A holographic bound for D3brane
Eur. Phys. J. C
A holographic bound for D3brane
Davood Momeni 1 2
Mir Faizal 0 4
Aizhan Myrzakul 1 2
Sebastian Bahamonde 3
Ratbay Myrzakulov 1 2
0 Irving K. Barber School of Arts and Sciences, University of British ColumbiaOkanagan , 3333 University Way, Kelowna, BC V1V 1V7 , Canada
1 Department of General Theoretical Physics, Eurasian National University , Astana 010008 , Kazakhstan
2 Eurasian International Center for Theoretical Physics, Eurasian National University , Astana 010008 , Kazakhstan
3 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
4 Department of Physics and Astronomy, University of Lethbridge , Lethbridge, AB T1K 3M4 , Canada
In this paper, we will regularize the holographic entanglement entropy, holographic complexity and fidelity susceptibility for a configuration of D3branes. We will also study the regularization of the holographic complexity from the action for a configuration of D3branes. It will be demonstrated that for a spherical shell of D3branes the regularized holographic complexity is always greater than or equal to the regularized fidelity susceptibility. Furthermore, we will also demonstrate that the regularized holographic complexity is related to the regularized holographic entanglement entropy for this system. Thus, we will obtain a holographic bound involving regularized holographic complexity, regularized holographic entanglement entropy and regularized fidelity susceptibility of a configuration of D3brane. We will also discuss a bound for regularized holographic complexity from action, for a D3brane configuration. In this paper, we will analyze the relation between the holographic complexity, holographic entanglement entropy and fidelity susceptibility for a spherical shell of D3branes. We shall also analyze the holographic complexity from the action for a configuration of D3branes. These quantities will be geometrically calculated using the bulk geometry, and the results thus obtained will be used to demonstrate the existence of a holographic bound for configurations of D3branes. It may be noted that there is a close relation between the geometric configuration involving D3branes and quantum informational systems [1]. It is well known that D3branes can be analyzed as a real threequbit state [2]. This is done using the configurations of intersecting D3branes, wrapping around the six compact dimensions. The T 6 provides the microscopic stringtheoretic interpretation of the charges. The most general real threequbit state can be parameterized by four real numbers and an angle, and the most general STU black hole can be described by four

D3branes intersecting at an angle. Thus, it is possible to
represent a threequbit state by D3branes. A system of
D3branes has been used to holographically analyze the quantum
Hall effect, as a system of D3–D7branes has been used to
obtain the Hall conductivity and the topological
entanglement entropy for the quantum Hall effect [3]. The mutual
information between two spherical regions in N = 4
superYang–Mills theory dual to type IIB string theory on AdS5×S5
space has been analyzed using correlators of surface
operators [4]. Such a surface operator corresponds to having a
D3brane in AdS5 × S5 space ending on the boundary along
the prescribed surface. This construction relies on the strong
analogies between the twist field operators used for the
computation of the entanglement entropy, and the disorderlike
surface operators in gauge theories. A configuration of
D3branes and D7branes with a nontrivial worldvolume gauge
field on the D7branes has also been used to holographically
analyze a new form of quantum liquid, with certain
properties resembling a Fermi liquid [5] The holographic
entanglement entropy of an infinite strip subsystem on the asymptotic
AdS boundary has been used as a probe to study the
thermodynamic instabilities of planar Rcharged black holes and
their dual field theories [6]. This was done using spinning
D3branes with one nonvanishing angular momentum. It
was demonstrated that the holographic entanglement entropy
exhibits the thermodynamic instability associated with the
divergence of the specific heat. When the width of the strip
was large enough, the finite part of the holographic
entanglement entropy as a function of the temperature resembles the
thermal entropy. However, as the width become smaller, the
two entropies behave differently. It was also observed that
below a critical value for the width of the strip, the finite part
of the holographic entanglement entropy as a function of the
temperature develops a selfintersection.
Thus, there is a well established relation between
different D3brane configurations and information theoretical
processes. Thus, it would be interesting to analyze different
information theoretical quantities for a configuration of
D3branes. It may be noted that entropy is one of the most
important quantities in information theoretical processes. This is
because entropy measures the loss of information during a
process. It may be noted that the maximum entropy of a
region of space scales with its area, and this observation has
been motivated from the physics of black holes. This
observation has led to the development of the holographic principle
[7,8]. The holographic principle equates the degrees of
freedom in a region of space to the degrees of freedom on the
boundary surrounding that region of space. The AdS/CFT
correspondence is a concrete realizations of the holographic
principle [9–11], and it relates the string theory in AdS to a
superconformal field theory on the boundary of that AdS. The
AdS/CFT correspondence in turn can be used to
holographically obtain the entanglement entropy of a boundary field
theory. The holographic entanglement entropy of a
conformal field theory on the boundary of an AdS solution is dual to
the area of a minimal surface defined in the bulk. Thus, for a
subsystem as A, we can define γA as the (d −1)minimal
surface extended into the AdS bulk, with the boundary ∂ A. Now
using this subsystem, the holographic entanglement entropy
can be expressed as [12,13]
where G is the gravitational constant for the bulk AdS and
A(γA) is the area of the minimal surface. Even though this
quantity is divergent, it can be regularized [14,15]. The
holographic entanglement entropy can be regularized by
subtracting the contribution of the background AdS spacetime from
the deformation of the AdS spacetime. Thus, for the
system studied in this paper, let A[D3(γA)] be the contribution
of a D3brane shell and A[ Ad S(γA)] be contribution of the
background AdS spacetime, then the regularized holographic
entanglement entropy will be given by
SA =
In this paper, we will use this regularized holographic
entanglement entropy.
The entropy measures the loss of information during a
process. However, it is also important to know how easy it is
for an observer to extract this information. The complexity
quantified this idea relating it to the difficulty to extract
information. It is expected that complexity is another fundamental
physical quantify, as it is an important quantity in
information theory, and the laws of physics can be represented in
terms of informational theoretical processes. In fact,
complexity has been used in condensed matter systems [16,17]
where R and V are the radius of the curvature and the volume
in the AdS bulk.
As it is possible to define the volume in different ways in
the AdS, different proposals for the complexity have been
made. If this volume is defined to be the maximum
volume in AdS which ends on the time slice at the AdS
boundary, V = V ( max), then the complexity corresponds to the
fidelity susceptibility χF of the boundary conformal field
theory [24]. This quantity diverges [25]. However, we will
regularize it by subtracting the contribution of the background
AdS spacetime from the contribution of the deformation of
AdS spacetime. So, let V [D3( max)] be the contribution of
a D3brane shell and V [AdS( max)] be the contribution of
the background AdS spacetime, then we can write the
regularized fidelity susceptibility as
V [D3( max)] − V [AdS( max)] .
It is also possible to use a subsystem A (with its complement),
to define a volume in the AdS case as V = V (γA). This is
the volume which is enclosed by the minimal surface used to
calculate the holographic entanglement entropy [26]. Thus,
using V = V (γA), we obtain the holographic complexity as
CA. As we want to differentiate between these two cases, we
shall call this quantity, defined by V = V ( max), the fidelity
susceptibility, and the quantity defined by V = V (γA)
the holographic complexity. The holographic complexity
diverges [25]. We will regularize it by subtracting the
contributions of the background AdS from the deformation of
the AdS spacetime. Now if V [D3(γA)] is the contribution of
a D3brane shell and V [AdS(γA)] is the contribution of the
background AdS spacetime, then we can write the
regularized holographic complexity as
and molecular physics [18,19]. Complexity is also
important in black hole physics, as it has been proposed that even
though the information may not be ideally lost during the
evaporation of a black hole, it would be effectively lost
during the evaporation of a black hole. This is because it would
become impossible to reconstruct it from the Hawking
radiation [20]. It has been proposed that the complexity can be
obtained holographically as a quantity dual to the volume
of a codimension one time slice in the antide Sitter (AdS)
[21–23] case,
C A =
It may be noted that there is a different proposal for
calculating the holographic complexity of a system using the
action [27,28]. According to this proposal, the holographic
complexity of a system can be related to the bulk action
evaluated on the Wheeler–DeWitt patch,
A(W )
CW = π h¯ , (6)
where A(W ) is the action evaluated on the Wheeler–DeWitt
patch W , with a suitable boundary time. To differentiate it
from the holographic complexity calculated from the volume
C, we shall call this quantity the ”holographic complexity
from action”, and denote it by CW (as it has been calculated on
a Wheeler–DeWitt patch). This quantity also diverges [29].
We shall regularize it by subtracting the contributions of the
AdS spacetime from the contributions of the deformation of
the AdS spacetime. So, if A[D3(W )] is the contribution of
a D3brane shell and A[AdS(W )] is the contribution of the
background AdS spacetime, then we can write the
regularized holographic complexity from the action
CW = A[D3(W )] −πh¯A[AdS(W )] . (7)
It may be noted that this proposal is very different from the
other proposals to calculate the complexity of a boundary
theory. This difference occurs as there are differences in the
definition of complexity for a boundary field theory. So, this
proposal cannot be directly related to the proposals where the
complexity can be calculated from the volume of a
geometry. In fact, it is possible to have the same volume for two
theories with different field content. In this paper, we will
first use calculate a bound for the D3brane geometries using
the volume of a shell of D3branes. Then we shall calculate a
different holographic bound for a configuration of D3branes
using the action of this system.
In this paper, we will analyze a specific configuration
of D3branes and discuss the behavior of these regularized
information theoretical quantities for it. It is possible to use a
static gauge and write the bosonic part of the action for such
a system in AdS5 × S5 background as [35]
−h −
F ∧ F,
Here hμν = φ2ημν , h = det hμν , with ημν being the
four dimensional Minkowski metric. Thus, we can write
√−h = φ4, where φ2 = (φ I )2, and φ I are six scalar
fields corresponding to the six dimensions transverse to the
D3brane geometry. It may be noted that the F ∧ F term
only contributes to the magnetically charged configurations.
The D3brane can be placed at a fixed position on S5, such
that the five scalar fields corresponding to the S5 geometry
will not have any contribution. We shall consider the
spherically symmetrical static solutions, centered at r = 0, for this
geometry. So, the electric field E and the magnetic fields B
will only have radial components, which we shall denote by
E and B. So, all fields of this system are only functions of
the radial coordinate r , E (r ), B(r ), φ (r ). Thus, we can write
det(−Gμν ) = φ6Grr = φ6[φ2 + γ 2(φ /φ)2], and
Grr −
So, the Lagrangian density for this system can be written as
where γ = 2Nπ2 = R2√TD3, TD3 is D3brane tension.
There are two BPS solutions for this geometry, φ± = μ ±
Q/r. The probe D3brane solution discussed here describes
a BIon like spike (either up to the AdS5 boundary or down
to the Poincaré horizon, depending on the sign in φ±). This
solution also breaks the translational symmetry in the field
theory, and it preserves the rotational invariance.
It is also possible to analyze a probe D3brane with Q =
0, E = 0, and B = 0. Now we will analyze such a specific
solution representing a D3brane configuration, and analyze
these quantities for that specific geometric configuration. It
is possible to study such a D3brane shell. The metric for the
near horizon geometry of D3brane shell is given by [30]
where the function h(z) is defined as
h(z) =
For this geometry, the entangled region is a strip with width
in the D3brane shell defined by the embedding A = {x =
x (z), t = 0}. The area functional can be expressed as
where x (z∗) = ∞. The Euler–Lagrange equation for x (z)
has the following form:
x (z) h(z∗) z 3
.
x (z)2 + h(z)2 = h(z) z∗
The total length can be obtained by
= 2 0 z∗ dzh(z) ⎡⎣ 1 −hh((zz∗h)h)((zz∗)z)z∗ z3z 3 2 ⎦
∗
We can also write the volume V (γA) as
0
We can solve Eq. (15) exactly and obtain
z∗ h(z)3/2
√−zh1(0zh∗()zz∗3)z20+2z010 dz , z ≥ z0.
Since h(z∗) > 0, we can express this as
⎧ z0h(z∗) ,
V (γA) − V ( max) ≥ ⎨ 4
z ≤ z0,
⎩ h4(zz0∗3) (z∞4 − z04) , z ≥ z0.
So, for a D3brane, we obtain a relation between V (γA) and
V ( max), V (γA) − V ( max) ≥ 0. However, as the
holographic complexity and fidelity susceptibility for a system
are obtained using V (γA) and V ( max), we obtain the
following bound for a D3brane:
So, we have demonstrated that for a D3brane the holographic
complexity is always greater than or equal to the fidelity
susceptibility. This was expected, as the fidelity susceptibility
where C1 and C2 are integration constants. The maximal
volume, which is related to the fidelity susceptibility, is given
by
0
Now we will use h(z), and split the integral into two parts,
0z∞ = 0z0 + zz0∞ , to obtain
V ( max) =
It may be noted that by setting C1 = C2 = L, the difference
of the volumes (17) and (20) is given by
⎧
V (γA) − V ( max) = ⎨⎪
0z0 √
z ≤ z0,
zz0∞ √−zh1(0zh∗()zz∗3)z20+2z010 dz , z ≥ z0.
CA ≥
is calculated using the maximum volume in the bulk, and the
holographic complexity is only calculated for a subsystem.
It is also possible to demonstrate that a relation exists
between the holographic complexity and the entanglement
entropy of D3brane. To obtain this relation between the
holographic complexity and entanglement entropy of a D3brane,
we note that SA is given by
SA =
Here cT is the proportionality coefficient in the definition
of the Tent [31,32], and C1, C1 are integration constants. As
the only dependence of c on the geometry is from the AdS
radius R, the value of the coefficient c does not depend on the
specific deformation of the AdS geometry, and so it cannot
depend on the specific configuration of the D3branes. It may
be noted that this bound can also be used to understand the
meaning of the holographic complexity for a boundary
theory, as all the other quantities are defined for boundary theory,
and thus this relation can be used to understand the behavior
of the holographic complexity for the boundary theory.
because z ∼ 0 is near the AdS boundary limit. So, now as
z∗ < z0, we obtain
(C1 − C1).
4√h(z∗)z∗ .
l ≈ 5 (29)
By defining the effective holographic temperature Tent ∼
l−1, we obtain the relation between the holographic
complexity and the holographic entanglement entropy,
√−zz31z00+2z010 dz , z ≥ z0,
CA =
SA ≈
where c is given by
Thus, we have obtained a relation between the holographic
complexity and holographic entanglement entropy for a
D3brane. However, as the holographic complexity is also related
to the fidelity susceptibility, we obtain the following
holographic bound for a D3brane:
Tent R =
CA ≥
It may be noted that a bound on the holographic
entanglement entropy for a fixed effective holographic temperature
can be translated into a bound on the holographic
complexity, and this in turn can be related to a bound on the fidelity
susceptibility. So, we have obtained a relation between the
holographic complexity, holographic entanglement entropy
and fidelity susceptibility for a D3brane. The holographic
entanglement entropy is directly proportional to the
holographic complexity, when the effective holographic
temperature is fixed. Furthermore, the holographic complexity is
always greater than or equal to the fidelity susceptibility,
so the fidelity susceptibility can also be related to the
holographic entanglement entropy.
As it has recently been proposed that the holographic
entanglement entropy can be calculated from the action
evaluated at a Wheeler–DeWitt patch [27, 28], we shall now
calculate the holographic complexity from the action for this
D3brane configuration. It may be noted that it is expected
that the holographic complexity from the action will satisfy
the bound [33]. This bound has been tested for different AdS
black hole geometries [27, 28, 34], and we will test it for a
D3brane configuration. Now the holographic complexity from
the action for this D3brane configuration can be obtained by
evaluating the bulk action on the Wheeler–DeWitt patch. The
full type IIB action cannot be used for such a calculation as
no action is known for the selfdual five form, which exists in
the full theory. So, we will evaluate the probe D3brane action
on the Wheeler–DeWitt patch, and not use the full type IIB
action. In fact, this solution will depend on Q, which exists in
the probe solution, and not the domain wall solution. So, this
only represents the probe D3brane action on the Wheeler–
DeWitt patch. Now we will calculate the contributions of the
probe to the complexity from the action. As this quantity is
divergent, we will also subtract the background AdS
contribution from this quantity. Thus, the regularized holographic
complexity from the action, for this D3brane contribution,
can be written as
CW =
1 Q2z02(z∞2 − z02)
where is an IR cutoff and z∞ is the replacement for a
UV cutoff. It may be noted that unlike the holographic
complexity or fidelity susceptibility, this holographic complexity
from the action does not only depend on the geometry, but
on details of the field content of the theory. Thus, it cannot be
related to the holographic complexity, or fidelity
susceptibility, or even the holographic entanglement entropy in a direct
way. This is because these quantities are purely geometric
quantities. The main reason for this difference is that unlike
the entropy, there is an ambiguity in the definition of the
complexity, and thus many alternative proposals have been
made to define the complexity of a boundary theory. Thus,
we cannot relate the holographic complexity from the action
to those other purely geometric quantities. However, we can
calculate a different kind of bound for this holographic
complexity from the action. Thus, using the Poincaré coordinate
z, such that z ≡ rr0 , r0 = Qv , we obtain
It has been demonstrated that the mass of the BPS solution
for this geometry is M = 4π Q2/r0 [35]. So, we can write
M = 4π v Q, and r0 = Q/v, and we obtain
d dCtW ≈ 0.92π0h4¯0M ≤ 2πMh¯ , (35)
where the chemical potential v is defined through the
coupling constant MW = gv. Here we applied numerical
techniques to obtain this holographic bound. So, we have
demonstrated that, for a configuration of D3branes, the holographic
complexity from the action also satisfies an interesting
holographic bound.
In this paper, we analyzed certain holographic bounds for
D3brane configurations. We analyzed the regularization of
the information theoretical quantities dual to such a
configuration to obtain such bounds. It may be noted that there
are other interesting brane geometries in string theory. It
would be interesting to calculate the holographic complexity,
holographic entanglement entropy, and fidelity susceptibility
for such branes. It might be possible to analyze such
holographic bounds for other branes, and geometries that occur
in string theory. In fact, the argument used for obtaining the
relation between the holographic entanglement entropy and
holographic complexity of a D3brane can easily be
generalized to other geometries. Thus, it would be interesting to
analyze if this bound holds for other branes in string
theory. In fact, even in Mtheory, there exist M2branes and
M5branes, and such quantities can be calculated for such
branes. It may be noted that recently, the superconformal
field theory dual to M2branes has also been obtained, and
it is a bifundamental Chern–Simonsmatter theory called
the ABJM theory [36–38]. A holographic dual to the ABJM
theory with unquenched massive flavors has also been
studied [39]. It is also possible to massdeform the ABJM
theory [40], and the holographic entanglement entropy for the
massdeformed ABJM theory has been analyzed using the
AdS/CFT correspondence [41]. The holographic complexity
for this theory can be calculated using the same minimum
surface, and the fidelity susceptibility for this theory can be
calculated using the maximum volume which ends on the
time slice at the boundary. It would be interesting to analyze
if such a bound exists for the M2branes. It would also be
of interest to perform a similar analysis for the ABJ theory.
It may be noted that the fidelity susceptibility has been used
for analyzing the quantum phase transitions in condensed
matter systems [42–44]. So, it is possible to holographically
analyze the quantum phase transitions using this proposal. It
would also be interesting to analyze the consequences of this
bound on the quantum phase transition in condensed matter
systems.
Acknowledgements SB is supported by the Comisión Nacional
de Investigación Científica y Tecnológica (Becas Chile Grant No.
72150066).
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