use it to study the black hole at a longer time scale.…”

“…The computation in (1.4) can be generalized to Z(β + it) [14] as well as Y E,∆ (t). Due to the existence of latent heat in a first order transition, the Hagedorn temperature T H corresponds to a wide range of energy of order N 2 in the microcanonical ensemble.…”

“…where the integration contour of g m is from the origin to infinity. The detail of this transformation can be found in for example [14,44,45]. The goal of this transformation is solely putting the matrix integral part into a simpler form, for which we already know the phase structure due to [46].…”

“…This is illustrated in figure 4(b). This phenomena is termed "delayed deconfinement transition" and was studied in detail in [14]. 10 This is analogues to the transition from black hole to thermal AdS we mentioned in the introduction, and for this reason, the quantity Z(β + it) is not suitable to study the long time behavior of high energy states.…”

“…(2.28) 12 One might be worried about that g1 becomes complex and some other phases of the matrix model might be important. This is discussed in detail in [14]. In general one might need to consider them.…”