Entanglement entropy and Page curve from the M-theory dual of thermal QCD above <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> at intermediate coupling

Abstract

Exploration of entanglement entropy (EE) and obtaining the Page curve in the context of eternal black holes associated with top-down nonconformal holographic thermal duals at intermediate coupling has been entirely unexplored in the literature. We fill this gap by obtaining the Page curve of an eternal neutral black hole from a doubly holographic setup relevant to the M-theory dual of thermal QCD-like theories at high temperatures (i.e., above ${T}_{c}$) and intermediate coupling (effected by inclusion of terms quartic in curvature in M theory). Remarkably, excluding the higher derivative terms, the EE of the Hawking radiation from the on-shell Wald entanglement entropy integral (for appropriate choices of constants of integration appearing in the embeddings) increases almost linearly due to dominance of entanglement entropy contribution from the Hartman-Maldacena (HM)-like surface ${S}_{\mathrm{EE}}^{\mathrm{HM},{\ensuremath{\beta}}^{0}},\ensuremath{\beta}\ensuremath{\sim}{l}_{p}^{6}$. This imparts a ``Swiss-cheese'' structure to ${S}_{\mathrm{EE}}^{\mathrm{HM},{\ensuremath{\beta}}^{0}}$ effecting a ``Large N scenario'' (LNS). Then after the Page time, the EE contribution from the island surface (IS) ${S}_{\mathrm{EE}}^{\mathrm{IS},{\ensuremath{\beta}}^{0}}$ dominates and saturates the linear time growth of the entanglement entropy of Hawking radiation and leads to the Page curve. The same is also obtained via the areas of the Hartman-Maldacena-like/island surfaces. Requiring the (IS) EE (${S}_{\mathrm{EE}}^{\mathrm{IS},{\ensuremath{\beta}}^{0}}$) per unit Hawking-Beckenstein entropy (${S}_{\mathrm{BH}}$) post the Page time to be around 2, and positivity of the Page time, set respectively a lower and upper bound on the horizon radius ${r}_{h}$ (the nonextremality parameter). Further, with the inclusion of the $\mathcal{O}({R}^{4})$ terms in M theory, the fact the turning point associated with the HM-like surface is in the deep IR, requires a relationship between ${l}_{p}$ and ${r}_{h}$ along with a conjectural $\ensuremath{\gamma}\ensuremath{\equiv}{e}^{\ensuremath{-}\mathcal{O}(1){N}^{1/3}}$-suppression (motivated by the aforementioned requirement $\frac{{S}_{\mathrm{EE}}^{\mathrm{IS},{\ensuremath{\beta}}^{0}}}{{S}_{\mathrm{BH}}}\ensuremath{\sim}2$ up to leading order). We obtain a hierarchy with respect to $\ensuremath{\gamma}$ in ${S}_{\mathrm{EE}}^{\mathrm{HM},{\ensuremath{\beta}}^{0}},{S}_{\mathrm{EE}}^{\mathrm{IS},{\ensuremath{\beta}}^{0}}$ and ${S}_{\mathrm{EE}}^{\mathrm{HM},\ensuremath{\beta}},{S}_{\mathrm{EE}}^{\mathrm{HM},\ensuremath{\beta}}$ [at $\mathcal{O}(\ensuremath{\beta})$]. This is due to the existence of massless graviton corresponding to a null mass eigenvalue of the Laplace-Beltrami equation in the internal space.

Topics & Concepts