Yang-Baxter Deformed Wedge Holography

In this paper, we construct the wedge holography in the presence of Yang-Baxter deformation. By doing so, we propose the co-dimension two holography in the context of deformed SYK model. We observe that the DGP term play a crucial role in obtaining the non-zero tension of the Karch-Randall branes in Yang-Baxter deformed wedge holography. The Yang-Baxter deformation introduce non-trivial island surfaces inside the black hole horizon whose entanglement entropy is lower than the twice of thermal entropy of the black hole. Therefore, we obtain the Page curve even without the DGP term on the Karch-Randall branes due to the Yang-Baxter deformation in the context of wedge holography. Finally, we compute the correction to the holographic complexity in Yang-Baxter deformed JT gravity.


Introduction
Duality is a robust proposal in physics.In string theory, the famous duality is the gauge-gravity duality.Gauge-gravity duality is helpful for exploring the strongly coupled gauge theories from the weakly coupled gravitational theories.J. Maldacena proposed the duality between type IIB string theory defined on AdS 5 × S 5 and N = 4 super Yang-Mills theory [1].The gauge-gravity duality has been used in many branches of physics, e.g., condensed matter physics, cosmology, quantum chromodynamics (QCD), black hole information paradox, etc.The current paper is focused on the application of an extended version of this duality (wedge holography) to the black hole information paradox.
A very long time ago, Hawking found that black holes do not follow the unitary evolution of quantum mechanics [2,3] which give rise to the information paradox.To recover the unitary evolution of black holes, entanglement entropy of Hawking radiation must follow the Page curve [4].There has been extensive study on the information paradox using holography, and many proposals have been made (e.g.island proposal, double holography and wedge holography).The island proposal is based on the inclusion of contribution to the entanglement entropy from the interior of black holes at late times [5].In early times, islands do not contribute to the entanglement entropy of the Hawking radiation, and one obtains the divergent entanglement entropy.These two phases together (no-island and island) reproduce the Page curve 1 .The doubly holographic setup is constructed from the two copies of the usual AdS/CFT duality.In doubly holographic setups, the black hole is living on the end-of-the-world brane, and the external CFT bath acts as the sink to collect the Hawking radiation, see [15] and references therein.Another proposal is based on the wedge holography where the bath is taken as gravitating [16][17][18].In doubly holographic setups and wedge holography, one obtains the Page curve from the entanglement entropies of the extremal surfaces: Hartman-Maldacena [19] and island surfaces.The wedge holography states the duality between classical gravity in the (d+1)-dimensional bulk and the (d−1)-dimensional CFT at the defect, i.e., wedge holography can be called co-dimension two holography.
In doubly holographic setup and wedge holography, some authors found that gravity is massive on the end-of-the-world brane [20][21][22][23], and some authors showed that one could get the Page curve for the massless gravity in these models [15,24,25].The authors in [24,[26][27][28] showed that the Page curve can be obtained in wedge holography by including the Dvali-Gabadadze-Porrati (DGP) term [29] on the Karch-Randall branes.However theories that includes DGP term on Karch-Randall branes belongs to the swampland [30] and these are not physical [31,32].The results of [15] have been obtained from a top-down approach for a non-conformal bath (thermal QCD bath) whose gravity dual is M-theory inclusive of O(R 4 ) corrections [33] and do not include the DGP term on the end-of-the-world brane.Recently one of the co-authors (GY) has figured out how to describe the Multiverse from the wedge holography [34] and discussed the application of [34] to the information paradox of black holes with multiple horizons and the grandfather paradox.
The purpose of the present paper is to explore the effects of the Yang-Baxter deformation in the context of information paradox and holographic complexity.The importance of the YB deformations stems from the fact that they preserve the integrability of the sigma model [35][36][37][38][39] 2 .Recently, the authors in [44][45][46] explore the effects of the novel YB deformations in the 2D-dilaton gravity system having a quadratic potential known as Almheiri-Polchinski (AP) model [47].Interestingly, the authors found that the YB deformed AdS 2 background [44] could be a consistent solution of AP model if one deformed the quadratic potential into the hyperbolic function.The YB deformed AdS 2 metric plays a crucial role in the construction of YB deformed wedge holography.More concretely, in this paper • We propose the co-dimension two holography for the deformed SYK model where the gravity dual is Yang-Baxter deformed AdS 3 .
• We explore the effect of the Yang-Baxter deformation on the Page curve when the gravity dual is three-dimensional.
• Further, we investigate the effect of the Yang-Baxter deformation on holographic complexity.
The paper is organized as follows.Section 2 has been divided into three subsections 2.1, 2.2, and 2.3.Subsections 2.1 and 2.2 have discussions on the vacuum and black hole solutions for threedimensional bulk, whereas subsection 2.3 is based on the explicit construction of Yang-Baxter deformed wedge holography.Section 3 compromises of two subsections: 3.1 and 3.2.In 3.1, we have shown that it is possible to get the Page curve in the presence of Yang-Baxter deformation even without DGP term.In subsection 3.2, we have discussed the DGP term and swampland criteria in the Yang-Baxter deformed wedge holography.We discuss the effect of Yang-Baxter deformation on holographic complexity in section 4 and then summarise our results in section 5.
2 Wedge Holography in Yang-Baxter Deformed AdS Background In this section, we construct Yang-Baxter (YB) deformed vacuum and black hole solutions in subsections 2.1 and 2.2 respectively.We further construct Yang-Baxter deformed wedge holography in subsection 2.3.
Working action for the wedge holography with AdS 3 bulk is written as follows: where the first term of the above equation ( 1) is Einstein Hilbert term with negative cosmological constant [48], the second term is the Gibbons-Hawking-York boundary term, and the third term is defined on the Karch-Randall branes with γ = 1, 2. To discuss the wedge holography in YBdeformed background, we have considered two Karch-Randall branes with induced metric h γ and tensions T γ .On varying the action with respect to space-time (g µν ), we obtain the Einstein equation:

Vacuum Solution
In this subsection, we discuss vacuum solution without the DGP term in 2.1.1 and with the DGP term in 2.1.2,respectively.

Vacuum Solution without DGP Term
Vacuum solution of the bulk Einstein's equation ( 2) of the action (1) is given by [48] 3 where where η is the Yang-Baxter deformation parameter.Notice that the above solution satisfies the equations of motion provided the following constraint holds: and we stay infinitesimally away from the boundary i.e. z B = ηα + , where ∼ O(α).
Next we compute the Neumann boundary condition (NBC) which can be obtained by varying (1) with respect to the metric h ij where 4 and θ = θ 1,2 denote the locations of the two Karch-Randall (KR) branes, see Fig. 1.For general metric, NBC is satisfied provided, . For the metric (3), K γ ij = 0, and hence K γ = 0, therefore (6) implying Notice that, h γ ij = 0, and hence T γ = 0, i.e., KR-branes are tensionless without DGP term.Moreover, the situation remains unaltered even when we consider the black hole solution ( 16) instead of the vacuum solution (3).

Vacuum Solution in the presence of DGP Term
In the presence of the DGP term on the KR-branes, gravitational action ( 1) is modified as [24]: where all the terms in ( 10) are same as defined in (1) except R hγ , which are intrinsic curvature scalars on two Karch-Randall branes.In this case, bulk metric satisfies the following Neumann boundary condition at θ = θ 1,2 : The bulk metric (3) satisfies Neumann boundary condition (11) provided tensions of the KR-branes in the presence of DGP term should be given as follows: For the bulk metric (3), equation ( 12) simplifies to the following form: 4 Where θ is the location of the KR brane and n θ is the unit normal to the KR brane.The unit normal is define through the following relation Using the above relation (7), we obtain where ± denotes the increasing and decreasing direction of the θ.
For the given metric (3), the Ricci tensor and Ricci scalar on the KR-branes are obtained as: Now we define κ = ηα, and using ( 14), tensions of the branes ( 13) turn out to be of the following form: Notice that when λ 1 = λ 2 then both KR-branes have same tensions.From ( 15), we conclude that, the presence of the DGP term generates the tensions on Karch-Randall branes in the Yang-Baxter deformed wedge holography5 .

Black Hole Solution
In this subsection, we compute the consistent Yang-Baxter deformed black hole solution associated with (1).On solving (2), we obtain where Here, z h denotes the location of the black hole horizon.It should be noted that the above black hole solution (16) satisfies the equations of motion (2) up to the following constraint: and the black hole horizon must be located far away from the αη, i.e., z h >> αη.
The black hole solution defined in ( 16) has the following thermal entropy: Alternatively, one can compute the thermal entropy of 3D black holes using the Wald method [49,50].Wald entropy or the Noether charge entropy for stationary black holes is defined as: where ab is the anti-symmetric tensor with the normalisation condition ab ab = −2 and the integral is evaluated on the D−2 dimensional bifurcation surface Σ.For our case, the bifurcation surface is at z = z h and t = t 0 , where z h is the location of horizon and t 0 is a constant.Furthermore, the derivative in ( 20) is evaluated using the on-shell condition.
Using ( 4), ( 18) and ( 19), the thermal entropy of the black hole phase can be estimated as where we set z h = 1 for simplicity.

Wedge Holography in the Presence of Yang-Baxter deformation
Wedge holography in the presence of Yang-Baxter deformation can be constructed as follows.
We have two Karch-Randall branes located at θ = θ 1,2 , and these branes are joined at the one-dimensional defect (P ) as shown in Fig. 1.In this setup, the non-conformal boundary 6 of the 3D bulk is located at the holographic screen placed at z = z B7 .The YB deformed wedge holography has the following three descriptions.
• Boundary description: Two-dimensional non-conformal field theory (NCFT) at the holographic screen (z = z B ) with one-dimensional boundary.We call the NCFT located at the holographic screen as "holographic screen non-conformal field theory (HSNCFT)".

𝑧 = 𝑧 𝐵
Karch-Randall Branes • Intermediate description: Gravitating systems with Yang-Baxter deformed AdS 2 geometries (KR-branes) are connected to each other via transparent boundary conditions at the defect8 .
• Bulk description: HSNCFT located at the holographic screen (z = z B ) has threedimensional gravity dual.
In our setup, the dictionary of wedge holography in the presence of Yang-Baxter deformation can be schematically expressed as: Classical gravity in three-dimensional Yang-Baxter deformed AdS 3 bulk ≡ (Quantum) gravity on two Karch-Randall branes with metric Yang-Baxter deformed AdS 2 ≡ deformed SYK model living at the one-dimensional defect.

Information Paradox in Yang-Baxter Deformed Wedge Holography
In this section, we use the wedge holography constructed in section 2 to discuss the Page curve of black holes in the Yang-Baxter deformed wedge holography in 3.1.and in 3.2 we discuss the DGP term and swampland criteria in YB deformed wedge holography.

Page Curve Due to the Presence of Yang-Baxter Deformation
In this subsection, we explore the Page curve of the eternal black hole in the absence of the DGP term on the Karch-Randall branes.For this purpose, we consider the two extremal surfaces: Hartman-Maldacena, and island surfaces, and calculate their respective entanglement entropies in 3.1.1and 3.1.2.

Hartman-Maldacena Surface
Let us write the bulk metric (16) in terms of the infalling Eddington-Finkelstein coordinate, dv = dt − dz f (z) as below: Hartman-Maldacena surface is parametrized by θ ≡ θ(z) and v ≡ v(z) which has the induced metric as given below, and the same can be obtained from (23): where θ (z) = dθ dz and v (z) = dv dz .Using (24), the area of the Hartman-Maldacena surface can be computed as: where z 1 and z max are the point on gravitating bath and turning point of the Hartman-Maldacena surface.From (25), we obtain the equation of motion for θ(z) as where C 1 is a constant.Solving the above equation for θ (z), we obtained: Substituting θ (z) from ( 27) into (25), we obtained: The equation of motion for the embedding v(z) from ( 28) is calculated as Using ( 28) and ( 29), we obtained: Using the condition v (z max ) → −∞ [65], equations ( 4) and ( 30), we obtained: where we retain the terms upto O(κ).
At the turning point, dE dzmax = 0 which implies To make sure that the turning point of the Hartman-Maldacena surface is outside the horizon and positive, we need to consider Time on the gravitating bath can be obtained after simplifying dv = dt − dz f (z) and the same is written below: Now we explore the late-time behavior (t → ∞) of the area of the Hartman-Maldacena surface as follows: Since, Therefore, Hence, the Hartman-Maldancena surface has the following entanglement entropy in Yang-Baxter deformed model:

Non-Trivial Island Surface in the Presence of Yang-Baxter Deformation
Island surface is parametrized by t = constant, z = z(θ), and hence induced metric of the same can be obtained from ( 16) as written below: The area of the island surface using the induced metric ( 37) is obtained as follows: Using ( 4), ( 17)(we used z h = 1 for the simplicity of the calculations) and ( 38), we obtained: The Euler-Lagrange equation of motion(EOM) for the island surface's embedding z(θ) upto O(κ 2 ) can be schematically expressed as where we defined EOM κ 0,1,2 as below: • EOM κ 0 : • EOM κ 1 : • EOM κ 2 : EOMs ( 41), (42), and ( 43) have a common physical solution which is given below.
As discussed earlier, z h = 1 is the black hole horizon.Therefore solution to the island surface's embedding z(θ) up to O(κ 2 ) is given by: We can see very easily that (45) satisfies the NBC on the branes, i.  45) implying z(θ) < 1, i.e., z YB deformed < z h .Since z(θ) is constant, therefore substituting z(θ) from ( 45) into (38), we get the minimal area of the island surface as: According to Ryu-Takayanagi's prescription the entanglement entropy of the island surfaces is obtained as [66]: The factor "2" in the above equation is due to the second island surface contribution from the thermofield double partner.Therefore, we get a non-trivial island surface inside the horizon due to the presence of Yang-Baxter deformation whose entanglement entropy is lower than twice of thermal entropy of the black hole.
κ affects the emergence of islands and information recovery of the Hawking radiation 9 .

DGP Term and Swampland Criteria in the Yang-Baxter Deformed Wedge Holography
In this subsection, we will discuss the effect of including the DGP term on the Karch-Randall branes.As discussed in [24], the Hartman-Maldacena surface entanglement entropy remains same.In the presence of the DGP term, the island surface's entanglement entropy receives an extra contribution from the boundary of the island surface [24].In AdS d+1 /CF T d correspondence, the same can be expressed as follows: where Γ being the RT surface with induce metric γ and boundary of Γ (∂Γ) has the induced metric σ.
Recently it has been pointed out that including DGP term on the Karch-Randall brane is non-physical [31,32].DGP term can leads to negative entanglement entropy because of negative effective coupling constant of one brane.To have a positive entanglement entropy in the Yang-Baxter deformed wedge holography we need to consider θ 2 > θ 1 .Another swampland criteria mentioned by authors in [31] is that for any extremal surface if S ext < S thermal then those theories will belong to the swampland.
We obtain the non-trivial island surface in Yang-Baxter deformed wedge holography due to the Yang-Baxter deformation.We can have positive entanglement entropy for the island surface provided θ 2 > θ 1 and hence we can avoid one of the swampland criteria given in [31].But in our setup S island < S thermal therefore the second criteria of swampland is unavoidable.Based on the results of [31] we can say that Yang-Baxter deformed wedge holography is also the part of swampland.

Holographic Complexity in AdS η
2 Background In this section, we compute the holographic complexity of Yang-Baxter deformed AdS 2 using the complexity equals volume proposal [72].This proposal states that the complexity of holographic dual is given by the volume of co-dimension one surface in the bulk.For our case, we 9 See [13,71] where the higher derivative terms affect the Page curves.
have AdS η 2 /deformed SYK duality, and hence we can write the expression for the holographic complexity as: where V, G N , and l are the volume of a one-dimensional surface in two-dimensional bulk, Newton constant in two dimensions, and length scale associated with Yang-Baxter deformed AdS 2 background respectively.
In order to proceed, we parametrize the volume slice by t ≡ t(z).The induced metric associated with co-dimension one surface in YB deformed AdS 2 has the following form: Using (50), the volume of co-dimension one surface is obtained as: Since there is no t(z) dependence in (51) therefore conjugate momentum of t(z) is constant (say C), and is given as below: On solving the above equation for t (z), we obtained: At the turning point dz/dt| z=zt = 0, and using f (z) = 1 − z 2 , F η (z) from (4), turning point from ( 53) is obtained as: Now substituting t (z) from ( 53) into the action (51), on-shell volume is obtained as: Using ( 49) and ( 59), we obtained the complexity growth in the presence of Yang-Baxter deformation as given below: The above equation implies that where a 1 and a 2 are constants.
As a consistency check of our results, we see that at the leading order, i.e., O(κ 0 ), our results reduce to the ones obtained in [73].In particular, leading order terms of ( 54), ( 57) and ( 55) match with [73] for z h = 1.In [73], the boundary of AdS 2 is located at z = 0 whereas in our case boundary of Yang-Baxter deformed AdS 2 is located at z = z B .The boundary, z = 0, of AdS 2 leads to divergent complexity growth because a 1 and a 2 are divergent when z = 0.However, the holographic screen located at z = z B makes the right-hand side of (61) finite, and we obtain the finite constant contribution to the complexity growth.The effect of Yang-Baxter deformation on the complexity growth will depend upon the sign of a 2 .When a 2 > 0, then this will increase the rate of complexity growth, and when a 2 < 0, then this will decrease the complexity growth.One can obtain a 2 from the appropriate choice of the constant "C" and the location of holographic screen "z B ".

Conclusion and Discussion
In this paper, we have constructed the wedge holography in Yang-Baxter deformed AdS 3 background.The same has been done by considering the two Karch-Randall branes located at θ = θ 1,2 and the Yang-Baxter deformed bulk AdS 3 metric that satisfies Neumann boundary condition on the branes.In our case, the defect that connects the two Karch-Randall branes can be identified as a "deformed SYK model".In this way, we proposed the co-dimension two holography for the "deformed SYK model".In other words, this is the, "duality between a Yang-Baxter deformed AdS 3 bulk and deformed SYK model living at the defect" 10 .
Let us summarise the key results of the paper.
• In this setup, the non-conformal boundary of the Yang-Baxter deformed AdS 3 bulk is located at the holographic screen placed at z = z B .Hence, the defect is also located at the holographic screen in this setup as shown in Fig. 2. In Yang-Baxter deformed wedge holography, two-dimensional field theory is dual to three-dimensional bulk located at the holographic screen which we termed as "Holographic Screen Non-Conformal Field Theory (HSNCFT)".Since we have Yang-Baxter AdS 2 on the KR-branes and hence the deformed SYK model is living at defect situated at the holographic screen because of "AdS η 2 /deformed SYK" duality.Overall, deformed SYK model living at the defect is dual to three-dimensional bulk.
• According to the wedge holography, the only possible extremal surface is the black hole horizon [22].The authors in [24,26,27] obtained non-trivial island surface with entanglement entropy lower than the thermal entropy of the black hole by including the DGP term on the KR-branes.This possibility has been ruled out by the swampland criteria discussed in [31,32].
• We used the Yang-Baxter deformed wedge holography to obtain the Page curve of the black hole and obtained the usual Page curve without DGP term in the presence of Yang-Baxter deformation in contrast to the flat Page curve as obtained in [22,24,26].Yang-Baxter deformation is the reason for the existence of non-trivial island surface inside the black hole horizon.
Let us compare our results with the literature.In [22], authors argued that we could not get the Page curve in usual wedge holography because the only possible extremal surface is the black hole horizon.In this paper, we showed that there exists a non-trivial island surface inside the horizon in our case if we include the Yang-Baxter deformation in our setup.If we switch off the Yang-Baxter deformation, then the extremal surface is the black hole horizon similar to [22].
We also discussed the effect of Yang-Baxter deformation on the complexity growth and computed the correction to the same using the complexity equals volume proposal [72].We found that Yang-Baxter deformation will enhance or decrease the complexity growth controlled by the parameter a 2 .Without Yang-Baxter deformation, our results reduce to ones obtained in [73].

Figure 1 :
Figure 1: Pictorial realization of Yang-Baxter deformed wedge holography.z B is the holographic screen and P is the defect.

Figure 2 :
Figure 2: Description of the Yang-Baxter deformed wedge holography.The presence of Yang-Baxter deformation makes it possible to have a non-trivial island surface (solid brown curve) inside the horizon.The green dotted line in the figure corresponds to the black hole horizon.In this figure, we have shown just one part of the wedge holography; the Hartman-Maldacena surface will join the defect located on the other side of wedge holography.