# A holographic bound for D3-brane

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## Abstract

In this paper, we will regularize the holographic entanglement entropy, holographic complexity and fidelity susceptibility for a configuration of D3-branes. We will also study the regularization of the holographic complexity from the action for a configuration of D3-branes. It will be demonstrated that for a spherical shell of D3-branes 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 D3-brane. We will also discuss a bound for regularized holographic complexity from action, for a D3-brane configuration.

In this paper, we will analyze the relation between the holographic complexity, holographic entanglement entropy and fidelity susceptibility for a spherical shell of D3-branes. We shall also analyze the holographic complexity from the action for a configuration of D3-branes. 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 D3-branes. It may be noted that there is a close relation between the geometric configuration involving D3-branes and quantum informational systems [1]. It is well known that D3-branes can be analyzed as a real three-qubit state [2]. This is done using the configurations of intersecting D3-branes, wrapping around the six compact dimensions. The \(T^6\) provides the microscopic string-theoretic interpretation of the charges. The most general real three-qubit state can be parameterized by four real numbers and an angle, and the most general STU black hole can be described by four D3-branes intersecting at an angle. Thus, it is possible to represent a three-qubit state by D3-branes. A system of D3-branes has been used to holographically analyze the quantum Hall effect, as a system of D3–D7-branes 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 \(\mathcal {N}= 4\) super-Yang–Mills theory dual to type IIB string theory on AdS\(_5 \times S^5\) space has been analyzed using correlators of surface operators [4]. Such a surface operator corresponds to having a D3-brane in AdS\(_5 \times S^5\) 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 disorder-like surface operators in gauge theories. A configuration of D3-branes and D7-branes with a non-trivial worldvolume gauge field on the D7-branes 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 R-charged black holes and their dual field theories [6]. This was done using spinning D3-branes with one non-vanishing 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 self-intersection.

*A*, we can define \(\gamma _{A}\) as the \((d-1)\)-minimal surface extended into the AdS bulk, with the boundary \(\partial A\). Now using this subsystem, the holographic entanglement entropy can be expressed as [12, 13]

*G*is the gravitational constant for the bulk AdS and \(\mathcal {A}(\gamma _{A})\) 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 \( \mathcal {A} [\mathrm{D3} (\gamma _{A})]\) be the contribution of a D3-brane shell and \( \mathcal {A} [AdS (\gamma _{A})]\) be contribution of the background AdS spacetime, then the regularized holographic entanglement entropy will be given by

*R*and

*V*are the radius of the curvature and the volume in the AdS bulk.

*A*(with its complement), to define a volume in the AdS case as \(V = V(\gamma _A)\). This is the volume which is enclosed by the minimal surface used to calculate the holographic entanglement entropy [26]. Thus, using \(V = V(\gamma _A)\), we obtain the holographic complexity as \( \mathcal {C}_A\). As we want to differentiate between these two cases, we shall call this quantity, defined by \(V = V(\Sigma _\mathrm{max})\), the fidelity susceptibility, and the quantity defined by \(V = V(\gamma _A)\) 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 \( V [\mathrm{D3} (\gamma _A)]\) is the contribution of a D3-brane shell and \( V [\mathrm{AdS} (\gamma _A)]\) is the contribution of the background AdS spacetime, then we can write the regularized holographic complexity as

*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 \(\mathcal {C}\), we shall call this quantity the ”holographic complexity from action”, and denote it by \(\mathcal {C}_W\) (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 \( A [\mathrm{D3}(W)] \) is the contribution of a D3-brane shell and \(A [\mathrm{AdS}(W)] \) is the contribution of the background AdS spacetime, then we can write the regularized holographic complexity from the action

*E*and

*B*. So, all fields of this system are only functions of the radial coordinate

*r*,

*E*(

*r*),

*B*(

*r*), \(\phi (r)\). Thus, we can write \(\det (-G_{\mu \nu }) = \phi ^6 G_{rr}= \phi ^6[\phi ^2 + \gamma ^2 (\phi '/\phi )^2], \) and

*h*(

*z*) is defined as

*x*(

*z*) has the following form:

*h*(

*z*), and split the integral into two parts, \(\int _0^{z_{\infty }}=\int _0^{z_0}+\int _{z_0}^{z_{\infty }}\), to obtain

*c*is given by

*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 D3-branes. 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.

*Q*, which exists in the probe solution, and not the domain wall solution. So, this only represents the probe D3-brane 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 D3-brane contribution, can be written as

*z*, such that \(z\equiv \frac{r_0}{r},r_0 = \frac{Q}{v}\), we obtain

*v*is defined through the coupling constant \(M_W = gv\). Here we applied numerical techniques to obtain this holographic bound. So, we have demonstrated that, for a configuration of D3-branes, the holographic complexity from the action also satisfies an interesting holographic bound.

In this paper, we analyzed certain holographic bounds for D3-brane 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 D3-brane 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 M-theory, there exist M2-branes and M5-branes, and such quantities can be calculated for such branes. It may be noted that recently, the superconformal field theory dual to M2-branes has also been obtained, and it is a bi-fundamental Chern–Simons-matter theory called the ABJM theory [36, 37, 38]. A holographic dual to the ABJM theory with un-quenched massive flavors has also been studied [39]. It is also possible to mass-deform the ABJM theory [40], and the holographic entanglement entropy for the mass-deformed 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 M2-branes. 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, 43, 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.

## Notes

### Acknowledgements

SB is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066).

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