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

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 $S_{EE}^{HM, \beta}, S_{EE}^{HM, \beta} (O(\beta))$.