LMU-ASC 14/22

MPP-2022-40

#  Islands and Light Gravitons in type IIB String Theory

Saskia<EMAIL_ADDRESS>, Alessandra
<EMAIL_ADDRESS>, Ioannis<EMAIL_ADDRESS>muenchen.de , Dieter<EMAIL_ADDRESS><EMAIL_ADDRESS>muenchen.de

(αMax-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Föhringer Ring 6, 80805, München, Germany
βArnold-Sommerfeld-Center for Theoretical Physics,
Ludwig-Maximilians-Universität, 80333 München, Germany
)

Abstract

We consider the setup of a black hole in AdS4 coupled to an external bath,
embedded in type IIB string theory. We study quantum extremal islands in these
backgrounds, in relation to the existence of a massive graviton. Using
explicit results of the microscopic embedding of AdS4 massive gravity in
string theory, we investigate whether it is possible to achieve backgrounds
with extremal islands, in which the lowest lying graviton is only slightly
massive. For certain regions of the microscopic parameters, the graviton mass
can be computed explicitly, and we explain how it directly affects the
existence and the properties of the islands. We also show that islands can in
principle exist within the regime of validity of the massive gravity effective
field theory. However we see via numerical computations that the existence of
quantum extremal islands at zero temperature is highly constrained, also when
the dilaton is allowed to vary, so that the mass of the graviton cannot be
made arbitrarily light. At finite temperature, we also identify a critical
parameter, above and below which islands still exist but exhibit a different
behavior. Our work supports recent proposals that the unitary evolution of
black holes in higher dimensions, and more precisely their Page curve, relies
on the presence of a massive graviton in the effective theory.

###### Contents

  1. 1 Introduction
  2. 2 Entanglement islands in higher dimensions: From braneworld models to type IIB embeddings
    1. 2.1 Karch-Randall braneworlds and Page curve
    2. 2.2 IIB string theory embeddings
  3. 3 Massive graviton, islands and EFT validity regime
  4. 4 Island surfaces and running dilaton
    1. 4.1 Critical parameters and graviton mass
    2. 4.2 Numerical setup for island surfaces at $\delta\phi\neq 0$
    3. 4.3 Island surfaces at zero temperature
    4. 4.4 Island surfaces at finite temperature
  5. 5 Conclusions and discussion

##  1 Introduction

Recent progress in the study of systems of black holes coupled to an external
bath has lead to important developments in deciphering the evolution of
information during black hole evaporation555This has fueled an intense
research activity over the last couple of years, which has been captured by
recent reviews and progress reports [1, 2, 3]., reproducing the Page curve of
an evaporating black hole [4, 5, 6]. Advances in this direction rely on a
deeper understanding of the calculation of the entropy of Hawking radiation,
which has been possible thanks to the identification of a new contribution to
the entanglement entropy, following from a quantum generalization [7] of the
holographic entanglement entropy prescription [8, 9]. This novel contribution
is due to _islands_ , quantum extremal surfaces physically disconnected from
the asymptotic radiation region, that dominate the entropy contribution after
the Page time, ensuring a unitary evolution of the system [7, 4, 5, 6]. The
role of quantum extremal islands in reproducing the Page curve of black hole
evaporation has been extensively studied in models of lower dimensional
gravity, where it corresponds to the contribution of wormhole geometries to
the gravity path integral [10, 11]. Not withstanding that methods to directly
compute the entanglement entropy of quantum fields in a conformal field theory
(CFT) are best developed in lower dimensional systems, the extension to higher
dimensional setups is one of the most significant problems that has emerged
from the islands prescription.

Proposals to investigate islands in higher dimensions have appeared [12, 13,
14, 15, 16], based on gravitational setups that include a Randall-Sundrum
brane [17, 18, 19]. These are models of localized gravity, which we will refer
to as Karch-Randall setups, in which a 4d end-of-the-world (EOW) brane is
added to the anti-de-Sitter AdS5 bulk. The brane ends on the asymptotic region
of the AdS5 space, cutting the boundary off at the intersection. A black hole
living on the brane will evaporate in a part of the AdS5 boundary, the
radiation region. In this framework, the entanglement entropy of quanta in the
radiation region can be computed holographically via the Ryu-Takayanagi
prescription: the contribution of quantum extremal islands is captured through
the identification of a corresponding extremal surface in the AdS5 bulk.

The Karch-Randall branes setup can be interpreted as gravitational
realizations of dual boundary CFTs in presence of a defect [20]. Holographic
string theory configurations realizing conformal field theories with boundary
and interfaces are known, and can be constructed via explicit uplifts to,
e.g., type IIB string theory [21, 22, 23, 24]. In this context, quantum
extremal islands and their contribution to the black hole entropy have been
studied in [25], making more precise the relation of these systems to Karch-
Randall configurations. Also bottom-up models of interface CFTs have been
investigated recently in [26, 27], which clarify the correspondence between
the gravitational system coupled to a CFT and the BCFT, and derive general
results for entaglement entropy in braneworld setups.

The main physical lessons emerging from higher dimensional studies of quantum
extremal islands are that 1) when the black hole is coupled to a non-
gravitating bath, there must be a massive graviton in the theory for islands
contribution to become dominant at some stage in black hole evaporation[13],
and 2) islands extremal surfaces undergo a phase transition related to the
variation of the underlying geometric parameters (brane tension, ratios of the
number of branes in the microscopic setup…) [13, 25].

In this paper, we will address the study of quantum extremal islands in a
family of the type IIB backgrounds. In particular we will consider the uplift
of a Janus-like geometry to type IIB string theory, extending the analysis of
[25] both presenting a quantitative discussion on the graviton mass in the low
energy theory, as well as considering more general backgrounds with varying
dilaton. In this analysis the black hole is coupled to a non-gravitating
bath666In the case of a gravitating bath, previous works have consistently
showed that the Page curve is not correctly reproduced in Karch-Randall setups
or in type IIB uplifts [28, 14, 25].. Crucially, in a particular regime of
these solution one disposes of a quantitative control over the mass of the
graviton. Indeed, by looking at a specific range of microscopic parameters,
through the study of the spin two wave equation it is possible to derive a
quantitative expression for the lightest spin 2 field [29, 30], both for non-
varying dilaton as well as for background where $\delta\phi\neq 0$. This will
enable us to clarify the role of the mass of the graviton on the appearance of
islands in this particular context.

We will investigate the properties of island surfaces both for empty Anti de
Sitter (_zero temperature_) and when the black hole is present (_finite
temperature_). We will see that islands at zero temperature are bound to a
region of parameter space, where the graviton mass is above a certain critical
value. This critical value must be contrasted with the natural ultraviolet
(UV) cutoff [31] of a massive gravity effective field theory (EFT), and we
will discuss the relevant regime of the microscopic parameters to ensure that
the range of validity of the EFT does not exclude the presence of islands.
Notice that the embedding of this EFT in the UV has been provided in [30], so
we will not have to worry about the validity of the massive gravity EFT which,
in general, can be subject to Swampland constraints [32, 33].

After ensuring the consistency of the EFT with the existence of islands, we
will study the properties of these surfaces in backgrounds with varying
dilaton.

Our main result is that increasing the dilaton variation affects the
microscopic parameter range, within which the islands still contribute (at
zero temperature), or they undergo a phase transition (at finite temperature).
This directly reflects on the mass of the graviton, which depends both on the
microscopic parameters and the dilaton variation. More precisely, we find
that, when allowing the dilaton to vary, one can find overlapping regimes
where the graviton is only slightly massive and at the same time islands
contribute at zero temperature. However, islands cannot be found for
arbitrarily large dilaton variation $\delta\phi$, where the graviton mass can
tuned to be arbitrarily low. This shows that the above is not possible, if one
also wants to preserve the properties of quantum extremal islands: the latter
enter in fact a critical regime as the graviton mass is increased.
Furthermore, this yields another confirmation that, in higher dimensions,
islands and black hole evaporation are physically related to the graviton
mass.

The present work is structured as follows. In Section 2 we review the Karch-
Randall setup of a black hole coupled to a thermal bath, as well as the
realization of such configuration in IIB string theory via an explicit
microscopic uplift. We consider in particular backgrounds with varying
dilaton, and we present the expression for the mass of the graviton. In
Section 3 we discuss the regime of validity of the EFT below the cutoff
$\Lambda_{EFT}$ of [31], and how this translates into a bound on the number of
D-branes of the 10 dimensional setup. Section 4 is devoted to the detailed
analysis of islands in background with varying dilaton. Our results are
discussed in comparison to the constant-dilaton setup [25], which one
retrieves in the $\delta\phi=0$ limit of our analysis. We end the paper with
conclusions in Section 5, where we stress once again how our results provide
evidence that the island contribution to black hole entropy is related to the
presence of a graviton with mass above a critical value.

##  2 Entanglement islands in higher dimensions: From braneworld models to
type IIB embeddings

In the present section, we summarize the main attributes of the basis of this
work, namely the braneworld models describing AdS black holes coupled to an
external, non-gravitating bath, and their realization via string theory
uplifts, consisting on a class of warped AdS4 backgrounds. We will deal with
special limits of the internal manifold, in which the lowest-lying graviton
has a parametrically small mass, and we also briefly review of the
corresponding massive gravity embedding.

###  2.1 Karch-Randall braneworlds and Page curve

The most promising realization of quantum extremal islands for evaporating
black holes in higher dimensions is the Karch-Randall (KR) braneworlds [17,
18]. They offer in fact important insights on the calculation of the
entanglement entropy of emitted radiation, thanks to their doubly holographic
nature: these configurations can be seen in fact either as $d$-dimensional
conformal field theories (CFTd) on half-space coupled to a $(d-1)$-dimensional
boundary theory, or a gravitational system on the AdSd+1 bulk sliced by an end
of the world (EOW) AdSd brane, or CFTd on the half space and AdSd coupled to
CFTd on the brane. These three description are related by holographic duality,
applied to either the full system, or just to the boundary of AdSd+1. The last
description, depicted in Fig. 1, is central for the study for the black
hole/bath system.

The present work focuses on a four dimensional funnel configuration 777Recall
that only in the funnel solution the Hawking radiation can be encoded in the
classical 5d bulk, as opposed to the droplet solutions, for details see
section 3 of [13] as well as [14]. , consisting of a black-hole AdS4-brane
embedded in an AdS5 bulk and intersecting its conformal boundary. This setup
provides a standard configuration of an evaporating black hole coupled to an
external, non-gravitating bath, in which two kinds of entanglement extremal
surfaces appear that contribute to the Page curve: the Hartman-Maldacena
surface ($\mathcal{S}_{\text{HM}}$) and the Quantum Extremal Island surface
($\mathcal{S}_{I}$) [13, 14]. At early times, the Page curve is characterized
by a linear growth reflecting the dominance of the $\mathcal{S}_{\text{HM}}$
over $\mathcal{S}_{I}$, while after some characteristic time the latter
dominates, and the entropy saturates. This is due to the fact that the area of
the $\mathcal{S}_{\text{HM}}$ increases with time while the area of the
$\mathcal{S}_{I}$ remains constant.

In the braneworld constructions, a relevant parameter characterizing the
configuration is the angle $\vartheta$ formed between the EOW brane and the
boundary. Its significance, at zero temperature, relies on the fact that it
introduces a regime of contribution of the islands to the generalized
entanglement entropy. It has been found in fact that island surfaces in empty
Anti de Sitter roughly disappear once the angle falls below a certain value,
referred to as the critical angle $\vartheta_{c}$. On the other hand, this
behavior is regulated by the horizon in the AdS black string background, and
islands contribute also for angles below the critical one [14]. Understanding
how island surfaces are related to a critical parameter in the type IIB uplift
corresponding to this braneworld scenarios is one of the main points we will
address in this work.

The graviton is massive due to the coupling of the brane to the bath, as a
result of transparent boundary conditions. The next section will be entirely
devoted to the analysis of the consequences a massive graviton has in the EFT
where islands are supposed to contribute to the entanglement entropy. What we
would like to review here is that from doubly holographic models, it can be
observed that the massless graviton limit corresponds to the vanishing angle
limit $\vartheta\to 0$ where the EOW-brane approaches the conformal boundary
of AdS5. In the massless limit, the islands cease to contribute. This can be
understood from the behavior of the island surface in this limit, both
endpoints of which now approach $\partial\text{AdS}_{5}$. The surface anchors
on the boundary of the radiation region and on the boundary of the island, as
seen below. In the limit where $\vartheta\to 0$, this surface falls into the
Poincaré horizon. The divergence of the surface is explained by the divergence
of the inverse of the gravitational constant $G$ on the brane [13].

Figure 1: Karch-Randall setup: funnel configuration with AdS4 black hole on
the EOW brane within the AdS5, bent at an angle with respect to the conformal
boundary of the bulk. The blue segment on the brane is the Island, on the
boundary of which is anchored the island surface, ending on the endpoint of
the radiation region, denoted by the green segment, on the conformal boundary.
The dashed line segments correspond to the two kinds of RT surfaces.

###  2.2 IIB string theory embeddings

Having reviewed the braneworld constructions, in this section we will
introduce the family of corresponding type IIB embeddings that will be the
protagonist of our study of quantum extremal islands.

#### Uplifting 4d/5d configurations

Karch-Randall double holographic models, as CFTs ending on a boundary, can be
actually explicitly embedded in IIB string theory. The embedding is described
by the warped background $AdS_{4}\times_{w}\mathcal{M}_{6}$, where the six
dimensional internal manifold is a warped product of two-spheres over a
Riemann surface. When the internal manifold $\mathcal{M}_{6}$ is compact,
these backgrounds are holographically dual to 3d $\mathcal{N}=4$ super
conformal field theories (SCFT) engineered in type IIB with D5, NS5 and D3
branes [23, 24]. In the case where the internal manifold $\mathcal{M}_{6}$ is
non-compact, the geometry asymptotically becomes AdS${}_{5}\times$S5 and the
solutions are dual to a 3d defect SCFT in 4d $\mathcal{N}=4$ super Yang-Mills.
The 10d geometry is parametrized by a pair of harmonic functions, with
logarithmic singularities at points on the boundary of the Riemann surface,
representing the locations of D5 and NS5 branes. The AdS4 fiber degenerates at
these points, with one of the two $S^{2}$’s shrinking to a point at each
singularity. Non-contractible three and five-cycles support non-vanishing
three-form and five-form fluxes, indicating that the background can be seen as
the near horizon limit of intersecting IIB brane configurations, where D3
branes are suspended between five-branes.

In this work we consider a particular case of the general setups introduced in
[23]. The left asymptotic region smoothly caps off and is substituted by an
interior point of the full ten-dimensional space while the geometry asymptotes
to global $AdS_{5}\times S^{5}$ on the right888Strictly speaking, these
backgrounds correspond to a realization of a “fat” Karch-Randall brane, given
by the NS5-D5-D3 bound state, while the original KR setup uses a thin brane;
both descriptions model the same defect CFT configuration [30].. The resulting
background is given by the following metric:

$ds_{10}^{2}=L_{4}^{2}ds_{\text{AdS}_{4}}^{2}+f_{1}^{2}ds_{S_{1}^{2}}+{f_{2}}^{2}ds_{{S_{2}}^{2}}+4\rho^{2}dzd\bar{z},$
(2.1)

where $z$ and $\bar{z}$, parametrize the Riemann surface:

$\displaystyle\Sigma$ $\displaystyle=\left\\{z=x+iy\ \large|\
x\in\mathbb{R}\,,\ y\in\left[0,\frac{\pi}{2}\right]\right\\}\ ,$ (2.2)

the warp factors are

$\displaystyle L_{4}^{8}$ $\displaystyle=16\frac{N_{1}N_{2}}{W^{2}}\
,\qquad\rho^{8}=\frac{N_{1}N_{2}W^{2}}{h^{4}\hat{h}^{4}}\ ,\qquad
f_{1}^{8}=16h^{8}\frac{N_{2}W^{2}}{N_{1}^{3}}\ ,\qquad
f_{2}^{8}=16\hat{h}^{8}\frac{N_{1}W^{2}}{N_{2}^{3}}\ ,$ (2.3)

and

$\displaystyle\begin{aligned} W&=\partial
h\bar{\partial}\hat{h}+\bar{\partial}h\partial\hat{h}=\partial\bar{\partial}(h\hat{h})\
,\\\ N_{1}&=2h\hat{h}|\partial h|^{2}-h^{2}W\ ,\\\
N_{2}&=2h\hat{h}|\partial\hat{h}|^{2}-\hat{h}^{2}W\ .\end{aligned}$ (2.4)

Finally, these functions are determined by two harmonic functions which, in
our setup, will take the form999In their standard form, the harmonic functions
are labelled by a set of parameters controlling the asymptotic regions of the
geometry, the variation of the dilaton between the asymptotic regions, the D5
brane charges and finally the position of the point singularities on the
boundaries of the strip. For a complete presentation and details of the
supergravity solutions, the reader is referred to [21, 22, 23].  (we set
$\alpha^{\prime}=1$):

$\displaystyle\begin{aligned}
h&=-\frac{i\pi}{4}e^{z}\kappa-\frac{N}{4}~{}\text{log}\,\text{th}\Big{(}\frac{i\pi}{4}-\frac{z}{2}\Big{)}+\text{c.c}\,,\\\
\hat{h}&=\frac{\pi}{4}e^{z}\hat{\kappa}-\frac{N}{4}~{}\text{log}\,\text{th}\Big{(}\frac{z}{2}\Big{)}+\text{c.c}\,.\end{aligned}$
(2.5)

This setup will generalize the one of [25], by considering a nonzero dilaton
variation in the $AdS_{5}$ throat as in [30], given by

$\displaystyle e^{2\delta\phi}=\frac{\hat{\kappa}}{\kappa}\ .$ (2.6)

The configuration with a black hole is obtained by substituting
$ds^{2}_{AdS_{4}}$ with the metric of a black hole in AdS4. Since the geometry
considered here is capped off on the left, the black hole will be radiating in
the right asymptotic region, playing the role of a non-gravitating bath.

The solution described by the harmonic functions in (2.5) includes two point
singularities localized on the strip boundaries at $(0,\frac{\pi}{2})$ and
$(0,0)$. For $\kappa=\hat{\kappa}\equiv K$ there are N-D5 and N-NS5 branes
respectively, and $2NK$ semi-infinite D3 branes along the asymptotic region on
the right. In [25] it was shown how the extremal island surfaces are crucially
dependent on the parameter $\alpha=N/K$, which is the corresponding critical
parameter of the KR setups. This parameter will be recovered in our setup
as101010If we define $\kappa=Ke^{-\delta\phi}$, and
$\hat{\kappa}=Ke^{\delta\phi}$, then $\alpha=N/K$ which holds for generic
$\delta\phi$, and reduces to the parameter of [25] when $\delta\phi=0$. :

$\displaystyle\alpha=\sqrt{\frac{N^{2}}{\kappa\hat{\kappa}}}\ .$ (2.7)

As presented above, the angle $\vartheta$ between the EOW brane and the
conformal boundary of AdS5 determines at zero temperature the contribution of
the islands to the entropy. In the string theory embedding at hand, the
quantity corresponding to the angle $\vartheta$ of the lower dimensional
setups, is the ratio $\alpha$ of the D3 branes suspended between the
fivebranes over the semi infinite D3s of the solution, see Fig. 2. Practically
this quantity is a ratio of “boundary” over “bulk” degrees of freedom and can
be interpreted as the “tension” of the composite brane (NS5-D5-D3 system).
Therefore, for large values of $\alpha$ the angle $\vartheta$ becomes small
and vice versa: $\alpha\leftrightarrow 1/\vartheta$. The behavior of islands
crucially changes upon variation of parameters above or below
$\alpha_{\text{crit.}}$.

Figure 2: The geometric ratio $\alpha$ corresponds to the relative number of
D3 branes that are either semi-infinite or suspended between the D5 branes in
the brane configuration realizing the IIB embedding with a gravitating AdS
bath.

The analysis of [25] indicates that at zero temperature, there exists a
critical value of this ratio below which the island surfaces dominate over the
Hartmann-Maldacena surface and the islands contribute to the entropy leading
to a Page curve.

In practice determining this critical value $\alpha_{\text{crit.}}$ is based
on numerical analysis that studies the divergence of the distance between the
anchoring points of the island surface on the black hole region and on the
external bath. The numerical computation of the critical ratio will be
explained and performed in Section 4. This behavior is analoguous to the low-
dimensional one where with respect to critical value of the angle parameter,
subcritical setups are characterized by island contribution and hence by Page
curve-like behavior of the entanglement entropy.

At this point, we underline the fact that the existence of a critical value
for the geometric ratio $\alpha$ for which the islands stop to contribute is
an attribute of the zero temperature solution, which is an uplift of the
corresponding empty-AdS braneworld model presented in the previous subsection.
On the contrary, as in the AdS-black string background, at finite temperature
this effect is regulated by the black hole horizon and islands continue to
contribute even for values of the geometric ratio higher than the critical one
determined by the zero temperature solution.

A central point to be noted is that in the above solution the lowest-lying
graviton is massive, as a result of the non-compactness of the internal
manifold [29]. Nevertheless an explicit computation of its mass is only
possible on the special limit of the geometry where the mass is parametrically
small.

#### Light gravitons

The type IIB string embedding corresponding to the metric (2.1) is a
consistent uplift of the corresponding braneworld models. In this
construction, the lowest-lying spin-2 mode acquires a mass due to the fact
that the internal manifold is non-compact. In the following we review how this
type IIB background can be appropriately adjusted in order to allow for such a
nearly vanishing graviton mass, which can then be computed explicitly. After
briefly presenting the corresponding setup we will continue by examining,
using the results of the previous section, whether islands contribute in the
limit where the graviton mass is nearly non-zero.

The AdS4 vacuum where the internal manifold $\mathcal{M}_{6}$ is compact,
includes a massless low-lying graviton and a KK tower of massive modes. The
geometry in which the graviton acquires a mass, includes on the contrary a
non-compact internal manifold. The spin-2 spectrum can be extracted
numerically by the corresponding eigenvalue problem for the Laplace-Beltrami
operator on the internal manifold [29]. Choosing to focus instead on the
lowest lying spin-2 mode one solves the variational problem giving the minimal
eigenvalue of the mass squared operator [30]. The background allowing for a
small mass for the lowest-lying spin-2 consists of a compact region in the
vicinity of the fivebrane singularities (bag) attached to a semi-infinite
AdS${}_{5}\times$S5 throat under the constraint that the radius of the throat
is much smaller than the size of the bag111111It has been proven recently that
the graviton mass is very light in generic compactifications consisting of a
small throat connecting two compact geometries [34].
($L_{5}<<L_{\text{bag}}\sim L_{4}$). This can be seen in the Fig. 3, where
$L_{5}$ is the throat radius and $L_{4}$ the AdS4 radius, bound to the bag-
size $L_{\text{bag}}$ as a result of the scale non-separation.

Figure 3: The geometry of interest: The compact bag is attached to a semi-
infinite throat with radius hierarchically smaller than the size of the six-
dimensional internal manifold. The fivebranes are found on the point
singularities located at $\delta,\hat{\delta}$.

The reason for this can be well understood holographically [30]: Consider a
CFT3, dual to the AdS4 solution above. This theory has a conserved stress-
tensor $T_{(3)}^{\mu\nu}$, with canonical conformal dimension, as a result of
the conservation. Coupling this three-dimensional theory to a four dimensional
$\mathcal{N}=4$ super Yang-Mills in the bulk, results to the violation of the
conservation of the stress tensor. Since the two theories are coupled, the
three-dimensional stress tensor is no longer conserved but rather leaks in the
bulk direction. In particular, we consider that the stress tensor is
dissipating weakly in the extra bulk direction, a fact that implies that its
(canonical) dimension acquires a small anomalous dimension. Holographically,
the anomalous dimension of the stress tensor corresponds to a small correction
to the $AdS_{4}$ graviton mass:

$m_{g}^{2}L_{4}^{2}=\Delta(\Delta-3)\xrightarrow{\Delta=3+\varepsilon}{}m_{g}^{2}L_{4}^{2}\sim\varepsilon$
(2.8)

The weak dissipation is ensured by the fact that the number of degrees of
freedom of the boundary theory is much greater than the one of the bulk
theory, hence

$\displaystyle m^{2}_{g}L_{4}^{2}\ \sim\ \frac{\text{bulk dof}}{\text{bdy
dof}}\ .$ (2.9)

In terms of the dual solutions, the scarcity of the bulk degrees of freedom
correspond to the small D3 brane number in the semi infinite throat as opposed
to the large number of D3 branes connecting the D5 and NS5 branes in the bag-
region. The graviton mass, obtained by solving the variational problem for the
minimal mass eigenvalue corresponding to the graviton wavefunction can be
written explicitly as [30]:

$m_{g}^{2}L_{4}^{2}\equiv\bar{m}_{g}^{2}\simeq\frac{3\pi^{3}}{4}\Big{(}\frac{L_{5}}{L_{\text{bag}}}\Big{)}^{8}\mathcal{J}(\text{ch}(\delta\varphi))$
(2.10)

at leading order in the radii ratio. The last factor gives the contribution of
the dilaton to the graviton mass121212In the case of varying dilaton the
background develops a Janus throat, a one-parameter deformation of
$AdS_{5}\times S^{5}$ geometry [21].: if it does not vary through the throat,
$\mathcal{J}(\text{ch}(\delta\varphi))$ reduces the graviton mass for
increasing dilaton variation, while its contribution becomes trivial in the
case of a constant dilaton. As we will see in the following analysis, the
dilaton variation plays a central role in the discussion of the island
surfaces. In the case of the solution at hand, parametrized by (2.5), the
radii are given by $L_{5}^{4}\sim NK$ and $L_{\text{bag}}^{4}\sim N^{2}$ ,
with the expression for the mass taking the form:

$m_{g}^{2}L_{4}^{2}\equiv\bar{m}_{g}^{2}\simeq\frac{3\pi^{3}}{4}\Big{(}\frac{K}{N}\Big{)}^{2}\mathcal{J}(\text{ch}(\delta\varphi)).$
(2.11)

Figure 4: The Janus correction function. Observe that for vanishing dilaton
variation the mass is not affected.

The above background is realized as a special limit of the bimetric gravity
embedding [35], where the throat connects two compact regions, with the
spectrum including also a massless graviton, while it is related under an
appropriate deformation to solutions with multiple massive gravitons [36].

In this work, by studying the uplift of the 4d/5d embedding, we analyse how to
tune the parameters of the geometry in such a way to allow for a slightly
massive graviton and study properties of the island surfaces under this
modification of the embedding. Finally, we focus on the effect of the dilaton
variation on the behavior of the island surfaces and on the condition for
reconciling small graviton mass and island contribution.

##  3 Massive graviton, islands and EFT validity regime

Let us start from the simple configuration of Section 2.2, with $N$-D5 branes
connected by D3 branes to $N$-NS5 branes, residing in the upper and lower
point singularities respectively. The background is engineered so that the
dilaton is constant $\delta\phi=0$. We are going to discuss, in this section,
the graviton mass and the EFT cutoff in relation to the critical value of the
geometric parameter, $\alpha_{\text{crit.}}$, characterizing the island
surfaces.

When the background develops a small-radius throat connecting the AdS4 to the
AdS5 geometry, as explained in the previous section and depicted in Fig. 3,
the background has a slightly massive graviton, whose mass is given from
(2.11), which for $\delta\phi=0$ takes the form:

$\bar{m}_{g}^{2}\simeq\frac{3\pi^{3}}{4}\frac{1}{\alpha^{2}}\ .$ (3.1)

The dependence of the mass on the parameter $\alpha$ is a feature expected
from the analysis in Section 2 and the holographic interpretation of the
graviton mass, as we already saw in eq. (2.8): the mass of the graviton
decreases for increasing number of bag-degrees of freedom, and hence for
increasing value of the parameter $\alpha$. Since the parameter $\alpha=N/K$
is the corresponding quantity of $\vartheta^{-1}$ in the KR setup, this
implies that the graviton mass is also directly related to the angle: a small
graviton mass $\bar{m}_{g}^{2}$ corresponds to small values of $\vartheta$ and
vice versa. Finally, in the KR setups island surfaces are not expected to
contribute for values $\vartheta<\vartheta_{\text{crit.}}$ or equivalently in
10d for $\alpha>\alpha_{\text{crit.}}$. This means that for a background with
fixed bag degrees of freedom (namely with fixed number of five-branes, $N$),
islands are expected to contribute where the graviton mass is above some
critical value:

$\bar{m}_{g}>\bar{m}_{g,\text{crit.}}\sim\frac{1}{\alpha_{\text{crit.}}}\qquad\leftrightarrow\qquad{K>K_{\text{crit.}}}\
.$ (3.2)

For a given 10d background, the value of $\alpha_{\text{crit.}}$ can be
determined numerically. However, for consistency with small graviton mass,
from eq. (3.1), $\alpha_{\text{crit.}}$ must be be sufficiently large in this
regime (this translates into $N\gg K$). For models where
$\alpha_{\text{crit.}}$ is not found to be sufficiently large to ensure a
hierarchy between $L_{4}$ and $L_{5}$, (or equivalently between the defect and
the bulk CFT degrees of freedom) the mass formula (2.10) is not valid.

For an AdS massive gravity theory in the regime $m_{g}\ll L_{4}^{-1}$ (i.e.
$K\ll N$), the EFT breaks down at a scale given by [31, 37]

$\Lambda_{*}=\frac{m_{g}^{\frac{1}{3}}M_{\text{Pl}}^{\frac{1}{3}}}{L_{(4)}^{\frac{1}{3}}}\,,$
(3.3)

where it is already implied that $m_{g}<\Lambda_{*}$, in an AdS background
with fixed $L_{4}$ ($M_{\text{Pl}}$ is the 4d Planck Mass that will be set to
1). Then, the critical values of the graviton mass for the island surfaces can
be outside (case a)) or inside the regime of validity of the EFT (case b) and
c) of Fig. 5). For the critical mass to be an energy scale at least below the
EFT cutoff, we must require

$\displaystyle m_{g\,\text{crit.}}<\Lambda_{*}\ \ \Rightarrow\ \
K_{\text{crit.}}<K^{1/3}N^{5/6}\ll N^{7/6}\ ,$ (3.4)

where we have averaged131313From non-separation of scales, recall that :
$\langle L_{(4)}^{2}\rangle_{\text{bag}}\sim L_{\text{bag}}^{2}\sim N$.
Averaging over the 6d internal manifold: $\langle
f\rangle_{\text{bag}}=\int\sqrt{g}f/\sqrt{g}$. the expression on ${\cal
M}_{6}$. Recall that, due to the no-scale separation, $\langle
L_{(4)}^{\frac{1}{2}}\rangle_{\text{bag}}\sim L_{\text{bag}}^{\frac{1}{2}}\sim
N^{\frac{1}{4}}$.

A more stringent regime corresponds to the requirement that the graviton mass
is above the critical mass value, which simply translates into

$\displaystyle m_{g\,\text{crit.}}<m_{g}\ \ \Rightarrow\ \ K_{\text{crit.}}<K\
.$ (3.5)

Notice that the first bound varies if one varies the AdS4 vacuum, while the
second case is independent on the scale of the Anti de Sitter curvature
$L_{4}$ of the background under consideration. Only if we require (3.5)
islands always exist in the massive gravity EFT, for consistency with (3.2).
These bounds reproduce once again the fact that islands contribution to
entanglement entropy are in tension with a parametrically small graviton mass.

Figure 5: For the range $m_{g\,\text{crit.}}<m_{g}<\Lambda_{*}$, shown in case
c), the island contributes to the entropy while the theory has a valid EFT
description.

We can compare the results of this general analysis with the numerical study
of [25] for backgrounds of the form considered in this section. There, a value
of $\alpha_{\text{crit.}}\sim{\cal O}(1)$ was found, which is way below the
value needed to reach a parametrically small graviton mass. The rest of the
paper will be dedicated to an analysis of a more general background by
introducing a varying dilaton through the harmonic functions $h$ and $\hat{h}$
with two different parameters $\kappa\neq\hat{\kappa}$, to investigate how the
presence of a $\delta\phi\neq 0$ affects the critical value of the microscopic
parameter $\alpha$.

Let us conclude this section with a comment on the $m_{g}\to 0$ limit. In
principle, in Karch-Randall double holographic models, one can smoothly and
continuously tune the angle $\vartheta$ to zero. From the string theory
microscopic embedding, this cannot happen since the critical parameter
$\alpha$ depends on the number of D3 branes, which is quantized. Another way
to tune the graviton mass to zero is to take $\delta\phi\to+\infty$, as can be
seen from eq. (2.10), which corresponds to a decoupling of the bulk theory.
However, this limit is singular and can be seen as an infinite distance limit
in moduli space [30], showing that the decoupled theory is no more consistent
with the 10d uplift: reaching that point in the dilaton moduli space would
bring down an infinite tower of modes rendering the EFT invalid. In this
limit, the compactness of the geometry is restored and the graviton becomes
massless, so we can interpret it as the limit of the gravitating bath regime.
In that case, however, the system has a flat Page curve [28, 25], in line with
the findings of [37] that one cannot smoothly connect the non-compact
background with the compact one, that would correspond to the
$\delta\phi\to+\infty$ limit.

##  4 Island surfaces and running dilaton

In this section we study entanglement islands in the type IIB solution
introduced in section 2.2, namely we consider the aforementioned 10d
background with a dilaton that is allowed to vary between the bag
$AdS_{4}\times{\cal M}_{6}$ region and the $AdS_{5}\times S^{5}$ bath. We
investigate how this variation affects the island surfaces. To this aim, we
perform a numerical study of the surface PDE from which we extract the
critical value $\alpha_{\text{crit.}}$, which characterizes the island
surfaces behavior. We discuss our findings in relation to the possibility of
being in a regime with sufficiently small graviton mass, either in order to
trust the massive gravity description from perturbative string calculations
[30] or to stay within the valid EFT regime, as we discussed in Section 3.

Notice that the massless graviton limit corresponds to the limit of large
dilaton variation across the throat (at fixed geometric ratio $\alpha$), from
(2.10). We have already mentioned that this decoupling limit is singular, thus
we expect that the extremal islands will be affected by a large dilaton
variation, as we confirm with our numerical analysis, both at zero and at
finite temperature.

In our numerical analysis we do not solve for other extremal surfaces, like
Hartmann-Maldacena surfaces, which we expect to behave similarly to the zero
dilaton case, derived in [25].

###  4.1 Critical parameters and graviton mass

The massive gravity regime which is under control in string theory requires
the graviton mass to be $\bar{m}_{g}^{2}\ll 1$. It would be ideal to have a
numerical evidence that puts non-critical islands in the same regime, but the
previous results of [25] have found the critical value of the geometric
parameter141414We thank C. Uhlemann for pointing out a technical issue in the
numerical analysis in an earlier version of this paper.
$\alpha_{\text{crit.}}\sim 4$ . From (2.10) we see that this value is not
large enough to create a small throat between the AdS4 and AdS5 region, thus
we are in the regime $m_{g\,,\text{crit.}}\sim L_{4}^{-1}$, which is outside
of the validity regime of the EFT: parametrically small graviton masses are
achieved for values of the geometric ratio $\alpha$ which lie way above its
critical value (see e.g. figure 6). In this section, we will see how the
critical value changes once the dilaton is allowed to vary in a setup
analogous to the one of [25], and how one can enter the regime of small
graviton masses.

By introducing a varying dilaton, the graviton mass not only depends on
$\alpha$, but also on $\delta\phi$. If the properties of island surfaces
physically depended on $\alpha$ only, and not on $\delta\phi$, we would be
able to choose a background with islands below the critical value
$\alpha_{\text{crit.}}$, and then increase the dilaton variation such that the
graviton mass becomes only slightly nonzero. This would be in contrast with
expectations that the islands are strongly related to the presence of a
massive graviton, and in fact what we find is that it is the combination of
$\alpha$ and $\delta\phi$, thus the graviton mass, which effectively governs
the island behavior.

Figure 6: Zero temperature configuration: it can be observed that
parametrically small masses $(\bar{m}_{g}^{2}\ll 1)$ are achieved in the
regime of $\alpha\gg 5.0$. Only the black part of the curve can be computed
using the expression for the mass.

###  4.2 Numerical setup for island surfaces at $\delta\phi\neq 0$

We consider backgrounds with running dilaton introduced in Section 2.2. The
dilaton value in the bag region is determined by the number of five-branes,
while in the bath $AdS_{5}\times S^{5}$ region its value is a free parameter
determining the Yang-Mills coupling constant of the dual theory. The geometry
is specified by the harmonic functions (2.5), and the dilaton variation (2.6).

The island surfaces in this family of type IIB backgrounds are minimal
surfaces that extend from the bag region to the AdS5 bath. As in [25], the
minimal surfaces are eight dimensional surfaces embedded in the ten
dimensional geometry. The surfaces completely wrap the two $S^{2}$’s, and
partially wrap both the Riemann surface $\Sigma$ and the AdS4 black hole,
whose geometry is parametrized as

$\displaystyle\mathrm{d}s_{4}^{2}=\frac{\mathrm{d}r^{2}}{b(r)}+e^{2r}(-b(r)\mathrm{d}t^{2}+\mathrm{d}s^{2}_{\mathbb{R}^{2}})\,,$

with $b(r)=1-e^{3(r_{h}-r)}$, where $r_{h}$ is the horizon radius while
$b(r)=1$ being the zero temperature limit. We take $r_{h}=0$ throughout our
numerical analysis.

The extremal surfaces are thus realized as embedding surfaces in terms of the
AdS4 coordinate of the black hole geometry, $r(x,y)$ where $x\in\mathbb{R}$
and $y\in[0,\pi/2]$ are the real and imaginary part of the coordinates on the
Riemann surface $\Sigma$. The extremality condition for this surface can be
expressed in terms of the functions characterizing the harmonic functions
(2.5) pinpointing the type IIB solution described in Fig. 3:

$\displaystyle f=|h\hat{h}W|\,,\quad
g=\frac{1}{2b(r)}\left|\frac{h\hat{h}}{W}\right|\,,$

where $W=\partial_{z}\partial_{\bar{z}}(h\hat{h})$, which yield the
differential equation [25]

$\displaystyle\frac{1}{1+g(\nabla r)^{2}}\left[2-\nabla(g\nabla
r)+\frac{1}{2}g\nabla r\cdot\nabla\ln\left(\frac{1+g(\nabla
r)^{2}}{b(r)f^{2}}\right)\right]=0\,,$ (4.1)

here $\nabla$ is the covariant derivative on $\Sigma$. The minimal surface is
obtained by letting the extremality condition (4.1) stabilise after waiting
for a sufficiently long ‘relaxation time’ $\tau$

$\displaystyle\begin{aligned} \partial_{\tau}r(x,y,\tau)&=-\frac{1}{1+g(\nabla
r)^{2}}\left[2-\nabla(g\nabla r)+\frac{1}{2}g\nabla
r\cdot\nabla\ln\left(\frac{1+g(\nabla
r)^{2}}{b(r)f^{2}}\right)\right]\,,\end{aligned}$ (4.2)

where now $r$ is not only a function of $x,y$ but also of $\tau$ (though not
$f$ and $g$). The accuracy of the numerical solution at a fixed relaxation
time $\tau_{\mathrm{max}}$ can be measured by the residual

$\displaystyle R=\left|\frac{1}{1+g(\nabla r)^{2}}\left[2-\nabla(g\nabla
r)+\frac{1}{2}g\nabla r\cdot\nabla\ln\left(\frac{1+g(\nabla
r)^{2}}{b(r)f^{2}}\right)\right]\right|_{\tau=\tau_{\mathrm{max}}}\,.$ (4.3)

The problem is completely defined when the boundary conditions are specified.
The Janus deformation of the throat geometry does not affect the original
problem and boundary conditions considered in [25], which we recall here for
completeness. In the bag region, the surfaces are smooth in the limit
$x\rightarrow-\infty$ which corresponds to a regular point of the 10d
geometry. At the singular points, located on the two boundaries of the Riemann
surface (strip) the surfaces fall into the horizon. The surfaces continue in
the bath and where the right $AdS_{5}\times S^{5}$ asymptotic emerges, they
anchor on the bath at a specific positive value of the radial coordinate. This
translates into a Dirichlet boundary condition at $x\rightarrow+\infty$ where
the embedding function $r(x,y)$ has a constant value, the anchoring point
$r_{R}$:

$\lim_{x\to+\infty}r(x,y)=r_{R}\ .$ (4.4)

This boundary condition is not affected by the dilaton variation, as the
surface still anchors on $AdS_{5}\times S^{5}$, where there is no
$y$-dependence and most importantly the dilaton has a constant value151515The
$\delta\phi$ dependence concerns in fact the throat region and not the bag,
nor the radiation region..

Summarizing, the boundary condition for the minimal surface are captured by

$\displaystyle\partial_{y}r(x,y)\mid_{y=0,\frac{\pi}{2}}=0\,,\quad\lim_{x\rightarrow+\infty}r(x,y)=r_{R}\,,\quad\lim_{x\rightarrow+\infty}e^{+x}\partial_{x}r(x,y)=0\,.$
(4.5)

To access the boundary conditions in the numerical scheme detailed in the
following paragraph, the $x$ direction of the Riemann surface $\Sigma$ is
compactified by the transformation $\xi=\tanh x$.

The partial differential equation (4.1), for given functions $f$, $g$ and
$b(r)$, is then numerically solved by means of the method of finite
differences. The domain is discretized into a lattice, whilst the partial
differential equation is turned into a set of coupled algebraic equations. The
equations are then solved with the relaxation method in (4.2), which at this
point is simply a set of first order differential equations. The numerical
data are obtained using the finite difference scheme at second order as then
the Neumann boundary condition take on particularly simple form, which proves
sufficient to obtain numerical results within a more than acceptable accuracy.
Generically, we have found it sufficient to use a grid with
$\\{\xi,y\\}\sim\\{120,100\\}$ rows each, yielding an approximation of the
solution at relaxation times of the order $\tau=10^{3}$ with residuals in eq.
(4.3) of order $10^{-9}$ or lower.

###  4.3 Island surfaces at zero temperature

Figure 7: On the left: for increasing relaxation time for $\delta\phi=1.9$,
$\Delta r$ becomes arbitrarily large as $\alpha$ becomes larger than a
critical value. On the right: the critical geometric ratio
$\alpha_{{crit}}=\sqrt{\frac{N^{2}}{\kappa\hat{\kappa}}}|_{{crit}}$ is found
to increase monotonically with the dilaton variation $\delta\phi$ across the
geometry.

Even in the absence of a black hole in the bulk geometry, the vacuum
configuration yields valuable information, determining crucial features of the
behavior of islands, which are relevant also at non-zero temperature.

Zero temperature surfaces are characterized by the difference $\Delta r$
between the anchoring point in the radiation region, $r_{R}$, and the radial
coordinate in the bag region $r_{L}$. For a given $\alpha$ and $\delta\phi$
and at zero temperature, the parameter $K=\sqrt{\kappa\hat{\kappa}}$ drops out
of the extremization equation.

For $\delta\phi=0$, it was shown in [25] that islands cease to exist above a
critical value $\alpha_{\text{crit.}}$. This can be seen numerically, also for
$\delta\phi\neq 0$, in the divergence of $\Delta r$. The latter, in fact,
stops settling to a well-defined value. If we keep increasing the relaxation
time in the solution of the PDE, once we pass a critical value of $\alpha$,
the anchoring points $r_{R}$ and $r_{L}$ are pushed further apart, as shown in
Fig. 7, on the left. Actually, this critical value is drastically affected by
the non-vanishing dilaton variation. By tracking the onset of the divergence
for different values of $\delta\phi$, we find that the value of the critical
geometric ratio $\alpha_{\text{crit.}}$ is pushed to higher values for
increasing $\delta\phi$, as we show in Fig. 7, on the right.

This has an important consequence. First of all, for certain values of
$\delta\phi\neq 0$ one can find islands at zero temperature for values of
$\alpha$ which are big enough (e.g. $\alpha\sim 10$) to be in the regime of
validity of (2.10), namely of small graviton masses. However, even on these
backgrounds, one cannot adjust the graviton mass indefinitely by tuning the
dilaton variation to high values, as the islands are bound to disappear for
big enough values of $\delta\phi$.

We have numerically seen, in fact, that for values of $\delta\phi$ large
enough $\delta\phi\gtrsim 5$, there is no more numerical convergence for any
value of $\alpha$ and island surfaces cannot be found. For values
$2.5<\delta\phi<5$ divergences at small values of $\alpha$ appear, even though
surfaces exist for larger (but still bounded) $\alpha$, which are a signal
that our code looses predictability. We thus present the numerical results for
$\delta\phi\leq 3$, which captures the behaviour of $\alpha_{crit}$ but still
do not show significant singularities for small $\alpha$.

The signal that the surfaces may develop a critical behavior can also be
inferred by looking directly at the minimal surfaces, for ranging values of
the parameters $(\alpha,\delta\phi)$. For a fixed value of the dilaton
variation, the islands will exist for a larger range of $\alpha$, as it is
illustrated on the left of Fig. 8, displaying how the island surfaces vary
with increasing value of the geometric ratio
$\alpha=\sqrt{\frac{N^{2}}{\kappa\hat{\kappa}}}$. On the other hand,
increasing $\delta\phi$, in a background with fixed $\alpha$, the minimal
surfaces are seen to gradually settle onto the horizon (see Fig. 8 on the
right), corresponding to a larger and larger $\Delta r$.

Figure 8: An illustration of the evolution of the minimal surface at zero
temperature and non-zero dilaton variation. On the right, (a family of)
backgrounds is fixed for a geometric ration $\alpha=5.5$. In both cases, the
surface is indicated by a dot at the location of its values
$(\delta\phi,\alpha)$ in the graph. On the left, one can see how, for a fixed
dilaton variation $\delta\phi=1.7$, the surface settles into the horizon for
backgrounds that approach the critical geometric ratio. At zero temperature
the surfaces are shift-invariant in the radial direction, so $\Delta r$ is the
meaningful quantity.

Summarizing, we conclude that for these type IIB uplifts, at zero temperature,
in presence of a nonzero dilaton variation, quantum extremal islands can be
found in a regime where the lowest-lying graviton has a small mass, but the
mass cannot be tuned parametrically small by going to larger values of
$\delta\phi$.

###  4.4 Island surfaces at finite temperature

At finite temperature the horizon is expected to regulate the critical
effects, and islands can always exist for values of $\alpha$ above the
critical value $\alpha_{\text{crit.}}$. This means one can reach a regime of
small graviton mass (by appropriately tuning the dilaton variation and
$\alpha$ parameters), where island surfaces can still be found.

Figure 9: The difference $\Delta r=r_{R}-r_{L}$ as a function of the anchoring
point $r_{R}$ in the $AdS$ bath at finite temperature. On the left, each curve
is evaluated for values of the geometric ratio
$\alpha\in\\{0.1,0.5,1,2,2.5\\}$, with fixed $\delta\phi=0.9$. On the right,
one can how the dilaton variation $\delta\phi\in\\{0,0.6,1.2,1.8\\}$ affects
the difference $r_{R}-r_{L}$ as a function of the anchoring point $r_{R}$.

However, $\alpha_{\text{crit.}}$ still has a meaning at finite temperature,
related to the behavior of the islands surface [14]. More precisely, in the
case of type IIB uplifts, let us look at the anchoring points of the islands
surface at finite temperature. As before, $r_{L}$ denotes the point where the
surface reaches in the $\mathrm{AdS}_{4}\times\mathcal{M}_{6}$ bulk, and the
right radial coordinate $r_{R}$ denotes the location where it is anchored in
the $\mathrm{AdS}_{5}\times S^{5}$ bath. In Fig. 9, on the left, we see that
at finite temperature $r_{L}$ is constrained to remain anchored near the
horizon until the value of $\Delta r$ at zero temperature is reached, after
which the difference $r_{R}-r_{L}$ remains constant, with the same behavior
noticed for $\delta\phi=0$ [25]. However, the constant values at which the
surface settles is affected by the dilaton variation, as we show in Fig. 9, on
the right. A larger $\delta\phi$ pushes the value of $\Delta r$ to larger
constants, consistent with the effect at zero temperature161616The critical
value $\alpha_{\text{crit.}}$, where $\Delta r$ diverges, at zero temperature,
is pushed to lower values by a larger $\delta\phi$.. As a consequence, for
larger values of $\delta\phi$, $r_{L}$ remains longer anchored to the horizon.

The island surface stretched on the horizon will have a constant area, leading
to a flat Page curve. This behavior resembles the one in the case of a
gravitating bath [25]. Indeed, the large $\alpha$ regime corresponds to light
graviton masses. One can imagine that increasing $\alpha$ will drive the
background closer to the gravitating bath, where the graviton is massless.
Notice that the massless graviton is a direct consequence of the compactness
of the geometry, which can be obtained for any $\alpha$ by tuning
$\delta\phi\to+\infty$. We can see already that our analysis suggests that
these surfaces won’t be the adequate objects to exhibit Page-like behavior
when the bath is coupled to gravity.

Figure 10: The curve illustrates the cross-section at constant $y$ of the
minimal surface at a fixed value of the dilaton variation
$\delta\phi=\delta\phi_{2}$. The value of the dilaton variation determines a
minimal anchoring $r_{L}$ in the bulk. Increasing the dilation variation
lowers this critical anchoring point for which the surface can still reach the
bath.

##  5 Conclusions and discussion

Black hole evaporation represents an important test of our understanding of
gravitational interactions. Thanks to recent developments, it is now possible
to directly compute the entanglement entropy of Hawking radiation in various
setups. Our work places itself within the first endeavors of this analysis in
a string theory embedding. One of the main features of our analysis is to
consider a string theory framework where a quantitative treatment of the
graviton mass is possible, which up to now has mostly been qualitative.

In the first part of this work, we discussed the regime of validity of the
massive gravity EFT, in relation with the critical value of the graviton mass
relevant for the island surfaces. We have found a tension between the
existence of an arbitrarily light graviton and the presence of islands below
$\alpha_{\text{crit.}}$. This supports the idea that in higher dimensions it
is necessary to have large enough graviton mass to correctly reproduce the
Page curve [15].

In the second part of this work, we have investigated quantum extremal islands
(QEI) in a 4 dimensional model of a black hole coupled to an external non-
gravitating bath, uplifted to type IIB string theory, with non-constant
dilaton field. We have considered a geometry that develops a throat of
AdS${}_{5}\times$S5 geometry, where the mass of the lowest-lying graviton is
parametrically small and can be explicitly computed. The evaporating black
hole is then a system in a theory of massive gravity, that has a built-in
cutoff and we have put bounds on the microscopic parameters to ensure the EFT
does not break down in the region of existence of QEI surfaces, like described
in the first part.

We have been able to confirm the existence of a critical value of microscopic
parameters,
$\alpha_{\text{crit.}}=\sqrt{\frac{N^{2}}{\kappa\hat{\kappa}}}_{\text{crit.}}$,
generalizing the one in [25], above which the behavior of QEI surfaces changes
both at zero as well as finite temperature. By studying island surfaces in
backgrounds with varying dilaton, we have shown how the critical value
$\alpha_{\text{crit.}}$ is modified when increasing $\delta\phi$.

Our analysis has been able to determine that the existence of the islands does
not only depend on the microscopic parameters in $\alpha$, but both on
$(\alpha,\delta\phi)$, where $\delta\phi$ is the variation of the dilaton
across the geometry. In particular, these two parameters determine the mass of
the graviton, thus it is effectively the mass of the graviton that directly
affects the behavior of the island surfaces. Our analysis has shown that there
are backgrounds where the mass of the graviton can be quantitatively computed,
but cannot be tuned to arbitrarily small masses without spoiling the behavior
of the island surfaces. This supports the lesson about islands and massive
gravitons that comes from Karch-Randall configurations [13].

We studied the critical parameters affecting the behavior of the island
surfaces in a family of string solution with varying dilaton. In empty Anti de
Sitter, one can numerically see that the surface stops existing above a
critical value, although for a higher value of geometric ratio than the value
at $\delta\phi=0$ (see figure 7). For the black hole, our numerical study
shows that, both for $\alpha$ as well as $\delta\phi$ large enough, the
surfaces completely flatten on the horizon, possibly signalling a change in
behavior of the island surface. In Karch-Randall setups, analogously, a
critical angle determines the properties of the QEI surfaces. In that case,
the behavior of the extremal surfaces in an over-critical regime is
interpreted as if the surface is required to be anchored on the AdS4 brane. In
the type IIB construction of [25] for $\delta\phi=0$, and of this work for
$\delta\phi\neq 0$, we see that the anchoring point is stuck at the horizon,
unless $r_{R}$ reaches far enough in the $AdS_{5}$ region. It also emerges,
from the shape of the island surfaces at larger dilaton variation, that they
relax on the horizon in approaching the critical values of the geometric
parameter $\alpha$. This motivates further, future investigations to
understand better what physical properties are related to this change of
behavior, and to characterize the phase transitions the island surfaces
undergo. It could be also interesting to discuss the Page curve in presence of
a gravitating bath in backgrounds with varying dilaton.

### Acknowledgedments

We would like to thank Costas Bachas, Marine De Clerck, Ruben Monten and
Christoph Uhlemann for interesting discussions and comments. The work of D.L.
and I.L. is supported by the “Origins” Excellence Cluster. This work is
supported also by the German-Israeli Project (DIP) “Holography and the
Swampland”.

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††thanks: The usual disclaimer applies

# Evolutionary Dynamics of Sustainable Blockchains

Marco Alberto Javarone<EMAIL_ADDRESS>Centro Ricerche Enrico Fermi,
Rome, Italy  University College London - Centre for Blockchain Technologies,
London, UK  Gabriele Di Antonio  Centro Ricerche Enrico Fermi, Rome, Italy
Istituto Superiore di Sanità, Rome, Italy  Università degli Studi Roma Tre,
Rome, Italy  Gianni Valerio Vinci  Istituto Superiore di Sanità, Rome, Italy
Università Roma Tor Vergata, Rome, Italy  Luciano Pietronero  Centro Ricerche
Enrico Fermi, Rome, Italy  Carlo Gola  Banca d’Italia, Rome, Italy

###### Abstract

The energy sustainability of blockchains, whose consensus protocol rests on
the Proof-of-Work, nourishes a heated debate. The underlying issue lies in a
highly energy-consuming process, defined as mining, required to validate
crypto-asset transactions. Mining is the process of solving a cryptographic
puzzle, incentivised by the possibility of gaining a reward. The higher the
number of users performing mining, i.e. miners, the higher the overall
electricity consumption of a blockchain. For that reason, mining constitutes a
negative environmental externality. Here, we study whether miners’ interests
can meet the collective need to curb energy consumption. To this end, we
introduce the Crypto-Asset Game, namely a model based on the framework of
Evolutionary Game Theory devised for studying the dynamics of a population
whose agents can play as crypto-asset users or as miners. The energy
consumption of mining impacts the payoff of both strategies, representing a
direct cost for miners and an environmental factor for crypto-asset users. The
proposed model, studied via numerical simulations, shows that, in some
conditions, the agent population can reach a strategy profile that optimises
global energy consumption, i.e. composed of a low density of miners. To
conclude, can a Proof-of-Work-based blockchain become energetically
sustainable? Our results suggest that blockchain protocol parameters could
have a relevant role in the global energy consumption of this technology.

##  I Introduction

The Blockchain satoshi01 ; antonopoulos01  is a distributed ledger able to
manage transactions of digital tokens, constituting a form of digital money,
without relying on financial institutions serving as trusted third parties.
The underlying mechanisms of this technology exploit cryptographic protocols,
so we refer to digital tokens as crypto-assets. Although all implemented
protocols have a role in the functioning of a blockchain, the fundamental one
is the Nakamoto consensus protocol. The latter uses the Proof-of-Work (PoW),
i.e. a cryptographic proof that relies on solving a crypto-puzzle for
validating transactions and offering solid protection from attacks, such as
double-spending javarone01 . More in detail, particular users defined as
miners collect crypto-asset transactions not yet recorded in the Blockchain in
data structures called blocks. The miners’ goal is to attach their blocks to
the head of the Blockchain. To this end, miners have to show a PoW that can be
obtained by performing a process called mining. The first miner to complete
the PoW receives a reward, in the form of crypto-asset, which constitutes an
economic incentive. Mining is a competition based on finding a random number
(called nonce) whose value depends on the block at the head of the Blockchain
and transactions contained in the newly generated block. Also, mining
resembles a lottery, requiring miners to use substantial computational
resources to succeed. Then, getting high success probabilities equal high
energy consumption in a mining competition, as winning depends on the number
of used resources. The whole mechanism entails that each new block attached to
the chain forms a ’computational shield’ preserving transactions recorded in
the previously mined blocks. In summary, miners are fundamental for
blockchains based on the PoW. On the other hand, the energy consumption of PoW
blockchains constitutes a non-trivial issue gola01 . The latter directly
impacts the miners’ electricity bill and, indirectly, the whole society since
a fraction of energy gets wasted for validating transactions. Even if there
are other consensus mechanisms, for instance, the Proof-of-Stake saleh01 ;
thin01  and the Proof-of-Authority xiao01 ; zhang01 , the PoW is currently the
most used aggarwal01 . Therefore, energy consumption is at the core of a
heated debate around the societal impact of blockchains —see for instance
keller01 ; teufel01 ; vranken01 . The relevance of the debate motivates this
investigation. On one side, we observe miners that compete to get high profits
and allow Blockchain to work. On the other, we see a waste of electric energy.
Can the need to curb energy consumption meet the individual interest of
miners? Stimulated by this question, we study the above-described scenario as
a social dilemma. To this end, using the framework of Evolutionary Game Theory
nowak01 ; moreno01 ; perc01 ; perc02 ; perc03 ; szolnoki01 ; szolnoki02 ;
szolnoki03 ; szolnoki04 ; javarone02 ; javarone04  we propose a stochastic
evolutionary game named Crypto-Asset Game (CAG). The proposed game has two
strategies, i.e. mining and using crypto-assets. On varying the game
conditions, e.g. the miners’ reward, we analyse the evolution of the strategy
distribution occurring in an agent population playing CAG. In particular, the
evolutionary dynamics of CAG entail agents can change their strategy depending
on the accumulated payoff via a stochastic process, usually defined strategy
revision phase. Hence, a crypto-asset user (or just user hereinafter) can
become a miner and vice versa at no cost. Rational agents take decisions (i.e.
select a strategy) to maximise a profit. We aim to study whether a population
of rational agents can reach a strategy profile that minimises global energy
consumption. Finally, let us report that game-based approaches, devised for
studying various aspects of blockchains, have been previously proposed, as in
wu01 ; li01 ; liu01 ; kim01 ; bai01 ; teixeira01 . The remainder of the paper
is organised as follows: Section II introduces the Crypto-Asset Game. Section
III presents results of numerical simulations. Then, Section IV highlights the
main finding and, eventually, Section V discusses future developments.

##  II The Crypto-Asset Game

The Crypto-Asset Game represents the essential dynamics of a simplified
economic system composed of two kinds of agents, i.e. miners and crypto-asset
owners that produce and consume crypto-assets, respectively. More in detail,
miners produce crypto-assets through their mining activity, while crypto-asset
owners consume crypto-assets by executing transactions tessone01 . For the
sake of brevity, from now on, we refer to crypto-asset owners as users.
Accordingly, CAG is a two-strategy game whose agents can choose between mining
($m$) and using crypto-assets ($u$). Both strategies are associated with a
payoff. The miner payoff includes a reward and an energy cost, while the user
payoff includes a gain and a contribution they have to pay. The users’ gain
represents benefits related to using a blockchain, while their contribution
represents various costs associated with this technology, as later described.
In addition, the overall energy consumption of the blockchain affects the
users’ payoff as their gain gets multiplied by the miners’ density. Following
the above prescriptions, the payoff structure of CAG reads

$\begin{cases}\pi_{u}=\frac{1}{\rho_{m}}\frac{N_{u}}{5}-c_{u}\\\
\pi_{m}=\langle R\rangle-C\end{cases}$ (1)

whose parameters have the following meaning. Namely, $\rho_{m}$ is the density
of miners, $N_{u}$ is the number of users locally involved in a game iteration
(later detailed), $c_{u}$ is the contribution paid by users for participating
in the Blockchain system, $R$ is the miners’ reward and, eventually, $C$ is
the electricity cost of mining. The miners’ reward is taken as $\langle
R\rangle=\frac{R}{N_{m}}$, with $N_{m}$ total number of miners in the
population. This choice is motivated considering that an equal amount of
resources in a lottery-like competition (i.e. the mining competition) entails
that, on average, a miner gets a reward $R$ divided by the total amount of
miners. Even if the number of miners is related to the amount of energy wasted
in a lottery-like competition, PoW-based blockchains cannot work without them.
Hence, the presence of miners is mandatory for our economic system, which gets
investigated considering a list of assumptions. Firstly, we assume miners
cannot take any profit if there are no crypto-asset transactions (i.e. if
there are no users), although real blockchains allow mining also empty blocks.
That assumption, combined with the need to have at least one miner in the
population, leads to constraining agents’ payoff to zero when $\rho_{m}=0$ and
$\rho_{u}=0$ (i.e. density of users). Then, we assume miners have the same
computational resources, and users can switch to the mining strategy at no
cost. Lastly, $C$ and $c_{u}$ are considered constants albeit, in real
scenarios, costs and contributions could vary depending on various internal
and external factors. Let us now briefly clarify the meaning of the
contribution $c_{u}$. To this end, we may refer to the Public Goods Game
javarone06  (PGG), whose payoff structure is similar to that of CAG. In the
PGG, there are two strategies, namely cooperation and defection. Cooperators
make a contribution typically of unitary value, while defectors contribute
nothing. CAG does not include cooperators or defectors, yet the contribution
$c_{u}$ resembles the coin paid by cooperators in the PGG, although it might
have a fractional value (see also javarone05 ). In this context, $c_{u}$
represents the adoption costs of new technology and the consequences of using
it, e.g. waiting for the validation of transactions. Now, observing that the
ratio between costs and benefits of new technology affects the related
diffusion in a population hall01 , we highlight the relevance of $c_{u}$ for
studying blockchain dynamics. Finally, for the sake of simplicity, we assume
the electrical cost $C$ is denominated using the crypto asset of the game,
whose market value is constant. Once clarified the meaning of parameters and
values constituting the game payoff, we describe how the agent population
evolves. The evolution process is the same as that usually considered in other
evolutionary games, such as the PGG. In particular, a population arranged over
a networked structure at the beginning is equally divided between users and
miners. A networked structure entails that agents have a limited number of
neighbours and, among them, the number of users corresponds to $N_{u}$. After
each game iteration, agents undergo a strategy revision phase defined by an
updating rule. The latter, in this work, is realised via a Fermi-like
distribution javarone06  and defined as

$P(s^{x}\leftarrow s^{y})=\frac{1}{1-e^{-\frac{1}{K}(\pi_{y}-\pi_{x})}}$ (2)

so that the probability the $y$-th agent imposes its strategy, i.e. $s^{y}$,
on the $x$-th agent hinges on the difference in their payoff, namely $\pi_{y}$
and $\pi_{x}$, and a parameter $K$ representing the temperature (or noise) of
the system. In real scenarios, some human factors can be represented by the
parameter temperature (or noise). For instance, the temperature can represent
the degree of rationality in individuals, or the uncertainty in economic
agents, when they take decisions. More in detail, the higher the temperature,
the higher the probability a strategy revision occurs randomly. Likewise, the
lower the temperature, the higher the strategy revision results from a
rational choice. At high temperatures, i.e. with a lot of noise, the strategy
revision phase becomes a coin flip. The strategy revision phase occurs at each
game iteration once agents have played CAG in all groups of belonging. For
clarity, considering a population arranged over a bi-dimensional square
lattice with continuous boundary conditions, each agent belongs to $5$
different groups identified as follows. Then, considering the $x$-th agent,
its first group is identified considering that agent with its four direct
neighbours. The other four groups are defined considering the $x$-th agent
being part of a group whose central agent is one of its neighbours, so the
$x$-th agent occupies a side position. In doing so, agents play the game in
five different groups collecting the related payoff, whose total amount is
compared with that received by miners via equation 2.

Figure 1: Density of miners over time on varying the miners reward $R$ as
indicated on the top of each plot. Each simulation trial lasts $N\times 100$
time steps. Red line indicates the average density value, while a few single
trials are reported by black lines. From left to right, results obtained with
the following rewards: $R=1600$, $R=3150$, $R=48000$, $R=60000$

Finally, there are three possible outcomes, two representing full-order (i.e.
full of users and full of miners) and one representing co-existence of
strategies. Assuming a payoff equal to zero when $\rho_{m}=0$ or $\rho_{u}=0$
leads to oscillatory dynamics, e.g. in the case of no miners (i.e.
$\rho_{m}=0$), the strategy revision phase becomes a coin flip and, on
average, half of the population turns the strategy to mining at the next
iteration. Hence, the two full-order configurations cannot constitute
equilibria of the strategy distribution. Therefore, our analysis focuses on
the emergence of oscillatory behaviours and assessing the phases of strategy
co-existence resulting from game dynamics under different conditions.

##  III Results

In this section, we study the Crypto-Asset Game considering populations of
different size $N$, from $10^{2}$ to $10^{4}$ agents, distributed over a
regular square lattice with continuous boundary conditions. Results are
averaged over $100\times N$ attempts, and performed considering a user
contribution equal to $c_{u}=0.1$ and $K=0.5$ if not stated otherwise. The
first analysis relates the value of the reward $R$ with the density of miners.
Figure 1 presents outcomes for a population of size $100$ obtained at
different reward values $R$. Interestingly, we find that rewards smaller than
$R=3100$ in a population with $10^{4}$ agents, lead to steady-states
characterised by an oscillatory behaviour. More in detail, as shown in Figure
1, a low reward entails that almost half of the population changes
continuously strategy over time, albeit the average behaviour suggests a
configuration more stable with only few miners. Then, increasing the reward
$R$ up to $3150$ for the same population, after an initial phase characterised
by several oscillations like in the previous case, the agents reach a steady
state with a very low density of miners. Higher values of $R$ lead to higher
densities of miners till reaching a unitary miners density for $R=48000$. As
soon as $R>50000$ an oscillatory behaviour, conceptually similar to the one
observed at very low $R$ values, emerges. In the last case, almost the whole
population oscillates between a miners density equal to $1$ and about $0.6$.
Such analysis is then repeated on varying the size of the population and
allows us to identify threshold values in range of the reward $R$ for each
specific case. In figure 2 we show the threshold values of reward, we name
critical rewards, on varying the population size.

Figure 2: Scaling of the critical reward $\hat{R}$ on varying the population
size. The $L$ indicates the size of a (lattice) square side. Thus, for
instance, $L=50$ refers to a population with $L^{2}=2500$ agents.

According to the model, CAG has several parameters that can be investigated to
assess their effect on the agent population, with particular interest for the
density of miners. Starting with the temperature $K$, we consider a population
with $2500$ agents, to analyse the outcomes at different degrees of
rationality. Such analysis is performed with special attention to two ranges,
the first one for $K\in[0,1]$ and the second range for $K\in[1,50]$. Previous
investigations on dilemma games, as the Prisoners’ dilemma and the Public
Goods Game, reported that for the range of low temperatures the density of
cooperators increases as the temperature increases from $0$ to $0.5$ and then
the beneficial effect of the temperature on supporting cooperation reduces
javarone03 . Then, as reported in javarone06 , increasing the temperature the
dynamics of an evolutionary game resembles that of the voter model liggett01 .
Namely, too high temperatures entail agents change strategies by a process
equivalent to a coin flip. In this case, we cannot map cooperation to the use
of a token and defection to mining, because as described before, proof-of-
work-based blockchains need miners, while populations are perfectly healthy
without defectors. The result of this analysis is shown in figure 3.

Figure 3: Density of miners at the steady-state (i.e. after $100\times N$) on
varying the temperature $T$ in a population composed of $N=2500$ agents,
playing with a reward $R=900$. The standard error of the mean is very small on
all samples, remaining hidden inside the circles on the curve.

We find that for temperatures between $0$ and $1$ the behaviour is similar to
the one we observe in social dilemma games. Notably, increasing the
temperature in that range, the density of miners reaches a small peak with
around $T=0.75$. Then, as the temperature goes towards high values, the
density of miners increases till $0.5$, i.e. as expected, as soon as agents
reduce their rationality, the game resembles a voter model, and both
strategies survive in the population almost with the same density (i.e.
$0.5$). The agents’ payoff is a relevant component of a game. Then, as shown
in Figure 4, we study the average payoff of both strategies by varying the
reward in a population with $N=10^{4}$ agents. Results indicate that miners
can reach the highest payoff, on average, as the reward is slightly higher
than the critical value for that population size —see the second plot from the
left-hand side in Figure 4. Then, increasing the reward, the emerging
configurations lead both strategies to become poorly convenient. That
justifies the behaviour we observed analysing the density of miners on varying
the reward $R$. When only very few miners survive, miners and users receive
the highest possible payoff they can reach, and the population benefits from a
virtuous (in terms of energy consumption) steady state.

Figure 4: Average payoff $\langle\pi\rangle$ over time, obtained by users
(blue) and miners (purple line) on varying the reward $R$ in a population with
$N=10^{4}$ agents.

We conclude the section studying how the user contribution $c_{u}$ affects the
miners density —see Figure 5.

Figure 5: Miners density on varying the user contribution $c_{u}$ in a
population with $N=2500$ agents. Error bars report the standard error of the
mean for each sample.

As reported, a too low user contribution entails the density of miners is not
optimal as too many miners survive in the population. That result is neither
trivial nor expected, and it resembles the behaviour we observe for too small
rewards $R$, i.e. smaller than a critical value $\hat{R}$ (for each population
size). Finally, as the $c_{u}$ gets close to $1$, the density of miners
slightly increases. In summary, a user contribution ranging from $0.1$ to
$0.75$ supports optimised strategy distributions.

##  IV Discussion

Energy sustainability is a critical issue of Proof-of-Work-based blockchains
and feeds a heated debate. This class of blockchains requires computational
resources for verifying transactions, which might lead to high energy
consumption. In an economic system composed of rational agents, the energy
sustainability of a PoW blockchain can be treated as a social dilemma. In the
proposed model, miners and crypto-asset users interact by playing a game and
getting a payoff, depending on their strategy. The miners’ payoff includes a
reward whose market value kristoufek01 ; blau01 ; javarone07  can make mining
highly profitable. Thus, the resulting dilemma on one end of the scale has the
individual interest of miners and, on the other, the collective interest in
curbing electricity consumption. The global energy consumption of a blockchain
is linked to the nature of the mining competition. Namely, the higher the
number of resources used for mining, the higher the average reward miners get
over time. Increasing the computational power entails paying higher
electricity costs. Yet, as in lotteries, whose chances of success increase by
buying more tickets, chances of success in the mining competition increase by
using more computational resources. Sadly, albeit miners pay their electricity
bills, such a competition entails society wastes a fraction of energy. The
overall scenario, characterised by the above-described dilemma, can be studied
by the framework of Evolutionary Game Theory. Therefore, motivated by all
these observations, we propose a two-strategy game called the Crypto-Asset
Game. The complexity of an economic system leads us to make several
assumptions in the proposed model. For instance, miners and users can change
strategy at no cost, all miners have the same computational resources, the
price of electricity is constant, and the value of the crypto-asset in the
market does not change. In summary, CAG simplifies real scenarios.
Notwithstanding, we deem CAG captures the essential dilemmatic aspects
affecting rational individuals using blockchains. The results of our model,
obtained via numerical simulations, show a rich spectrum of possible steady
states that the agent population can reach. More specifically, depending on
the game parameters, the agent population can converge toward steady-states
characterised by a small number of miners or toward oscillatory behaviours
characterised by a fraction of agents continuously switching between the two
strategies. Such an oscillatory behaviour agrees with previous studies related
to energy aspects of PoW blockchains fiat01 . We recall that the parameters of
CAG represent the protocol parameters of PoW blockchains. Finding steady-state
configurations corresponding to optimal strategy distributions, i.e. with a
limited number of miners in the population, suggests that the protocol
parameters may have a relevant role in blockchain sustainability. Among such
parameters, the miners’ reward is probably the most important. More
specifically, results indicate that rewards too low and too high are
detrimental to blockchain energy sustainability, as the density of miners is
relatively high in both cases. Remarkably, we identify a range of values that
rewards can take to ensure the evolution of a population towards a virtuous
strategy distribution. Such a range of values depends on the population size.
The smaller the population size, the smaller the value a reward can take for
being beneficial (for the society itself). In real blockchains, the miners’
reward reduces over time antonopoulos01 , so the mining activity gets
preserved by other mechanisms, such as the transaction fees that users can add
(voluntarily) to their transactions. To conclude, we observe that the
definition of a range of beneficial rewards appears, in terms of
sustainability, in agreement with real-world blockchain dynamics. Notably, as
more users adopt and use a crypto-asset, the miners’ reward gets adapted
according to various factors, such as those mentioned before. That does not
entail rewards change to comply with environmental needs however, the miners’
reward and the community size in a real blockchain do not seem uncorrelated
parameters.

##  V Conclusions

In summary, we deem that the proposed model captures fundamental aspects of
PoW blockchains. Notwithstanding, several model features constitute relevant
elements for future research. Just to cite a few, elements deserving further
attention relate to the introduction of switching costs for representing the
efforts required for mining thum01 , the volatility of the crypto-asset value
in the market aalborg01 , and other mechanisms, such as the halving of the
miners’ reward. In addition, in light of the structure of transaction networks
kondor01 ; tessone01 ; tessone02 ; tessone03  we observe in various
cryptocurrencies, the interaction topology of the proposed model deserves
additional investigations. For instance, to study whether network structures
estrada01 , such as scale-free networks and small-world networks, can affect
the strategy profile of a population playing the Crypto-Asset Game, as
observed in social dilemmas santos01 ; santos02 ; cardillo01 ; amaral01 . Let
us conclude by remarking that the optimal steady states reached by agents in
some game conditions suggest that protocol designers may have a relevant role
in the energetic impact of PoW-based blockchains. In particular, the miners’
rewards could get designed to promote virtuous user behaviours, making, in the
long run, this technology sustainable and environmentally-friendly.

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