Entanglement entropy and Page curve from the ${\cal M}$-theory dual of thermal QCD above $T_c$ at intermediate coupling

TL;DR

This paper explores the entanglement entropy and Page curve in a top-down M-theory dual of thermal QCD above the critical temperature, revealing a transition from linear growth to saturation due to island contributions, with implications for black hole information.

Contribution

It introduces a novel holographic setup for studying the Page curve in non-conformal thermal QCD at intermediate coupling within M-theory, including higher derivative corrections.

Findings

Entanglement entropy grows linearly before Page time

Island surfaces cause entropy saturation after Page time

Hierarchy established between different entanglement entropy contributions

Abstract

Obtaining the Page curve in the context of eternal black holes associated with top-down non-conformal holographic thermal duals at intermediate coupling, has been entirely unexplored in the literature. We fill this gap in the context of a doubly holographic setup relevant to the M-theory dual of thermal QCD-like theories at $T>T_c$ at intermediate coupling. Remarkably, excluding the higher derivative terms, the entanglement entropy(EE) of the Hawking radiation from the on-shell Wald EE (for appropriate choices of constants of integration appearing in the embeddings) increases almost linearly with the boundary time due to dominance of EE contribution from the Hartman-Maldacena(HM)-like surface $S_{EE}^{HM, \beta^0}, \beta\sim l_p^6$. Curiously, this imparts a "Swiss-Cheese" structure to the surface $S_{EE}^{HM, \beta^0}$ at a given time (less than the Page time $t_{P}$), in $\mathbb{R}_{\geq0} \times \mathbb{C}$ effecting what could be dubbed as a "Large N Scenario" (LNS). Then, after $t_{P}$, the EE contribution from the Island Surface (IS) $S_{EE}^{IS, \beta^0}$ dominates and saturates the linear time growth of the EE of Hawking radiation and leads to the Page curve. Requiring $S_{EE}^{IS, \beta^0}/S_{BH}\sim2$ up to LO in the non-conformal analog of "$c G_N^{(11)}/r_h^9$", and positivity of $t_{P}$, set respectively a lower and upper bound on the horizon radius $r_h$ (the non-extremality parameter). With the inclusion of the $O(R^4)$ terms in M theory, the turning point associated with the HM-like surface/IS being in the deep IR, results in a relationship between $l_p$ and $r_h$ along with a conjectural $e^{-{\cal O}(1) N^{1/3}}$-suppression (motivated by $S_{EE}^{IS, \beta^0}/S_{BH}\sim2$). We obtain a hierarchy with respect to this N-dependent exponential in $S_{EE}^{HM, \beta^0}, S_{EE}^{IS, \beta^0} (O(\beta^0))$ and $S_{EE}^{HM, \beta}, S_{EE}^{HM, \beta} (O(\beta))$.