An Exact Construction of Codimension two Holography

Recently, a codimension two holography called wedge holography is proposed as a generalization of AdS/CFT. It is conjectured that a gravitational theory in $d+1$ dimensional wedge spacetime is dual to a $d-1$ dimensional CFT on the corner of the wedge. In this paper, we give an exact construction of the wedge holography. Interestingly, we find an exact map from the gravitational solution to vacuum Einstein equations in AdS/CFT to the one in wedge holography. By applying this map, we prove the equivalence between wedge holography and AdS/CFT for vacuum Einstein gravity, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. The equivalence to AdS/CFT can be regarded as a"proof"of wedge holography in a certain sense. As an application of this powerful equivalence, we derive easily the holographic Weyl anomaly, holographic Entanglement/R\'enyi entropy and correlation functions for wedge holography. Besides, we discuss the general solutions of wedge holography and argue that they correspond to the AdS/CFT with suitable matter fields. Interestingly, we notice that the intrinsic Ricci scalar on the brane is always a constant, which depends on the tension. Finally, we generalize the discussions to dS/CFT and flat space holography. Remarkably, we find that AdS/CFT, dS/CFT and flat space holography can be unified in the framework of codimension two holography in asymptotically AdS. Different dualities are distinguished by different types of spacetimes on the brane.


Introduction
The holographic principle [1,2] reveals a deep connection between a higher dimensional gravity theory and a lower dimensional quantum field theory (QFT). As an exact realization of holographic principle, the AdS/CFT correspondence [3,4,5] conjectures that the quantum gravity theory in an asymptotically anti-de Sitter space (AdS) is dual to the conformal field theory (CFT) on the boundary. It provides a powerful method to study the non-perturbative phenomena in gauge theories, and has a wide applications in QFT [6,7,8], quantum information [9] and condensed matter physics [10].
There are many interesting generalizations of AdS/CFT, such as dS/CFT [11,12,13,14,15], Kerr/CFT [16,17], flat space holography [18,19], brane world holography [20,21,22], surface/state correspondence [23,24] and AdS/BCFT [25,26,27,28,29,30]. Recently, a codimension two holography, called wedge holography, is proposed by [31] between the gravitational theory in a d + 1 dimensional wedge spacetime and the d − 1 dimensional CFT on the corner of the wedge. See also [32] for a similar proposal of codimension two holography. The geometry of wedge holography is showed in figure 1 (left). It is conjectured that the following dualities hold Classical gravity on wedge W d+1 (Quantum) gravity on two AdS d (Q 1 ∪ Q 2 ) CFT d−1 on Σ where the first equivalence is due to the brane world holography [20,21,22] and the second equivalence originates from AdS/CFT. It is closely related to the so-called doubly holographic model, which is recently developed for the resolution of information paradox, in particular, for recovering Page curve of Hawking radiation [33,34,35]. The main difference from the wedge holography is that the double holography focus on a larger region of AdS instead of only the wedge. See also [36,37,38,39,40,41,42,43,44,45,46] for related topics. The gravitational action of wedge holography is given by [31] where N denotes d+1 dimensional wedge space, Q 1 and Q 2 denote two d dimensional branes. [31] propose to impose Neumann boundary condition (NBC) on the two end-of-world branes where K ij are the extrinsic curvatures, h ij are the induced metrics on the brane and T is the brane tension. Notice that the wedge holography can be regarded as a limit of AdS/BCFT [25] with vanishing width of strip [31]. See figure 1 (right) for example.
In [31], the authors test wedge holography by studying free energy, Weyl anomaly of 2d CFT, entanglement entropy, two point functions of scalar operators and so on. For simplicity, they mainly focus on (locally) AdS in the bulk N and on the branes Q. In this paper, we provide more supports for wedge holography. For simplicity, we focus on the codimension two correspondence, i.e., classical gravity on wedge W d+1 CFT d−1 . Remarkably, we find an exact map from the solution to vacuum Einstein equations in AdS d /CFT d−1 to the solution in wedge holography AdSW d+1 /CFT d−1 . By using this map, we prove that wedge holography with this novel class of solutions is equivalent to AdS/CFT with vacuum Einstein gravity. Assuming that AdS/CFT is correct, which is widely accepted, this equivalence is actually a proof of wedge holography in a certain sense. Now most of the results of AdS/CFT apply directly to wedge holography. For examples, we calculate holographic Weyl anomaly, holographic Rényi entropy and correlation functions for wedge holography and find that them all take the correct forms. We also discuss the more general solutions in wedge holography and argue that they are equivalent to AdS/CFT with matter fields. Finally, we generalize the discussions to de Sitter space (dS) and flat space (Minkowski) on the branes. Remarkably, we find that AdS/CFT, dS/CFT and flat space holography can be unified in the framework of codimension two holography in asymptotically AdS.
The paper is organized as follows. In section 2, we give an exact construction of the gravitational solutions in wedge holography from the ones in AdS/CFT. By using this construction, we prove that wedge holography is equivalent to AdS/CFT, at least for a novel class of solutions. In section 3, we test wedge holography by studying the holographic Weyl anomaly, holographic Rényi entropy and correlation functions. In section 4, we discuss the general solutions in codimension two holography for wedges. In section 5, we generalize our results to dS/CFT and flat space holography. Finally, we conclude with some open problems in section 6.

Construction of codimension two holography
In this section, we construct the gravitational solution for wedge holography (or AdS d+1 /BCFT d generally) in d+1 dimensions from the one in AdS d /CFT d−1 in d dimensions. In other words, we find a map between the solutions in wedge holography and in AdS/CFT. By using this map, we prove that the gravitational action of wedge holography is equivalent to that of AdS/CFT, which means classical gravity on wedge W d+1 classical gravity in AdS d . ( Assuming that AdS d /CFT d−1 holds we conclude immediately that the wedge holography also holds classical gravity on wedge W d+1 CFT d−1 .
Note that following [31] we focus on the classical gravity in this paper. In general, AdS/CFT and wedge holography also apply to quantum gravity.

Exact constructions of solutions
Let us start with the following ansatz of the metric where x µ = (x, y i ), g µν and h ij are the metrics in d + 1 dimensions and d dimensions, respectively. Note that h ij (y) are independent of the normal coordinate x. Without loss of generality, we set the AdS radius L = 1 in this paper. When h ij is the AdS d metric, g µν turns out to be AdS d+1 metric [22,31]. The key observation of this paper is that actually h ij can be relaxed to be any metric obeying Einstein equations in the framework of codimension two holography, or generally, the AdS/BCFT [25]. Let us present the above statements in two theorems.
Theorem I The metric (6) is a solution to Einstein equation with a negative cosmological constant in d + 1 dimensions provided that h ij obey Einstein equation with a negative cosmological constant in d dimensions Here R h ij denote the curvatures with respect to the metric h ij and we have set the AdS radius L = 1 for simplicity.
Proof I The proof of theorem I is straightforward. From (8), we get Substituting the above equation into (93) of the appendix, after some simple calculations we obtain which are the equivalent expressions of Einstein equations (7), i.e., R µν = −dg µν . Now we finish the proof that the metric (6) is a solution to Einstein equation in d + 1 dimensions as long as h ij is a solution to Einstein equation in d dimension.
Proof II Note that the metric (6) is written in Gauss normal coordinates. For this special kind of coordinates, the extrinsic curvatures take simple form which indeed obeys NBC (2). Please see the appendix for more discussions of extrinsic curvatures. Now we finish the proof.
Some comments are in order. 1 According to theorem I and theorem II, it is clear that the metric (6) is a solution to AdS/BCFT [25] with range −∞ ≤ x ≤ ρ, and a solution to wedge holography [31] with range −ρ ≤ x ≤ ρ 1 . Note that the wedge holography can be regarded as a special limit of AdS/BCFT [31]. 2 The metric (6) defines a map from the gravitational solutions in d dimensions to those in one dimension higher. By using this map, one can obtain non-trivial exact black hole solutions in AdS/BCFT. We leave a careful study of black holes in AdS/BCFT in [47]. 3 For simplicity, we focus on vacuum Einstein gravity in this paper. The map (6) can be generalized to the case with matter fields [48]. 4 Of course, (6) is not the most general solution to Einstein equations in d + 1 dimensions. That is because the solution space in higher dimensions is larger than the one in lower dimensions. As we will shown below, the solution (6) is of special interest due to the evident relation to AdS/CFT [3]. We leave the discussions of other kinds of solutions in section 4 and section 5.

Equivalence to AdS/CFT
Now let us prove that the wedge holography AdSW d+1 /CFT d−1 with the solution (6) is equivalent to AdS d /CFT d−1 with vacuum Einstein gravity. The AdSW d+1 /CFT d−1 claims that the partition function of CFT d−1 in large N limit is given by the classical gravitational action in d + 1 dimensional asymptotically AdS spacetime with a wedge Similarly, the AdS d /CFT d−1 assumes that To prove the equivalence between wedge holography and AdS/CFT, we only need to prove that I AdSW d+1 = I AdS d . Since the solution (6) is symmetrical for x, for simplicity we focus on half of the wedge spacetime 0 ≤ x ≤ ρ below. Please keep in mind that the total gravitational action is double of the result below.
Substituting the solution (6) into the action (1) and applying NBC (2) together with the formula (94), we derive which is equal to the gravitational action I AdS d with Newton's constant given by Note that we take h ij off-shell in the above derivations. In other words, we do not need to impose Einstein equations in the bulk N (7) or on the brane Q (8). In fact, Einstein equations (8) on Q can be derived by the variation of I AdSW d+1 with respect to h ij . Besides, we have used −2Λ = d(d − 1), K − T = tanh ρ and the following formula In the above discussions, we focus on the non-renormalized action (15), which is divergent generally. To get the finite result, one can perform the holographic renormalization [49,50] by adding suitable counterterms on Σ. Following [49,50], we choose the following counterterms on the corner of the wedge Σ which makes the equivalence still holds after renormalization. Note that, in order to have a well-defined action variation, one need to add a Hayward term [51,52] at the corner of the wedge where Θ is the angle between two branes. As a constant, the Hayward term (20) can be absorbed into the counterterms (18) of holographic renormalization. Now we finish the proof of the statement that AdSW d+1 /CFT d−1 with the solution (6) is equivalent to AdS d /CFT d−1 , at least at the classical level for gravity, or equivalently, in the large N limit for CFTs.
Some comments are in order. 1. The equivalence (19) is quite powerful which enables us to derive many interesting physical quantities such as Entanglement/Rényi entropy for wedge holography directly following the approach of AdS/CFT. See sect. 3

for examples. 2.
Assuming that AdS/CFT holds which is widely accepted, the equivalence (19) is actually a proof of the wedge holography in a certain sense. 3. Although we only prove the equivalence (19) for vacuum Einstein gravity, the equivalence (19) is expected to hold generally with matter fields. We leave the generalization to matters in future works. 4. As we have mentioned in above section, the solution (19) is not the most general solution to vacuum Einstein equations in d + 1 dimensions. There are other kinds of solutions, which may correspond to different kinds of dualities. For example, the wedge holography can also be equivalent to dS/CFT if the spacetime on the brane is asymptotically dS instead of AdS. Please see section 5 for more details. 5. Even for the case with asymptotically AdS branes, (19) is not the most general solution. See sect. 4 for the discussion of more general solutions. We argue that they are equivalent to AdS/CFT with suitable matter fields. That is reasonable since there are Kaluza-Klein modes after the dimensional reduction in the x direction.
interesting that holographic g-theorem for BCFTs becomes holographic c-theorem for CFTs in the framework of wedge holography. The discussions of this section are quite similar to those of AdS/CFT. Readers familiar with the following topics of AdS/CFT can skip this chapter.

Holographic Weyl anomaly
Weyl anomaly measures the violation of scaling symmetry of conformal field theory (CFT) due to quantum effects. For the CFT without boundaries, Weyl anomaly appears only in even dimensions. In general, it takes the following form [53] where 2p = d−1 is even, A, B n are central charges, I n are the Weyl invariant terms constructed from curvatures and their covariant derivatives, and E 2p is the Euler density defined by which is normalized so that integrated over a d-dimensional sphere: S d √ σE d = 2 [54]. It is conjectured that the central charges related to Euler density obey the c-theorem [55] So far the c-theorem is proved only for 2d CFTs [56] and 4d CFTs [57] . In higher dimensions, there are interesting holographic proofs and generalizations [58]. See also [59,60,61,62,63] for related works.
Let us consider some examples of Weyl anomaly. For 2d CFTs and 4d CFTs, the more conventional nomenclature is where C Σ ijkl are the Weyl tensor on Σ.
In the holographic theory, Weyl anomaly can be obtained from the UV logarithmic divergent term of the gravitational action [64]. We assume that the spacetime on the brane is asymptotically AdS where . Solving Einstein equations (7) in d + 1 dimenions, we get Note that σ (1) ij can also be obtained from the asymptotical symmetry of AdS [65]. Substituting the metric (26) with (27,28) into the action (1) and selecting the UV logarithmic divergent term, we can derive Weyl anomaly (24,25) for 2d and 4d CFTs with the central charges given by Note that we consider only half of the wedge in the above calculations, the total central charges for the whole wedge are double of the results above. Note also that we do not need to know σ (2) ij of order O(R 2 Σ ) in the above derivations for 4d CFTs. That is because the logarithmic terms including σ (2) ij vanish for 4d CFTs in the so-called 'off-shell' method [66].
Let us go on to consider the Weyl anomaly in higher dimensions. For simplicity, we focus on the A-type anomaly E d−1 . For our purpose, it is sufficient to consider CFTs living on the sphere so that all of the B-type anomaly I n of (21) vanish. Thus we take the the following metric where dΩ 2 d−1 denote the volume element of (d − 1)-dimensional unit sphere. Substituting (31) into the action (1) and selecting the UV logarithmic divergent term, we easily get the Weyl anomaly Comparing (32) with (21) and noting that S d−1 √ σE d−1 = 2, I n = 0 for spheres, we read off the central charge From the Weyl anomaly (21), one can derive the universal term (log term) of entanglement entropy as [54]  when the entangling surface is a sphere. Later, we will see that (34) agrees with the results of Entanglement/Rényi entropy of wedge holography.
Let us end this subsection with a holographic proof of the c-theorem (23) for wedge holography. From the null energy condition on the brane, [26] find that ρ is a monotonically decreasing function under the RG flow. There is a natural geometric interpretation of the monotonicity of ρ. See figure 2. Note that ρ is related to the angle between the brane Q 1 and AdS boundary M as [25] φ = arctan csch(ρ).
As a result, the smaller the tension ρ is, the larger the angle φ is, and the deeper the brane bends into the bulk. Recall that the AdS boundary M corresponds to UV, while the deep bulk N corresponds to IR. It is clear that ρ decreases under RG flows. Since ∂ ρ A ≥ 0 from (33) , the central charge A also decreases under RG flows. Thus the c-theorem (23) is indeed obeyed in wedge holography. Note that, in the viewpoint of BCFT, (23) is the g-theorem for boundary central charges. It is interesting that the holographic g-theorem [26] for BCFTs becomes holographic c-theorem [58] for CFTs in the framework of wedge holography.

Holographic Rényi entropy
Rényi entropy is a complete measure of the quantum entanglement of a subsystem, which is defined by where n is a positive number, ρ A = trĀ ρ is the induced density matrix of a subregion A. Herē A denotes the complement of A and ρ is the density matrix of the whole system. In the limit n → 1, Rényi entropy becomes the von Neumann entropy, which is also called entanglement entropy In the gravity dual, Rényi entropy can be calculated by the area of cosmic brane [67] where the cosmic brane n is anchored at the entangling surface ∂A. Notice that since the brane tension is non-zero generally, the cosmic brane backreacts on the bulk geometry. In the tensionless limit n → 1, the cosmic brane becomes a minimal surface and (38) reduces to the Ryu-Takayanagi formula for entanglement entropy [68] The elegant formula of Rényi entropy (37) can be straightforwardly generalized to wedge holography. Similar to the case of holographic entanglement entropy [31], the new characteristic in wedge holography is that now the cosmic brane ends on the end-of-world brane Q 1 and Q 2 . See figure 3 for example. Note that the cosmic brane is codiemsnion two, while the brane Q 1 and Q 2 are codiemsnion one. Please do not confuse them. The location of cosmic brane can be fixed by solving Einstein equations with backreactions [67]. In the case that the solution is not unique, one choose the cosmic brane with the minimal area.
In principle, by applying (37) one can investigate the Rényi entropy for general cases. However, the actual challenge is that it is difficult to solve Einstein equations with nontrivial backreactions due to the cosmic brane. So far the known solutions are limited. The hyperbolic black hole is an exact solution, when the entangling surface is a sphere [69]. There are also perturbation solutions around the hyperbolic black hole [70,71,72]. Inspired by [69], we choose the following bulk metric for wedge holography where and dH 2 d−2 = dr 2 + sinh 2 rdΩ 2 d−3 is the line element of (d − 2)-dimensional hyperbolic space with unit curvature. The Rényi index n is related to the temperature of hyperbolic black hole which yields For the solution (41), the cosmic brane is just the horizon of hyperbolic black hole, whose area is given by where V H d−2 is the volume of hyperbolic space. From (38,44,45), we finally obtain the holographic Rényi entropy with sphere entangling surface as Let us discuss briefly the result (46). First, for 2d CFTs, the Rényi entropy (46) becomes which has the correct n dependence. Second, by taking the limit n → 1, we get the entanglement entropy which agrees with the RT formula (40). Since V H d−2 includes a log term [69] (48) yields the correct universal term of entanglement entropy (33,34). This is a double check of our results of holographic Weyl anomaly and holographic Rényi entropy, which is also a support for the wedge holography.

Holographic correlation function
In this subsection, we discuss correlation functions in wedge holography. For simplicity, we focus on the two point functions of stress tensors. We find that the two point functions take the expected form, which can be regarded as a test of wedge holography.
Following [73], we consider the metric fluctuations H ij on the AdS brane Q 1 and choose the gauge H zz (z = 0, y) = H za (z = 0, y) = 0 (51) on the corner of the wedge Σ. Since the contributions from Q 1 and Q 2 are the same, we only need to consider the contributions due to Q 1 and double the final results. Here x µ = (x, y i ) denote coordinates in the bulk N and y i = (z, y a ) are coordinates on the brane Q. Imposing the Dirichlet boundary condition on Σ we solve the bulk solution where S 2 = z 2 + (y a − y a ) 2 , From the renormalized action (15) plus (18), one gets the on-shell quadratic action for H ij [73], where G (d) N is given by (90). Substituting (53) into (55), we derive where From (56), we finally obtain the holographic two point function of stress tensor for CFT d−1 < T ab (y)T cd (y ) >= C T I ab,cd s 2(d−1) , with the central charge Recall that the total central charge is double of the above C T . It is clear that the two point function (59) takes the correct form. This is a strong support of wedge holography.

More general solutions
As mentioned in above sections, although it is novel and powerful, the metric (6) is not the most general solution to vacuum Einstein equation with NBC (2) in d + 1 dimensions. In this section, we discuss the property of general solutions and find a perturbation solution beyond the novel class of solution (6) . We argue that the wedge holography with general solutions is equivalent to AdS/CFT with suitable matter fields. A natural origination of the matter fields are Kaluza-Klein modes from the dimensional reduction.

General spacetime on brane
Let us first discuss the most general spacetime allowed on the brane Q. For simplicity, we focus on vacuum Einstein gravity (1) without matters in the bulk N . The generalization to the theory including matter fields is straightforward.
The induced metric h ij on the brane should obey the Momentum constraint and Hamiltonian constraint where D i is the covariant derivative on the brane. Imposing the NBC (2), i.e., K ij = T d−1 h ij , we find that the Momentum constraint (61) is satisfied automatically, and the Hamiltonian constraint (62) yields This is the only constraint of the spacetime on the brane. It is remarkable that it is the spacetime with a constant Ricci scalar. Thus the novel solution (6) obeying Einstein equations is a special case of it. In general, there are three kinds of spacetime allowed on the brane In this section, we focus on the first case, in particular, the asymptotically AdS on the brane. We leave the discussions of the other two kinds of spacetimes to the next section.

Perturbation solution
It is difficult to solve Einstein equations with the NBC (2) on both branes Q 1 and Q 2 . To gain some insight of the solutions, let us first study a simpler case, the perturbation solution near one of the brane. At the end of this section, we give a perturbation solution which satisfies NBC (2) on both branes.
In Gauss normal coordinates, the metric near the brane takes the following form where h ij are the induced metric at x = 0. Note that h ij of (65) is cosh 2 (ρ)h ij of sect. 2.1. Note also that the above O(x) term of the metric is chosen carefully so that it obeys NBC (2). Solving Einstein equations (7) at the leading order of x, we derive provided the constraint (63) is satisfied. Similarly, we can solve h (n) ij order by order for every given h ij obeying the constraint (63). Let us rewrite (66) into a more enlightening form where h N T ij and T ij = 0 for the novel solution (6). Recall that the above h ij denote cosh 2 (ρ)h ij of (6). We suggest that T ij are the effective matter stress tensors on the brane for the most general solutions. In other words, the special solution (6) corresponds to vacuum Einstein gravity on the brane, while the most general solution corresponds to Einstein gravity coupled with matters. It is interesting that the constraint (63) yields which implies that the effective matter fields are CFTs. This is consistent with the proposal of doubly holographic models [32,33,34,35]. Note that the effective matter fields can only be regarded as CFTs approximately. That is because, in general, the Kaluza-Klein modes from the dimensional reduction are massive.
The above solutions apply to the region near to one of the brane. Now let us go on to discuss the general solution which works well in the whole region. For simplicity, let us focus on four dimensions. We apply the method first developed by [74] for AdS/BCFT. The main challenge for wedge holography is that the solution should satisfy NBC on two branes instead of only one. Inspired by [74], we choose the following ansatz of metric and embedding functions of Q 1 and Q 2 where denote a small perturbation parameter and λ is a constant to be determined. We require f z (x), f y (x) to be even functions of x. As a result, if the brane Q 1 obeys NBC (2), so does the brane Q 2 . Substituting (69,70) into Einstein equations (7) and imposing the NBC (2), we solve From the above solution, we obtain the induced metric h ij on both branes where Substituting the induced metric (73) into (67), we get the effective stress tensors which are traceless.
As we have shown above and in sect. 4.1, the induced metrics on the brane need not to satisfy the vacuum Einstein equations (7). Instead, the only the constraint is that the Ricci scalar is a constant (63). This means that the solution space for wedge holography is larger than the one for AdS/CFT with vacuum Einstein equations. On the other hand, we have showed that the CFTs in wedge holography and AdS/CFT have the same central charges, Rényi entropy and correlation functions in sect. 2 and sect. 3. In other words, they must be the same kind of CFTs (up to some background fields) 2 . To resolve the 'contradiction', a natural guess would be that the wedge holography is equivalent to AdS/CFT with suitable "matter fields " turned on. This is reasonable, since after the dimensional reduction along the x direction, there are Kaluza-Klein modes on the brane, which can be regarded as effective matter fields. For example, the original 5d Kaluza-Klein theory consists of a massless graviton, a massless vector, a massless scalar, and a tower of charged massive gravitons. Apart from the massless graviton, we can regard the other compositions as effective "matter fields". Another quick guess for the effective theory is the higher derivative gravity. However, this is not a good candidate for reasons below. First, the higher derivative gravity is usually non-unitary, unless the higher derivative terms are chosen carefully. Second, the effective theory should yield the same Weyl anomaly as wedge holography. However, this is not the case for higher derivative gravity generally. Thus the effective theory is not likely to be a higher derivative gravity. Even if there were higher curvature terms, similar to the case of f (R) gravity they can be rewritten as matter fields formally.
For the reasons above, we propose that the wedge holography with vacuum Einstein gravity in d + 1 dimensions is equivalent to AdS/CFT with suitable "matter fields" coupled to gravity in d dimensions generally. It is interesting to work out exactly the effective theory on the branes. We leave a careful study of this problem to future works.

Information from Weyl anomaly
In this subsection, we discuss the general expression of Weyl anomaly for wedge holography. For simplicity, we take AdSW 4 /CFT 2 as an example below.
According to [31], the codimension two holography AdSW 4 /CFT 2 can be regarded as a special limit of AdS 4 /BCFT 3 . See figure 1 (right). By taking this limit, we can obtain the Weyl anomaly of AdSW 4 /CFT 2 from that of AdS 4 /BCFT 3 [74] wherek ab is the traceless part of extrinsic curvature for BCFTs, and θ = π 2 + 2tan −1 (tanh ρ 2 ) is the angle between the brane Q 2 and AdS boundary M [74].
Some comments are in order. 1 In the viewpoint of CFT 2 , the second term of Weyl anomaly (77) can be regarded as the contribution from some kinds of background fields. Similar to the gravitational field, the background matter fields such as vector fields and scalar fields can also produce contributions to Weyl anomaly. Take Dirac field with Yukawa couplings as an example By applying heat-kernel method [75], we obtain the Weyl anomaly in two dimensions which takes a similar form as (77). In particular, the second terms of (77) and (79) are both positive. The Weyl anomaly for the theory (78) in four dimensions can be found in [76].
2 In the viewpoint of AdSW 4 , the information ofk ab is encoded in the bulk metric g µν , or equivalently, the brane metric h ij . 3 Comparing (77) with (24) of sect.3.1, we notice that only the solutions different from (6) could produce the second term of Weyl anomaly (77) . 4 For even-dimensional CFT 2p , there are non-trivial contributions to Weyl anomaly such as trk 2p . While for odd-dimensional CFT 2p+1 , the Weyl anomaly vanish as expected. The potential contribution such as trk 2p+1 cancel, since the extrinsic curvature k ab differs by a minus sign on Q 1 ∩ Σ and Q 2 ∩ Σ in order to have a well-defined limit from AdS/BCFT. 5 The above comments support the proposal that AdSW d+1 /CFT d−1 is equivalent to AdS d /CFT d−1 with suitable matter fields generally. These matter fields on the brane would yield the non-trivial trk 2p terms in holographic Weyl anomaly.
As a summary, the most general solutions of wedge holography should obey the constraints (63,66) and should produce the general Weyl anomaly (77).

Generalization to dS/CFT and flat space holography
The wedge holography proposed in [31] assume that the spacetime on the brane is AdS, which is equivalent to AdS/CFT as we showed above. In fact, the spacetime on the brane can also be dS and Minkowski spacetime [22]. We study these cases in this section, and show that they are equivalent to dS/CFT [11] and flat space holography [16,17], respectively.
Inspired by (6), we choose the following ansatz of the metric Substituting (80) into the normal component of Einstein equations, i.e., R xx = −d, we get which can be solved as where c 1 and c 2 are integral constants. There are three kinds of f (x), which correspond to three types of spacetimes on the brane. Let us summarize them below. It should be mentioned that the metrics with (83) have been discussed in [22]. [22] study only the cases that h ij are the metrics of AdS, dS and Minkowski spacetimes. Here we find that actually h ij can be relaxed to be any metric obeying Einstein equations in d dimensions. The case of asymptotically AdS has been investigated in above sections. Let us go on to study the cases for asymptotically dS and asymptotically flat spacetime below.

Equivalence to dS/CFT
Unlike AdS/CFT, dS/CFT is more subtle. For instance, the CFT dual to dS is non-unitary [11,12]. In this paper, we do not aim to resolve this non-trivial problem. Instead, we just show that AdS/CFT and dS/CFT can be formally unified in the framework of codimension two holography in asymptotically AdS. It is widely believed that the holography defined in AdS is well-behaved, thus codimension two holography may shed some light on dS/CFT. We hope we could address this problem in the future. Now let us discuss codimension two holography with the dS brane. Since the discussions are similar to the AdS brane of sect. 2, we do not repeat the calculations below. Instead, we just list the main results. Let us start with some theorems.

Theorem III
The metric (80) with f = sinh 2 (x) is a solution to Einstein equation (7) with a negative cosmological constant in d + 1 dimensions, provided that h ij obey Einstein equation with a positive cosmological constant in d dimensions Here we have set the dS radius L = 1 for simplicity.
Note that the dS brane tension T = (d − 1) coth ρ is larger than the AdS brane tension T = (d − 1) tanh ρ.
The theorem III and theorem IV claim that the metric (80) with f = sinh 2 (x) is a solution to AdS/BCFT with range −∞ ≤ x ≤ ρ and a solution to condimension two holography with range −ρ ≤ x ≤ ρ.
Similar to the case of AdS brane, we can prove that the codimension two holography AdSW d+1 /CFT d−1 with asymptotically dS space on the brane is equivalent to dS d /CFT d−1 . Following the approach of sect. 2, we obtain the gravitational action with an asymptotically dS brane as which is equal to the gravitational action I dS d for asymptotically dS provided that the Newton's constants are related by Similar to the case of AdS brane, one can perform holographic renormalization to make finite the gravitational action. Now we finish the proof of the equivalence. Following the approaches of sect. 3, we can derive holographic Weyl anomaly, holographic Rényi entropy, correlation functions and so on.

Equivalence to flat space holography
Let us go on to generalize our discussions to the flat space holography [16,17]. Similarly, we have the following two theorems.
Theorem V The metric (80) with f = exp(±2x) is a solution to Einstein equation (7) with a negative cosmological constant in d + 1 dimensions, provided that h ij obey vacuum Einstein equation in d dimensions Theorem VI The metric (80) with f = exp(±2x) satisfies NBC (2) with |T | = (d − 1) on the branes at x = constant.
The above two theorems shows that the metric (80) with f = exp(±2x) is a solution to AdS/BCFT and codimension two holography. The codimension two holography with flat branes can be obtained as suitable limit ρ → ∞ from those with AdS brane and dS brane. In the large ρ limit, the tension of three kinds of branes coincide One can prove the equivalence between codimension two holography with flat branes and flat space holography by taking the limit ρ → ∞ and L → ∞ for AdS brane action (15) and dS brane action (85). Recall that we have set L = 1 in (15) and (85). Recover L, (d − 1)(d − 2) of (15) and (85) should be replaced by (d − 1)(d − 2)/L 2 which vanish in the large L limit. Thus the AdS brane action (15) and dS brane action (85) indeed become the flat brane action in the limit ρ → ∞ and L → ∞. However, the Newton's constant G (d) N vanish in such limit.
To get non-zero Newton's constant, let us consider a different case. We choose f = exp(2x) with −∞ ≤ x ≤ ρ, or f = exp(−2x) with −ρ ≤ x ≤ ∞. It means that we put one of the brane to infinity. In fact, we have a large freedom to choose the location of brane. It is not necessary to set them at x = ±ρ. Without loss of generality, let us take the first choice as an example. We get the gravitational actioñ which is equal to the gravitational action I flat d for asymptotically flat space provided that the Newton's constants are given by Now the Newton's constant G We leave a careful discussion of flat space holography to future work. It is beyond the main purpose of this paper. Formally, now we have unified AdS/CFT, dS/CFT and flat space holography in the framework of codimension two holography in asymptotically AdS.

Conclusions and Discussions
In this paper, we construct a class of exact gravitational solutions for wedge holography from the the ones in AdS/CFT. We prove that the wedge holography with this novel class of solutions is equivalent to AdS/CFT with vacuum Einstein gravity. Assuming that AdS/CFT is correct, this equivalence can be regarded as a proof of the recently proposed wedge holography, at least in the classical level for gravity, or equivalently, in large N limit for CFTs. By applying this powerful equivalence, we derive directly holographic Weyl anomaly, holographic Rényi entropy and correlation functions for wedge holography. They all take the expected expressions, which is a support of the codimension two holography for wedges. We also discuss the general solutions for wedge holography. We find that the intrinsic Ricci scalar on the brane is always a constant and there are three types of spacetimes (AdS, dS, flat space) depending on the value of brane tension. We argue that the general solutions with asymptotically AdS branes correspond to AdS/CFT with suitable matter fields (Kaluza-Klein modes) coupled to gravity. In particular, the effective matter fields should produce the trk 2p like contributions to holographic Weyl anomaly. Finally, we generalize our discussions to dS/CFT and flat space holography. It is remarkable that AdS/CFT, dS/CFT and flat space holography can be unified in the framework of codimension two holography in asymptotically AdS. Since the holography in AdS is well-defined, it may shed some light on dS/CFT and flat space holography.
There are many interesting open problems worth exploring. We just list some of them.
1. Find solutions different from (6) and study their properties in wedge holography. This is the most interesting part of wedge holography, which "distinguishes" it from AdS/CFT.

2.
Investigate the wedge holography with matter fields both in the bulk and on the brane. In this paper, we mainly focus on vacuum Einstein gravity on both sides. As a result, we can discuss only the correlation functions for stress tensors instead of scalar operators and currents. The generalization to wedge holography with matter fields is an interesting and important problem. See [48] for some progresses.
3. Study the property of black holes in wedge holography and AdS/BCFT [47].

5.
In this paper, we focus on the classical limit of gravity. It is interesting to study the quantum corrections and see if the equivalence between the wedge holography and AdS/CFT still hold.
6. Apply wedge holography to discuss the information paradox such as Island and the Page curve of Hawking radiations.
We hope these interesting problems could be addressed in the future.
where g ij = cosh 2 (x)h ij and ( ) h denotes the quantity defined by the metric h ij .