Entanglement Entropy and Page Curve From the M-Theory Dual of Thermal Qcd at Intermediate Couplingby@multiversetheory
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Entanglement Entropy and Page Curve From the M-Theory Dual of Thermal Qcd at Intermediate Coupling

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Explore the intricate relationship between entanglement entropy and Page curves in the context of M-theory's dual description of thermal QCD. Gain insights into the impact of higher derivative terms, the behavior of Hartman-Maldacena-like and island surfaces, and the hierarchy of entanglement entropies. This chapter delves into the entanglement entropy and Page curves in the M-theory dual representation of thermal QCD, examining the effects of intermediate coupling and higher derivative terms. It explores the behavior of Hartman-Maldacena-like and island surfaces, elucidating a hierarchy of entanglement entropies and the role of massless gravitons in shaping the curves. Additionally, it discusses the implications of O(R^4) corrections and their impact on the Page curve's behavior.
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(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute.




Chapter 1: Introduction

Chapter 2: SU(3) LECs from Type IIA String Theory

Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation

Chapter 4: Conclusion and Future Outlook


Chapter 5: Introduction

Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity

Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling

Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times

Chapter 9: Multiverse in Karch-Randall Braneworld

Chapter 10: Conclusion and Future outlook






7.1 Introduction and Motivation

In doubly holographic approaches, as described in 5.3.2 of chapter 5, a gravitational dual black hole is connected with an external CFT bath [202]. As an illustration, gravity in d-dimensions is connected with the external bath in d-dimensions, wherein d-dimensional exterior bath possesses a corresponding holographic dual in (d + 1)-dimensions. In such models, we take into consideration two variants of the previously mentioned configuration. We look at two distinct types of extremal surfaces in these models. The very first one corresponds to the Hartman-Maldacena-like surface [144], that begins from the position at which gravity interacts with with an external bath, i.e., at the defect, and traverses the black hole horizon to get to the defect of the thermofield double associated with doubly holographic setups, and the entanglement entropy involvement via the aforementioned surface possesses linear time dependency, leading to information paradox at later times. The other type of surface includes the island surface, that begins at the exterior bath and ends on the KarchRandall brane [142, 143]. The contribution of entanglement entropy coming from the island surface becomes time independent and takes over beyond the Page time. The Page curve is obtained through the combination of the entanglement entropy contributions obtained from both the extremal surfaces.

As we discussed in 5.3.2, in some cases, the gravity becomes massive on the Karch-Randall branes, and one was unable to get the Page curve with massless gravity localized on the KarchRandall branes. In this chapter, we have looked at this major issue of doubly holographic setup from M-theory perspective in the presence of O(R4 ) terms in the supergravity action. We have a different situation from the literature that we have a non-conformal bath (QCD bath), and the doubly holographic setup is being constructed from a top-down approach[1] with the inclusion of higher derivative terms in the supergravity action. Interestingly, the issue of the massive graviton is absent in our setup, and one can get the Page curve with massless gravity on the end-of-the-world(ETW) brane.

We cannot utilize the island approach for non-conformal backgrounds because currently there is no recognized Cardy-like formula for the non-conformal theories. As a result, we used the previously mentioned doubly holographic frameworks technique to arrive at the Page curve. In the context of our scenario, on the gravity dual side, there is a M-theory dual consisting of higher derivative terms, whereas on the gauge theory side, we find nonconformal thermal QCD-like theories at the intermediate coupling, the model is discussed in 1.4 of chapter 1. In the present chapter, we created a doubly holographic setup analogous to [7,203], where the researchers utilize a conformal theory on their gauge theory side. There are two types of potential surfaces to think about: Hartman-Maldacena-like and island surfaces. The Hartman-Maldacena-like surface has been accountable for the gradual linear temporal growth of Hawking radiation’s entanglement entropy. Above the Page time, the entanglement entropy component via the island surface takes over and this is not dependent upon time, so we extract the Page curve using the M-theory dual, which includes O(R4 ) corrections. As a remark, we used the formula of [127] to compute the entanglement entropy with the inclusion of higher derivative terms and when higher derivative terms are absent then we used the Ryu-Takayanagi formula to compute the entanglement entropy [107].

7.2 Doubly Holographic Setup in M Theory

Figure 7.1: Doubly holographic setup in M-theory dual. ETW-black “brane” is coupled the thermal bath (at the tip of seven-fold of G2-structure which is a cone over a warped non-Kähler resolved conifold), where the Hawking radiation is collected, which effectively is thermal QCD2+1 along x1,2,3 after Wick-rotation along x 3 at r = 0. Blue curves correspond to the island surfaces and red curve corresponds to the Hartman-Maldacena-like surface.

From the work of [127], we could compute the entanglement entropy associated with holographic dual theories with the inclusion of higher derivative terms. We could express the analogue of the generalised entropy functional described in equation (5.17) using [127] for M-theory dual in the presence of O(R4) terms as follows:

and inserting it into (7.20) yields:

7.3.1 At O(β)

and we assume the constant is zero.

7.3.2 No Boundary Terms Generated at O(R4)

The eleven-dimensional supergravity action, which includes O(R4 ) terms, is presented by the following:


As a result, we are able to identify that a usual boundary term including covariant derivatives of metric fluctuations that would have to be cancelled out by a suitable boundary term (using Stokes theorem) is:

The (dual to the) unit normal vector to W provided by:



7.4 Page Curve without Higher Derivative Terms

Here, we obtained the Page curve using the doubly holographic setup constructed in 7.2. In this part, we do not consider the O(R4) terms in the supergravity action. Therefore, we used the Ryu-Takayanagi formula to compute the entanglement entropies of Hartman-Maldacena-like and Island surfaces in 7.4.1 and 7.4.2. The Page curve has been obtained in 7.4.3. In the absence of O(R4) terms, the N = 1, D = 11 supergravity action has been provided as follows:

7.4.1 Entanglement Entropy Contribution from Hartman-Maldacena-like Surface

To calculate the time-dependent entanglement entropy associated with a Hartman-Maldacena-like surface, we first look at the induced metric on a constant x 1 slice, which was calculated utilizing (7.20) as follows:

The area density functional for a Hartman-Maldacena-like surface could be calculated using (7.23) as follows:


Using (7.27) and (7.30), we obtained:

Now we identify time via the integral:

As a result, the Hartman-Maldacena-like surface’s entanglement entropy is: Hartman-Maldacena-like Surface Analytics/Numerics

After integrating all of the angular coordinates and consequently including a (2π) 4 emerging from integration w.r.t. ϕ1,2, ψ, x10, the area corresponding to the Hartman-Maldacena-like surface is obtained as:

As a result, the entanglement entropy related with a Hartman-Maldacena-like surface grows linearly with time.

7.4.2 Entanglement Entropy Contribution from Island Surface

We take a constant t slice to calculate the entanglement entropy for an island surface. As a result, utilizing equation (7.20), we expressed the induced metric associated with the island surface as follows:

where C is the constant. We simplified the expression (7.43) utilizing equation (7.41) as given below:

When we solved the preceding equation for x˙(r), we were provided with:

Constant C could be calculated using the equations (7.45) and (7.46), as shown below:

Taking the previously stated expression of C, the equation (7.44) reduces to the following structure:

Now, the area density functional associated with the island surface has been simplified using the equations (7.41) and (7.48).

Hence, the entanglement entropy associated with the island surface is obtained as follows:

Since, there is no time dependence in (7.50), and hence entanglement entropy of the Hawking radiation for the island surface is constant. Island Surface Analytics

When (7.49) is evaluated, the entanglement entropy is as follows:


7.4.3 Page curve at LO from Areas of Hartman-Maldacena-like and Island Surfaces

7.5 Page Curve with Higher Derivative Terms

We obtained the Page curve corresponding to a neutral black hole in with the inclusion of higher derivative terms that are quartic in the Riemann curvature tensor in this section.

Figure 7.2: Page curve up to O(β0) of an eternal neutral black hole in M-theory dual via doubly holographic setup. The blue line in the figure indicates the area of the Hartman-Maldacena-like surface, and the orange line refers for the area of the island surface; E in (7.39) is set to 1.1 to obtain the same order of the magnitude of tPage as in Fig. 7.7.

This section has been broken into four subsections. We calculated the entanglement entropy associated with the HM-like surface in subsection 7.5.1, discussed the “Swiss-Cheese” structure of the identical surface in subsection 7.5.2, computed the entanglement entropy of island surface in 7.5.3, and at last obtained the Page curve corresponding to an eternal black hole in subsection 7.5.4 utilizing the outcomes that were obtained from previous subsections.

We conducted the same analysis as in 7.4. [127] was used to compute the entanglement entropy associated with Hartman-Maldacena-like surface and island surface in with the inclusion of higher derivative terms. In generic higher derivative gravity theories, we are able to calculate the holographic entanglement entropy (5.8) as follows:

• When the holographic dual consists of a (d+ 1) dimensional gravitational background, then with the help of the embedding function, compute the induced metric associated with the co-dimension two surface.

• Using the above-mentioned induced metric, compute (5.8). This yields the holographic entanglement entropy as an expression of the embedding function and its derivatives.

• Find the solution to the embedding function’s equation of motion.

• In higher derivative gravity theories, putting the solution found in the previous step inside the action yields the holographic entanglement entropy.

To determine the second part in the formula (5.8), we must first compute the four types of derivatives for the Hartman-Maldacena-like and island surfaces, which are listed below:

The numerator and denominator coefficients in the preceding equation are x = xR for the Hartman-Maldacena-like surface and x = x(r) for the island surface. All four types of variations occurring in equation (7.54) for the Hartman-Maldacena-like surface in appendix C.5.1 and island surface in appendix C.5.2 have been computed and listed.

7.5.1 Entanglement Entropy Contribution from Hartman-Maldacena-like Surface

Hartman-Maldacena-like surface in M-theory dual corresponds to a co-dimension two surface situated at x 1 = xR. Utilizing equation (5.8), we are able to write the mathematical expression that describes the entanglement entropy associated with a Hartman-Maldacena-like surface as follows:

where g denotes determinant of the induced metric defined in (7.23) for the co-dimension two surface. For the metric (7.23), the entanglement entropy associated with the Hartman-Maldacena-like surface using (7.55) and appendix C.5.1 after the angular integrations is obtained as:


The equation (7.58) has the following solution:

substitution of (7.59) in (C.20) yields:

Keeping the term up to the leading order in N, we could write the embedding function t(r) as follows:

The entanglement entropy of the Hawking radiation corresponding to the Hartman-Maldacenalike surface is computed as given below:

utilizing (C.13) and (7.59), (7.66) is expressed as:


7.5.3 Entanglement Entropy Contribution from Island Surface

The island surface is a co-dimension two surface that exists at a continuous time slice, much like the Hartman-Maldacena-like surface. Therefore, we can formulate the formula for the holographic entanglement entropy as follows:

Hence, we found that:


Thus, the island surface’s equation of motion associated with embedding x1(r) is as follows:

(7.89)’s solution is provided as:


The above equation has the following solution:

The black hole entropy for the metric (7.19) is obtained as:

Therefore, the entanglement entropy for the island surface after substitution of β simplified as:

From Fig. 7.5 and utilizing (7.101), it is obvious that (7.97) ∼ (7.98) means that a lower constraint will be placed on rh, the non-extremality parameter in the M-theory dual of large-N thermal QCD.

Figure 7.6: The Page curve obtained from the use of Dong’s proposal to compute entanglement entropies of the Hartman-Maldacena-like (blue line) and island (orange line) surfaces.

7.5.4 Page Curve and An Exponential(-in-N) Hierarchy of Entanglement Entropies up to O(R4) Before/After the Page Time

Page time: At the Page time, the entanglement entropies for the Hartman-Maldacena-like surface and the Island Surface are equal. This results in:

Utilizing (7.99), (7.104) implies,

Similarly, for the island surface O(β) contributions to the entanglement entropy is obtained as:

Utilizing (7.99), (7.107) results in:

From the equations (7.71), (7.100), (7.105) and (7.108), we obtained the following hierarchy:

7.6 Massless Graviton - the Physical reason for Exponentially Suppressed Entanglement Entropies

Similar to [209], we can write the eleven-dimensional metric (7.19) as given below:

We found that:

the solution of above equation is:

Since (7.121) is not well defined in the UV when m ̸= 0. Therefore, we need to consider m = 0 for which the graviton wave function is:

Figure 7.8: Volcano potential for massless graviton from (7.125)

Figure 7.8 plots the aforementioned potential for massless graviton (m = 0); this potential is “volcano”-like, with the massless graviton localized at the horizon on the ETW “brane”. This is analogous to [142], in which gravity can be localized on the end-of-the-world (ETW) brane with non-zero brane tension owing to the emergence of a “crater” in the “volcano” potential in the Schrödinger-like equation of motion of the graviton wave function.

Figure 7.9: Graviton Wave Function localization at the horizon via the m = 0-limit of the solution (7.119) of (7.115).

7.7 Summary

In this chapter, we constructed the doubly holographic setup from a top-down approach as discussed in 7.2. We have taken the external bath as thermal QCD in three dimension whose holographic dual is M-theory inclusive of O(R4) corrections [1], see chapter 1 for more detail of the same. The intermediate description of the doubly holographic setup constructed by us couples the black hole on the ETW-“brane” at x = 0 to thermal QCD bath via transparent boundary condition at the defect. We have addressed the effect of higher derivative terms in the context of top-down construction of double holography for the first time. First we obtained the Page curve without inclusion of O(R4) terms in the eleven-dimensional supergavity action in 7.4. To achieve the same, we computed the entanglement entropies associated with the Hartman-Maldacena-like (HM-like) and island surfaces and obtained the Page curve with the help of these entropies because the HM-like surface has

Figure 7.10: Localization of the Graviton Wave Function near the horizon from the solution of the Schrödinger-like radial wave equation (7.123).

Based on the viability of the Islands scenario, the M-theory dual of large-N thermal QCD at high temperature obtained in [15], [1] (based on [14]) produces the previously mentioned Page curve. The above-mentioned top-down M-theory dual, on the other hand, provides a number of new conceptual information:

• Due to the lack of details of precise structures for boundary terms on the ETW-“brane” along with the presence of higher derivative terms, there exists relatively few works (e.g., [147]) in which the authors look into the doubly holographic setup in higher derivative theory of gravity. Remarkably we showed in 7.3.2 that the presence of the O(R4) terms generates no boundary terms.

• As for as we know, our setup/(above-mentioned) M-theory dual appears either the sole one in existence (from M-theory) or one among the comparatively few top-down approaches that generate(s) the Page curve for massless graviton.

• In our top-down M-theory dual, we find that ETW-“brane” to be a fluxed hypersurface W that is a warped product of an asymptotic AdS4 and a six-fold M6 where M6 is a warped product of the M-theory circle and a non-Einstenian generalization of T 1,1 ; the hypersurface W, can also be thought of as an effective ETW-“brane” corresponding to fluxed intersecting M5-brane wrapping a homologous sum of S 3×[0, 1] and S 2×S 2 in a warped product of R 2 and an SU(4)/Spin(7)-structure eight-fold. The ETW-“brane”, W, then has non-zero “tension” and a massless graviton localized near the horizon by a “volcano”-like potential.

• Unlike largely other works in the literature, which compute the Page curve with a CFT bath, the external bath in our model is a non-CFT bath (thermal QCD). • Entanglement entropy contribution generated by a Hartman-Maldacena (HM)-like surface, which has been causing the growth of the Einstein-Rosen bridge in time, shows a Swiss-Cheese structure in the Large-N scenario (7.5.2) as discussed earlier.

• By the presence of O(R4 ) terms in the action, our previously discussed M-theory dual produces a hierarchy in the entanglement entropies of the HM-like and Island surface (IS) with respect to a large-N exponential suppression factor, resulting physically via the presence of massless graviton mode on an ETW-brane. This suppression also suggests the fact that the addition of higher derivative terms - O(R4) in particular - has no effect on the Page curve.

• To regulate the IR- and large-N enhancement in the IS entanglement entropy per unit BH entropy, a relationship involving the Planckian length and the non-extremality parameter (the horizon radius) has been shown to occur.

• The positivity of the Page time, calculated in (refPage-time), has been showed to set an upper constraint on the non-extremality parameter, the black-hole horizon radius rh.

This paper is available on arxiv under CC 4.0 license.