Finding $AdS^{5} \times S^{5}$ in 2+1 dimensional SCFT physics

We study solutions of type IIB string theory dual to ${\cal N}=4$ supersymmetric Yang-Mills theory on half of $\mathbb{R}^{3,1}$ coupled to holographic three-dimensional superconformal field theories (SCFTs) at the edge of this half-space. The dual geometries are asymptotically $AdS^5 \times S^5$ with boundary geometry $\mathbb{R}^{2,1} \times \mathbb{R}^+$, with a geometrical end-of-the-world (ETW) brane cutting off the other half of the asymptotic region of the would-be Poincar\'e $AdS^5 \times S^5$. We show that by choosing the 3D SCFT appropriately, this ETW brane can be pushed arbitrarily far towards the missing asymptotic region, recovering the"missing"half of Poincar\'e $AdS^5 \times S^5$. We also show that there are 3D SCFTs whose dual includes a wedge of Poincar\'e $AdS^5 \times S^5$ with an angle arbitrarily close to $\pi$, with geometrical ETW branes on either side.

: Schematic of geometries dual to N = 4 SYM theory on half of R 3,1 coupled to a 3D SCFT at the boundary. The geometry contains a region that approximates a range Θ ∈ (Θ * , π/2) of Poincaré AdS 5 × S 5 , and an end-of-the-world brane region where the S 5 smoothly degenerates. When the 3D boundary SCFT has many more local degrees of freedom than the N = 4 theory, the internal space typically grows to a large volume before pinching off.
A particularly interesting case occurs when an ETW brane cuts off the asymptotic region of an asymptotically AdS spacetime [13]. In this case, gravity can localize on the ETW brane such that over a significant range of scales, gravity on the brane appears to be four dimensional. Such ETW branes can have a microscopic description when the brane intersects the asymptotic boundary of AdS. As explained by Karch and Randall [14,15] (see also [16]), in this case the full system can be dual to a boundary conformal field theory (BCFT). The localization of gravity can arise in the situation where there are many more boundary degrees of freedom than bulk degrees of freedom.
Often, such ETW branes are considered in bottom-up models where the brane is described as a codimension-one boundary hypersurface with some simple action. In this case, gravity localization can occur when this brane intersects the boundary at a large angle, so that it removes a region Θ < Θ * = −π/2 + of AdS, where Θ is the polar angle in Poincaré coordinates formed by the radial direction and the field theory direction perpendicular to the CFT boundary. The limit → 0 corresponds to the tension of the brane increasing to a critical value.
There are also fully microscopic models which realize ETW brane physics, e.g. [17,18]. In these cases, the ETW brane often corresponds to a region of a higher-dimensional geometry where the internal space degenerates smoothly. In [19], examples were provided of such microscopic models where gravity is localized to the ETW brane. In this paper, we further study these models, showing that the bulk geometry away from the ETW brane can include a region Θ > Θ * = −π/2 + of Poincaré AdS with arbitrarily small . That is, we can push the ETW brane arbitrarily far towards the missing asymptotic boundary.
We further show that there exist solutions with two ETW branes such that the dual contains a region well-approximated by the −π/2 + < Θ < π/2 − of AdS, again with arbitrarily small .
In the first case, we conclude that the physics of the missing half of the bulk CFT can be reproduced by a set of boundary degrees of freedom. In the second case, the physics of a higher-dimensional CFT can be reproduced by a carefully chosen lower dimensional theory. This is reminiscent of the "deconstructing dimensions" story [20].

The microscopic setup
In the microscopic setups we consider, the BCFT is U (N ) N = 4 supersymmetric Yang-Mills theory on R 2,1 × R + with boundary physics preserving half supersymmetry and an OSp(4|4) superconformal symmetry. This boundary physics can generally be understood as a set of boundary degrees of freedom coupled to the N = 4 fields in some way. These theories arise in string theory from the low energy limit of D3-branes ending on stacks of D5-branes and NS5-branes, with additional D3-branes stretched between the fivebranes. In many cases, the boundary physics can be considered independently and describes a three-dimensional superconformal theory with OSp(4|4) symmetry.
The vacuum states of these field theories on a half-space are dual to known solutions of type IIB supergravity. These solutions have an asymptotically AdS 5 × S 5 asymptotic region whose boundary geometry is half of R 3,1 . The full geometry has a part that is well approximated by a portion Θ > Θ * of Poincaré AdS 5 × S 5 , where Θ ∈ (−π/2, π/2) is the angle in Poincaré coordinates that labels different AdS 4 slices and Θ = π/2 corresponds to the asymptotic region that is present. 2 The remaining part of the geometry can be understood as a geometrical "end-of-the-world brane": this is a region of the ten-dimensional geometry where the internal space smoothly degenerates, so that we have a spacetime boundary from the five-dimensional point of view. This ETW brane emerges from the CFT boundary where the SCFT lives. Such geometries are illustrated schematically in Figure 1.
For a fixed set of parameters in the N = 4 theory, different choices of the boundary physics (i.e. the choice of 3D SCFT and how this is coupled to the N = 4 theory) give supergravity solutions with the same asymptotically AdS 5 × S 5 region but a different behavior for the ETW brane, and in particular, a different brane angle Θ * . The main goal of this paper is to show that by choosing the boundary physics appropriately, we can find examples with Θ * arbitrarily close to −π/2. In other words, with the right choice of boundary degrees of freedom, we can, to an arbitrarily good approximation, reproduce the physics of the missing half of the N = 4 theory.
At the level of type IIB supergravity, it is trivial to exhibit families of such solutions that recover all of Poincaré AdS 5 × S 5 in a limit. However, the flux quantization conditions of the full type IIB string theory imply that the parameters present in the supergravity solutions cannot be varied continuously, but instead correspond to discrete solutions of a family of nonlinear equations. These parameters correspond to the discrete data used to specify the choice of boundary SCFT to which we couple the N = 4 theory. The non-linear constraints on the supergravity parameters are complicated enough that it is not possible to find a general solution analytically. Nevertheless, we are able to exhibit the existence of sequences of such solutions with the behavior that Θ * → −π/2.
On the field theory side, the theories that give Θ * ∼ −π/2 correspond to boundary theories with many degrees of freedom. These arise from string theory brane constructions where we have D3-branes ending on stacks of D5-branes and NS5-branes where both N D5 and N N S5 are taken large. The SCFTs describing these boundary degrees of freedom correspond to the IR limit of quiver gauge theories where the quiver generally has many nodes; we provide some explicit examples below.
Three-dimensional duals to arbitrarily large wedges of AdS 5 × S 5 For a give choice of boundary physics, we can also consider introducing a second boundary with the same physics (arising from an equivalent configuration of branes) so that supersymmetry is preserved. This theory, now on a strip, will flow to some SCFT in the infrared. The gravity dual for this theory will correspond to a wedge −|Θ * | < Θ < |Θ * | of AdS 5 × S 5 with ETW branes on either side. Such solutions were considered in [21] and provide a microscopic example of the "wedge holography" discussed in [22]. Our results in this paper show that the wedge can actually be arbitrarily large, i.e. with an angle that is arbitrarily close to π. Thus, we can have a 2+1 dimensional theory whose dual geometry contains an arbitrarily large wedge of AdS 5 × S 5 .

End-of-the-world brane geometries
The ETW branes in these constructions have a ten-dimensional geometry that was compared by Bachas et al. [19] to a bagpipe. Here, the "bag" is a small perturbation to the AdS 4 M 6 geometry dual to the decoupled 3D SCFT, where M 6 is a compact internal space. When the SCFT is coupled to the higher-dimensional N = 4 SYM theory, the previously compact internal space M 6 is perturbed to include a narrow semi-infinite "pipe" with the geometry of S 5 times a non-compact direction [19]. The perturbation is small since the N = 4 theory has many fewer local degrees of freedom than the SCFT.
The curvature scale of the internal space M 6 is generally of the same order of magnitude as the scale L 4 describing the non-compact AdS 4 geometry of the ETW brane, and these are both much larger than the AdS 5 scale L 5 . The lack of scale separation between the AdS 4 scale and the curvature radius of the M 6 has been noted in the past [19,23]; we provide a direct argument for it in Appendix A.

Outline
In the remainder of the paper, we review in Section 2 the field theories that we consider and their gravity duals in type IIB supergravity. In Section 3, we derive conditions on the parameters describing the boundary SCFT such that the dual theories include a region that is well-approximated by a region Θ > −π/2 + of AdS 5 × S 5 to an accuracy δ. In Sections 4 and 5, we find explicit examples of sequences of theories (with fixed g YM and N for the N = 4 theory) that satisfy our conditions with parameters and δ both approaching zero. In Section 6, we describe 3D SCFTs whose duals include arbitrarily large wedges of AdS 5 × S 5 (|Θ| < π/2 − with arbitrarily small ). We end with a brief discussion in Section 7.

Background
The field theories we consider and the corresponding supergravity solutions were reviewed in detail in our recent paper [24]. We refer the reader to Sections 2.2 and 3 of that paper for the details, or to the earlier references [25,26] for a discussion of theories with half-maximal supersymmetry N = 4 on a half-space and [17,18,27,28] for a discussion of the supergravity solutions. Here, we summarize only the basic information that we will use.
The set of supergravity solutions that we discuss take the form of AdS 4 × S 2 × S 2 fibred over a two-dimensional space Σ that we can take to be the positive quadrant of a plane.
Explicitly, the metric takes the form where θ ∈ [0, π/2] and ds 2 AdS 4 and ds 2 S 2 i are metrics for AdS 4 and two-spheres with unit radius. Here, f i and ρ are functions of r and θ which are given explicitly in terms of a pair of harmonic functions h 1 , h 2 on Σ.
The general expressions for the harmonic functions corresponding to vacua of N = 4 SYM on a half space with various choices for the boundary physics are given as Here the sets {l A } and {k B } represent the locations of poles on the x-axis and y-axis respectively. These correspond to throats in the ETW brane region of the geometry that are sources of D5-brane flux and NS5-brane flux respectively. The parameters c A and d B control the amount of D5 and NS5-brane flux emerging from these throats.
In string theory, the fivebrane flux is quantized; this gives the constraints that 3) The throats also have D3-brane flux, and there are additional constraints related to the quantization of this. These are (2.4) Here, the integer parameters L A and K B can roughly be thought of as the number of units of D3-brane charge per D5-brane associated with the l A throat or NS5-brane associated with the k B throat respectively.
N S5 , L A , K B ) are directly related to the parameters specifying the field theory. The connection is described most easily by referring to the string theory brane constructions from which the field theory arises. It is convenient to define where both sets are ordered from left to right. Then, in the setup of Figure 2,K i is the net number of D3-branes ending from the right on the i th NS5-brane plus the number of Removing the semi-infinite D3-branes on the right, we have a brane configuration that gives rise to a 3D SCFT in the infrared.
D5-branes to the left of this NS5-brane, andL i is the net number of D3-branes ending from the right on the i th D5-brane plus the number of NS5-branes to the left of this D5-brane. 3 3 Obtaining a large AdS 5 × S 5 region The solutions dual to OSp(4|4)-preserving BCFTs we consider can be thought of as having two general geometrical regions with distinct features: • Region I: An asymptotically AdS 5 × S 5 region occurring at large values of the radial coordinate r l A , k B on Σ, where O(l A /r), O(k b /r) corrections due to the 5-brane throats are small; and • Region II: An "end-of-the-world" brane region at r l A , k B where the geometry caps off smoothly except at the locations of the 5-brane throats.
We are interested in considering whether certain allowed choices for the supergravity parameters are able to produce a geometry where region (I) is large and approximates pure AdS 5 × S 5 ; by "large", we mean that the AdS 5 × S 5 region extends to Poincaré angle Θ * ≈ −π/2.

Conditions for a large AdS 5 × S 5 region
Consider the harmonic functions (F. 19) that determine the metric and other fields. Expanding these in 1/r, we can write are the harmonic functions that give pure AdS 5 × S 5 , with AdS length L given by For the pure AdS 5 × S 5 solution, the plane r = r 0 is an AdS 4 slice perpendicular to the boundary that divides the space in half.
We note that for r ≤ r 0 , the first term in square brackets will be small compared to the terms in h AdS if and only if 3) The ratio ∆/N gives the fractional size of the corrections (which do not have a significant dependence on r for r < r 0 ).
The remaining corrections, involving higher powers of 1/r, become larger (relative to the leading terms) for smaller r. It is straightforward to check that these corrections will be small relative to the leading terms provided that r l A and r k B . For example, when this is true, we have where the term on the right is the leading term in h AdS 1 in the r < r 0 region.
To summarize, we expect that provided the condition (3.3) holds, the solutions will be well approximated by pure AdS 5 × S 5 in a region r > r * where the coordinate r is much larger than any of the l A or k B . For h i = h AdS i , the coordinate r is related to the Poincaré angle Θ by [24] r r 0 = tan Θ 2 + π 4 (3.5) so the geometry includes a region well-approximated by the Θ > Θ * region of Poincaré AdS, where In particular, having Θ * close to −π/2 requires r * r 0 , which requires Thus, we have arrived at the two conditions (3.3) and (3.7). In Appendix B, we provide a more detailed justification that these give solutions with small Θ * .

Satisfying the conditions within string theory
In the context of type IIB supergravity, it is trivial to find solutions satisfying the constraints (3.3) and (3.7) for a given fixed N . We are free to take the individual l A and k B as small as we like, and then choose c A and d B so that However, in string theory, the solutions obey flux quantization conditions (2.3) and (2.4). Below, we will investigate, for fixed (g, L AdS ) (or equivalently fixed parameters (g YM , N ) in the N = 4 theory), the space of parameters {l A , k B , c A , d B } that satisfy both the quantization conditions and the constraints (3.3) and (3.7). We will demonstrate discrete families of solutions for which we obtain an arbitrarily large region 4 of AdS 5 × S 5 , approximated arbitrarily well, within the family.

Solutions with single D5-pole and NS5-pole
It is not possible to obtain a large AdS 5 × S 5 region when we have a boundary condition corresponding to a D-brane configuration with only D5-branes or only NS5-branes, since this manifestly violates (3.3) in our constraints. Thus, the simplest possibility is a solution with a single D5-brane throat and a single NS5-brane throat. We consider this case in the present section.
We fix the parameters N and g. Then, in terms of the integer parameters N D5 , N N S5 , the relation (3.2) and the constraint (2.4) demands that l, k satisfy In Appendix C, we show that the allowed (l, k) are in one-to-one correspondence with positive parameters (N D5 , N N S5 , L, K) such that 2) and The latter equation always has at least one solution with positive integers (L, K) provided that (4.2) is satisfied.
In this section, we will understand the space of parameters (N D5 , N N S5 , L, K) which can realize constraints (3.3) and (3.7), and therefore give rise to supergravity solutions with a large region of AdS 5 × S 5 .
The main results of this section are as follows: • If we would like a solution that is well approximated by AdS 5 ×S 5 to an accuracy δ 1 in some range r > r 0 (meaning that |cl−dk| • When these are satisfied, the additional condition is sufficient to ensure the existence of suitable (L, K) to give a solution with the desired properties. In particular, if we choose N D5 , N N S5 such that gcd(N D5 , N N S5 ) = N and gN D5 /N N S5 = O(1), the approximation accuracy δ will be of order 1/ √ N .
• We explicitly construct sequences of solutions labeled by a parameter n ∈ Z + with thus obtaining an arbitrarily good approximation to an arbitrarily large AdS 5 × S 5 region for large n. For example, in the case of self-dual coupling g = 1, this occurs for the choice (or exchanging N D5 ↔ N N S5 and L ↔ K in these expressions), where n ∈ N + is an integer parameter (and we must also require that n is odd if N is odd). More generally, we construct such families for any string coupling g and any choice of relative scaling

Necessary conditions for solutions with large AdS 5 × S 5 region
Suppose we would like a solution that is well approximated by AdS 5 × S 5 to an accuracy δ in some range r > r 0 . Then according to the conditions (3.3) and (3.7) we require that we may combine (4.10) and (4.11) to find that Combining these with (4.8) and (4.9), we see that Consequently, we see that both N D5 and N N S5 must be sufficiently large for (3.3) and (3.7) to simultaneously be satisfied, in addition to the previous requirement G | N . Notably, this implies that if we would like to construct a family of solutions which can achieve an arbitrarily large AdS 5 × S 5 region, then we will need to take both N D5 and N N S5 to be increasingly large within this family.

Sufficient conditions for solutions with large AdS 5 × S 5 region
Given N D5 , N N S5 satisfying G | N and (4.13), we will now investigate the additional conditions which guarantee a choice of (l, k) in the range (4.12) for which L and K are integers.
For N D5 and N N S5 satisfying constraints (4.13) and G | N , we have from (4.12) that (4.14) Using (4.1) together with (4.12) and (4.14), we have that More precisely, taking into account the allowed range of l and k/l, L must lie in a range of values with half width We can show that the second term here is larger when (4.13) is satisfied, so we can take the range as We need the range [L 0 − ∆L, L 0 + ∆L] to be large enough to contain an integer value. More specifically, we need a value for which K − N D5 = (N − LN D5 )/N N S5 is also an integer. This requires that G | N , in which case, a range of L of length N N S5 /G will lead to at least one integer value of K.
Thus, for fixed g and N , and some chosen N D5 and N N S5 satisfying the constraints (4.13) and that G | N , we will get a solution provided that the range (4.16) is at least N N S5 /G; that is, it should be sufficient that Since the term in brackets is larger than or equal to 2 and G < N , we expect that our sufficient condition can be satisfied provided that δ is at least 1/ √ N . However, we will see below that for fixed N , arbitrarily small values of and δ are possible for carefully chosen parameters.

One-parameter families with arbitrarily large AdS 5 × S 5 region
For simplicity, we will begin with the case of self-dual coupling g = 1. We consider a sequence of parameters labeled by n ∈ N + (and further imposing that n is odd for odd N to satisfy (4.2)), defining and 21) or alternatively, using the same expressions but with N D5 ↔ N N S5 and L ↔ K. In this case, we can check that (4.2) and (4.3) are satisfied, so our results in Appendix C show that there will be a unique choice (l, k) satisfying (4.1).
For large n, we can write this solution perturbatively as From these, we find that so we can indeed make max{l, k} and |cl − dk| arbitrarily small within this particular class of solutions, by choosing sufficiently large n. Thus, we can have an arbitrarily large region of AdS 5 × S 5 arbitrarily well-approximated by our solution.
To emphasize that these choices of parameters indeed give rise to a large AdS 5 × S 5 region, we show in Figures 3 and 4 the metric functions obtained for particular choices of these parameters, as well as the metric functions of AdS 5 × S 5 for reference. We find that these metric functions agree to good approximation for r above some r * which becomes small as the parameter n is taken to be large.

General construction of one-parameter families
Next, we consider a more general case where the string coupling takes the form where m ∈ Z + and a < b are relatively prime. The set of such string couplings is dense in [0, ∞). Taking (α, δ) to be any solution to the Diophantine equation 5 we define a sequence 6 We can also consider a similar sequence with the replacements N D5 ↔ N N S5 , L ↔ K, g ↔ 1/g. This choice is motivated in Appendix D. 5 A simple explicit case is to take b = a + 1 (so that g = m tan(π/(2b))), α = N , and δ = 0. 6 Note that different choices for (α, δ) lead to the same sequence with a redefinition of n.
For these choices, it is straightforward to check that (4.3) is satisfied. Also, constraint √ gN D5 l + 1 √ g N N S5 k = N implies that both l and k are at most O n −1 , so these go to zero in the limit n → ∞. Finally, we need to verify that |cl − dk| also vanishes in this limit.
From the definitions of L(n), N N S5 (n), and g, we see that The equations (4.1) yield Thus, we have It follows that as desired. Thus, an arbitrarily large region of AdS 5 × S 5 becomes arbitrarily well approximated for solutions corresponding to large enough n.
The construction so far applies to a particular dense set of string couplings of the form (4.24), and leads to a scaling of parameters where m is an integer. In Appendix D, we generalize the construction to arbitrary real string coupling and find families of solutions that exhibit a more general scaling N N S5 ∼ zN D5 for arbitrary z > 0.
For general z, conditions (3.3) and (3.7) then fix the scaling for the linking numbers to be We would now like to understand from the field theory perspective what boundary physics for the N = 4 SYM theory gives rise to the solutions with arbitrarily large regions of AdS 5 × S 5 (Θ * arbitrarily close to −π/2). In each case, we are coupling the N = 4 SYM theory on a half space to a particular 3D SCFT 7 that can be understood as arising from the low-energy physics of a particular brane configuration in string theory, or as the IR limit of a quiver gauge theory.
To understand the brane construction corresponding to parameters (N D5 , N N S5 , L, K), we note that the parametersL i introduced in Section 2 are simply L with multiplicity N D5 , while the parametersK i are K with multiplicity N N S5 . From the relation between these parameters and the brane configuration, we can check that this set corresponds to having N N S5 NS5-branes which we can initially think of as being separated along a direction x 3 (the direction in which the D3-branes are semi-infinite), with a stack of N D5 D5-branes between the Lth and (L + 1)st NS5-brane from the left. We additionally have n i D3-branes streteched between the ith and (i + 1)st NS5, where To the right of the final NS5-brane, we have the N semi-infinite D3-branes.
Stripping off the semi-infinite D3-branes gives a brane setup whose low-energy physics is a SCFT that corresponds to the IR limit of the quiver gauge theory shown in Figure 5. Such a quiver consists of N N S5 − 1 nodes, with N D5 fundamental hypermultiplets coupled to the L th node. For nodes to the left of the L th node, the gauge group rank increases in increments of K as we read the quiver from left to right; for nodes to the right, the gauge group rank decreases in increments of N D5 − K.
So far, this construction is completely general within boundary conditions involving a single D5-brane throat and a single NS5-brane throat; we now restrict to boundary conditions within the families considered in this section. For the one-parameter family introduced at the beginning of Section 4.3 (with g = 1 and z = 1), we see that the corresponding quiver is approximately "left-right symmetric" for large n; given that our family has L N N S5 ≈ 1 2 for large n, the hypermultiplets are coupled to a single node which is roughly in the middle of the quiver, after which the gauge group rank decreases in increments of More generally, we find that, if we parametrize the quiver by its 7 We recall that the general OSp(4|4)-invariant boundary condition of this theory can be specified by a triple (ρ, H, B) [25,26]; here, ρ : su(2) → g is a homomorphism into the Lie algebra of the gauge group (in our case U (N )) which specifies the "Nahm pole" boundary condition for the scalars in the bulk 4D hypermultiplet, H is the residual symmetry group at the field theory boundary, and B is the 3D SCFT coupled at the boundary. For the boundary conditions in the one-parameter families that we are currently considering, we are imposing a simple Dirichlet boundary condition on the bulk hypermultiplet (and a Neumann condition on the 4D vector multiplet), and there is no reduction in gauge symmetry; our boundary conditions are then entirely specified by the SCFT B. length N N S5 − 1 ≈ N N S5 , then we will have N D5 ≈ 1 z N N S5 fundamental hypermultiplets coupled to a node whose placement in the quiver grows proportionally to the length of the quiver to enforce the ratio L N N S5 ≈ 2 π arctan(z/g). In particular, we note that in the case of small coupling g z, the fundamental hypermultiplets will be roughly at the right end of the quiver, while in the case of large coupling g z they will be at the left end.
The fact that all the hypermultiplets are attached to the same gauge group factor (or that the D5-branes in the brane construction come in a single stack) is an artifact of our simplifying assumption that the harmonic functions leading to the supergravity solution have only a single D5-brane pole and a single NS5-brane pole. We expect that there are many other choices with additional poles that lead to more general quivers but still give Θ * → −π/2 in a limit. In Appendix E, we will verify that such cases can be obtained by small deformations of the boundary conditions in this section. In particular, we construct examples where we couple in additional hypermultiplets to an additional node of the quiver; this corresponds to adding in additional D5 and NS5-brane poles. We also consider deforming our single-pole boundary conditions by coupling the corresponding quivers to an additional small quiver at the left endpoint. In both of these contexts, we find more general sequences of solutions that still yield Θ * → −π/2. We will consider a further generalization with multiple D5-brane poles in the following section.

Solutions with multiple poles
In this section, we consider a more general case where we still have only a single NS5-brane pole in h 2 at location y = k with multiplicity N N S5 , but we allow arbitrary numbers of D5-brane poles in h 1 at locations x = l A .
These poles will correspond to some linking numbers K with multiplicity N N S5 and N D5 linking numbers {L A }, such that Given linking numbers satisfying this, the corresponding pole locations k and l A must satisfy We can determine k and l A as follows. Defining and noting that for any k, F k is a monotonic map from [0, ∞) to [0, ∞), we have that The actual value of k is determined by solving 8 To see which linking numbers satisfy our conditions for having a Θ * close to −π/2, we note that the requirements that (which follows from the first three equations of this section) and our condition require that both terms in each expression are close to N/2 so In order that k √ N , the latter condition implies Then the l A are approximately related to L A by The condition l A √ N gives that From the condition (5.8), we have in the sum is required to be much less than √ N but also greater than or equal to F −1 , we note that the number of D5-brane poles (including multiplicity) must satisfy πg . (5.14) Our choice of the L A must be such that is an integer. To see when this is possible, we note that for L A N N S5 , F is linear and Thus, adding an additional pole with L = 1 or varying one of the L A by 1 leads to a change in the left side of (5.13) of π 4 Given any set of Ls, changing the sum by an amount less than N N S5 will be enough to give an integer K. If we add or change Ls in the linear regime of F , the change in √ g l A will be less than π 4 We can satisfy (5.13) for integer K provided that this quantity is much less than N/2, so we have the additional condition N N S5 g .
So far, we have assumed that k = k (0) . The actual value of k corresponding to our chosen L A s and K is determined by We need to check that for this actual value, |k/ √ gN N S5 − N/2| N so that (5.7) is still satisfied. Since we know that the function varies by an amount much less than N as k is varied from k (0) to its actual value. This gives 23) so kN N S5 / √ g will change by an amount much less than N provided that the right side here is less than √ gN/N N S5 , or G (k) > N N S5 / √ g. This is clearly true, since the k derivative of the first term in G is N N S5 / √ g and the k derivative of the second term is positive.
To summarize, given N and g, the following procedure will yield a set of linking numbers that satisfy our conditions: • Choose some N N S5 satisfying N N S5 √ gN and N N S5 g and N D5 satisfying (5.14).
• Choose a set {L A } of N D5 Ls such that (5.12) and (5.13) are satisfied and is an integer. This will be possible provided the conditions on N N S5 and N D5 are satisfied.
• Once the linking numbers are fixed in this way, the precise k and l A are determined by the procedure described at the beginning of this subsection.
For this more general class of SCFTs, the corresponding quiver gauge theory will have fundamental matter distributed among the nodes of the quiver, with the number of distinct L A s determining the number of nodes with fundamental matter.
If we require that l A < √ N to satisfy (5.12), we get As there are N N S5 nodes in the quiver, it seems possible in some cases to have matter uniformly distributed throughout the quiver, with order one fundamentals per node.

Microscopic wedge holography
In this section we describe a generalization of the previous construction in which we have two ETW branes bounding an arbitrarily large wedge Θ ∈ (−Θ * , Θ * ) of AdS 5 × S 5 . In this case, only a R 2,1 of the original asymptotic region R 3,1 of AdS 5 × S 5 remains, and the dual theory is a three dimensional SCFT.
6.1 A 3D dual to an arbitrarily large wedge of AdS 5 × S 5 .
We have seen that for an appropriate choice of 3D SCFT coupled to N = 4 SYM theory on a half space, the ETW brane region of the dual geometry can be pushed to a Poincaré angle that is arbitrarily close to −π/2. We next consider the situation where we introduce another such boundary parallel to the first so that the N = 4 theory now lives on a strip. We can choose this second boundary SCFT to preserve the same set of supersymmetries as the first one. The brane construction of this SCFT involves the same set of branes as for the first SCFT, with the same orientations, but arranged in the opposite order in the spatial direction in which the D3-branes have a boundary; 9 see Figure 6.
We expect the dual of this theory to have two ETW branes, bounding a wedge of AdS 5 × S 5 whose asymptotic region has the geometry R 2,1 times an interval. The solutions of [17,18,27,28] are not general enough to describe this, since they correspond to theories with a 3D superconformal symmetry, while the interval in our construction introduces a scale. However, we expect that the IR limit of the theory on a strip will be a certain superconformal theory; this is the theory whose brane construction combines that of the original BCFT with that of the second SCFT, so that the initial semi-infinite D3-branes now connect the brane configurations describing the two SCFTs. The gravity dual of this IR SCFT is wedge of AdS 5 × S 5 with two ETW branes. This wedge geometry can be described explicitly as Figure 6: Illustration of procedure used to define families of solutions realizing arbitrarily large wedges of AdS 5 × S 5 ; here, D3-branes are black, D5-branes are blue, and NS5-branes are red. To pass from the second to the third configuration, we rearrange the fivebranes so that all NS5-branes are to the left of all D5-branes, while D3-branes between these fivebranes are created or annihilated to maintain fixed linking numbers. The third configuration is convenient for defining the quantities N in (6.2): they represent the net number of D3-branes ending on branes in the A th D5-brane stack or the B th NS5-brane stack respectively. particular cases of the solutions in [17,18,27,28] and were considered previously in [18,21]. These geometries are microscopic realizations of the "wedge holography" discussed in [22].
The new element in our work is that we can, by the choices described in the previous section, arrange for the wedge of AdS 5 × S 5 between the ETW branes to be arbitrarily large.
To verify this, we note that making the change of coordinates z = r 0 e w = r 0 e x+iy so that the positive quadrant is mapped to the strip 0 ≤ (w) ≤ π/2, the single boundary geometries correspond to harmonic functions where we have defined α A = − ln(l A /r 0 ) and β A = − ln(k A /r 0 ).
The pole of h 1 at −α and the pole of h 2 at iπ/2 − β lie at large negative values of x for the single-pole cases of interest. The corresponding solution with two ETW branes is given by where N

=N
(2) 5 = N N S5 are the number of D5-branes and NS5branes in the initial boundary condition, and now we have poles of h 1 at δ 1/2 and of h 2 at iπ/2 +δ 1/2 whose leading order behaviour is given by Solutions corresponding to more general 3D SCFTs are obtained by allowing the poles to be at more general locations.
To demonstrate this claim, we will proceed by analyzing the D-brane constructions for these theories. We must first revisit the families of boundary conditions from the previous section, choosing for convenience a string coupling g in the boundary case to be of the form g = m cot π 2 a b , as we have done above, and defining the parameters (N D5 , N N S5 , L, K) using (4.26). As in [18], when we pass to the dual of the 3D theory, we may consistently set g = 1 (while the dilaton is left arbitrary).

The doubled theory is described in the language of [18] by parameters
where the supergravity parameters δ a ,δ b are related to the D3-brane charges by  so that without loss of generality we may take leading order behaviour Comparing with the supergravity parameters from the boundary case , (6.6) 10 Our notation is actually slightly different from that of [18]: the N we find the leading behaviour of the poles δ 1/2 andδ 1/2 mentioned above.
One can consider h 1 , h 2 at leading order, and show that they give rise to an AdS 5 × S 5 region when |x| ln n. Indeed, we find in this region where L 2 = √ 4πN 2 s . We recognize these as corresponding to pure AdS 5 × S 5 . As n is increased, the curvature scale of the AdS 5 × S 5 region approaches a constant value, while the size of this region increases.
In Figure 7, we show the metric functions for such solutions (as well as those of AdS 5 ×S 5 for comparison) in the vicinity of the locally AdS 5 × S 5 bridge between the two ETW branes, for various increasing values of n. We see that for increasing n, the bridge connecting the two ETW brane regions corresponds to an increasingly large wedge of AdS 5 × S 5 .

Multi-wedge geometries
We have given a specific class of constructions describing arbitrarily large wedges of AdS 5 ×S 5 as the dual of a 3D SCFT. For concreteness, we focused on the case obtained by doubling of a brane configuration considered earlier in the context of boundary conditions for the 4D N = 4 theory. More generally, we may consider 3D SCFTs which descend from linear quivers arising from "gluing" together several large sub-quivers of the type discussed in Section 4.4 by coupling the first and last nodes of consecutive sub-quivers with bifundamental matter to some additional U (m A ) nodes with small m A . This procedure is in the spirit of the "quantum gate" solutions described by Bachas and Lavdas in [21], but the result here is a spacetime description involving multiple wedges of AdS 5 × S 5 separated by interface branes.
This "multi-wedge" construction suggests further generalizations for holographic theories realizing the same OSp(4|4) symmetry as the 3D SCFTs, including the OSp(4|4)-preserving BCFTs and 3D SCFTs descending from circular quiver gauge theories. In the former case, the holographic description involves a large AdS 5 ×S 5 region in the vicinity of the asymptotic boundary, but this region is connected to an additional multi-wedge region by an interface brane. In the latter case, we again obtain a multi-wedge geometry whose boundary is only a R 2,1 subset of the asymptotic R 3,1 of AdS 5 × S 5 , but in this case, the first and last AdS 5 × S 5 wedges are connected by another interface brane, so that we have non-contractible loops in = N 2 (n + 1) with the values of n given, while the metric functions shown in light blue are for pure AdS 5 × S 5 (with L AdS fixed by N ). We are displaying the metric functions with respect to complex coordinates (w,w) = (w 1 + iw 2 , w 1 − iw 2 ) = ln r r 0 e iθ , ln r r 0 e −iθ , and setting θ = π 4 in the figures. Note that the Jacobian of this coordinate change modifies ρ 2 from the expression provided. the internal space which traverse all of the wedges. We leave a more detailed analysis of multi-wedge solutions to Appendix F.

Discussion
We have provided a number of microscopic constructions of 4D BCFTs enjoying a holographic description with an arbitrarily large AdS 5 ×S 5 region terminating on an ETW brane, as well as 3D SCFTs which correspond to an arbitrarily large AdS 5 × S 5 wedge. While the possibility of realizing similar features by considering limits of the supergravity solutions provided in [17,18,29] has been discussed previously (e.g. in [18,19,21,29]), we have provided an important check that the required limits can indeed be realized in string theory, where the various charges are subject to quantization requirements, and we have characterized the appropriate boundary conditions explicitly in terms of the corresponding field theory data.
The simplest such BCFT boundary conditions arise in string theory from a single stack of N D5 D5-branes and N N S5 NS5-branes; choosing N D5 , N N S5 sufficiently large with gN D5 /N N S5 = O(1) ensures a large AdS 5 ×S 5 region, and a judicious choice of these parameters and the linking numbers L, K can make this region arbitrarily large. While these "single-pole" boundary conditions are especially easy to analyze, we have indicated several generalizations involving multiple fivebrane throats in the ETW brane region, including small perturbations to the single-pole boundary conditions, boundary conditions which redistribute the fundamental matter throughout the defining quiver diagram, and boundary conditions involving extended quivers which give rise to "multi-wedge" duals. By invoking similar D-brane constructions to generate supersymmetric boundary conditions for the 4D N = 4 SYM theory or 3D SCFTs describing the IR physics of linear or circular quiver gauge theories, we are able to produce holographic duals for these theories in type IIB supergravity that possess similar local features, including one or more AdS 5 × S 5 wedges. This suggests a precise sense in which the physics of these degrees of freedom can be associated to the wedge. In all of our examples, such wedges are necessarily accompanied by a large ETW brane region.
There are a number of further directions which remain interesting to explore. While we have studied a large class of solutions with large AdS 5 × S 5 regions, it would be desirable to provide a general characterization of theories which possess this feature. It is also interesting to understand if there is a relationship between our work and the "dimensional (de)construction" story [20,30]. In this context, it is shown that certain quiver gauge theories may admit a low energy effective description with emergent extra dimensions; for example, this may occur in superconformal theories moved onto the Higgs branch, with the spectrum of massive vectors obtained via the Higgs mechanism organizing precisely into the Kaluza-Klein modes of the higher dimensional theory. Our results also suggest a relationship between 3D and 4D supersymmetric theories, in the sense that the physics of large wedges of AdS 5 × S 5 can either be described by degrees of freedom in the 4D N = 4 SYM theory or in a suitably chosen 3D SCFT capturing the low energy behaviour of a quiver gauge theory.

Acknowledgments
This research is supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Simons Foundation though a Simons Investigator award and the "It From Qubit" Collaboration grant.

A Size of the internal space in the ETW brane region
The fact that the compact spherical directions in the "bag" or "ETW brane" region of the geometries of interest cannot be suppressed relative to the AdS 4 scale has already been noted by Bachas and Lavdas in [19] (following previous related comments by Bachas and Estes in [23]). As remarked by these authors, this property is related to the issue of scale separation in the context of flux compactifications (see e.g. [31][32][33]). More generally, it is a broad prediction that in holographic theories with supersymmetry, the R-symmetry is geometrized at the AdS scale (see e.g. [34]). For the sake of completeness, we will here provide a direct argument for these assertions in the context of the supergravity solutions considered in this note, based on the formulation of the reduced BPS equations by D'Hoker, Estes, and Gutperle in [27,28]. Our conclusions will apply to the solutions dual to the 3D N = 4 SCFTs of Gaiotto-Witten [25,26], first studied in [18], as well as the boundary and interface solutions studied in [17].
Our goal is to show that it is not possible to simultaneously have f 2 1 /f 2 4 1 and f 2 2 /f 2 4 1 in any region of the spacetime unless that region is locally AdS 5 × S 5 ; the conclusion is therefore that at least one of the S 2 factors of the internal space remains large relative to the AdS 4 scale in the ETW brane region.
In the following, we will be relying on the conventions of [27], introducing only the ingredients necessary. We may write the complex axion/dilaton P and connection Q oneforms as P = p a e a , Q = q a e a , (A.1) and the anti-symmetric five-form and three-form tensors F (5) and G as where the e are wedge products of the appropriate vielbeins; the indices a, b are summed over the Riemann surface Σ directions. It is demonstrated in [27] that, for solutions with 16 supersymmetries, one can always apply an SU (1, 1) S-duality transformation to a frame where the axion field vanishes and the dilaton is real; this corresponds to the reality conditions The metric functions f 1 , f 2 , f 4 may be expressed in terms of a (Grassmann-even) spinor degree of freedom (equation (6.18) of [27]) in terms of which we have (equation (6.26) of [27]) where ν ∈ {±1} (the sign will be irrelevant when we compare ratios of metric functions f 1; we will restrict to considering this neighbourhood for the remainder of the subsection. In this case, we must have either |α| |β| or |β| |α| throughout the neighbourhood. Indeed, using polar coordinates we have and since min we must have 2ab On the other hand, the dilatino BPS equation (equation (6.28) of [27]) gives with z,z frame indices. These two equations together imply either that p z = g z = h z = 0 or with the latter contradicting the conclusion that |α| |β| or |β| |α|. We therefore must have that the special condition p z = g z = h z = 0 holds throughout the neighbourhood we are considering. 11 But as shown in Section 6.9 of [27], that condition alone necessarily implies that the geometry is pure AdS 5 × S 5 , with (subject to a particular choice of normalization) and metric functions where w = x + iy is a complex coordinate on the strip Σ. (We should note that the argument provided applies to the case where p z , g z , h z are presumed to vanish everywhere, but the nature of the argument is local, and can be repeated to demonstrate that the geometry within the neighbourhood we are considering must be AdS 5 × S 5 .) In particular, this can be consistent with our assumption |α| |β| or |β| |α| near the asymptotic boundary x → ±∞, where the metric function f 2 4 diverges. We have therefore shown that the only case in which one can simultaneously have f 2 1 /f 2 4 1 and f 2 2 /f 2 4 1 is when the geometry is locally AdS 5 × S 5 ; as a corollary, we clearly cannot have the scale of the internal S 2 dimensions be small compared to the curvature scale of the non-compact dimensions.

B Justification of condition (3.3)
In general, the region I introduced in Section 3 is only asymptotically AdS 5 × S 5 , and may deviate from pure AdS 5 × S 5 significantly before the O(l A /r), O(k B /r) corrections become large. For example, considering the large-r asymptotics of the metric functions for our general 11 Note that we could have avoided this condition by requiring that one of f 2 1 /f 2 4 or f 2 2 /f 2 4 but not both was small; in this case, we would not necessarily require that |α| |β| or |β| |α|, but could instead have that αβ was almost pure real or pure imaginary. solution, we find Evidently, if we would like the terms subleading in large r to be suppressed for any r r 0 , then in addition to (3.7), we require We claim that conditions (3.7) and (B.2) are sufficient to ensure a large region of approximately pure AdS 5 × S 5 .
To further motivate this fact, let us fix N from the beginning, and recall that r 2 0 ≡ N π . Suppose we would like to have a geometry well-approximated by pure AdS 5 × S 5 down to some radial coordinate r * r 0 . Our approach will be to write down the metric functions in the limit l A r , k B r → 0 with A c A l A and B d B k B held fixed, and to understand how these functions depend on the quantity we find that when l A r , k B r → 0 in a way that keeps N and ε fixed, we have We then find the metric function Of course, in the limit ε → 0, we recover the metric function for pure AdS 5 × S 5 . One can demonstrate directly from the above expressions that these metric functions can be made uniformly close to those of pure AdS 5 × S 5 in r ∈ [r * , ∞) and θ ∈ [0, π 2 ] for sufficiently small ε; we have plotted some examples in Section 4.3.

C Space of solutions for the single pole case
In this section, we will understand the space of solutions to the constraints (4.1).
First, taking a linear combination of the last two equations, one obtains Choosing any N N S5 and N D5 satisfying this constraint, the linear diophantine equation (C.1) for K and L will always have multiple integer solutions of the form with (L 0 , K 0 ) some nominal solution.
There will be at least one solution for positive K and L, since for real m, (C.3) parameterizes a line that intersects the positive quadrant of the (K, L) plane, and the equal spacing between the (K, L) values for integer m is less than the length of the line segment in the positive quadrant: The number of solutions for (K, L) is evidently of order so for N D5 , N N S5 N we typically have only a single solution. The conditions that K and L are positive combined with (C.1) mean that any solution will satisfy Now, given any choice of (N D5 , N N S5 ) satisfying (C.2) and positive (K, L) satisfying (C.1), we will show that there is a unique positive (k, l) satisfying the constraints (4.1). We do so by combining these constraints to yield The right side increases monotonically from 0 to 1 as k/l increases from 0 to ∞. The left side varies monotonically from K/N D5 > 0 at k/l = 0 to 1 − L/N N S5 for large k/l. Thus, there is exactly one solution for k/l. Call this k/l = m.
We then have a unique solution (k, l) that is the intersection between the line k = ml and the line k In terms of m, the result is D General families with single D5-pole/NS5-pole and arbitrarily large AdS 5 × S 5 region We will here provide a significant generalization to the one-parameter family initially introduced in Section 4.3. Our construction of a one-parameter family analogous to the one appearing in that section occurs most simply when g is such that there exists m ∈ N + with , with m, a, b positive integers and 0 < a b < 1 in reduced form. In this case, we will take N D5 (n) = bf n + α , N N S5 (n) = bmf n + β , where f n is a sequence which we leave undetermined for now. We then see that so to ensure that (C.1) holds, we would like to ask whether or not it is possible to choose α, β, γ, δ such that 0 = (a − b)mα + bγ − aβ + bmδ N = αγ + βδ − αβ .

(D.4)
In fact, these equations are solvable for any (a, b, m). In particular, substituting the former into the latter yields We know that gcd ((a − b), b) = 1, since a and b were chosen to be relatively prime, so this linear diophantine equation has an integer solution (α, δ). We may then define which is manifestly integral.
We thus define the sequence of parameters (N D5 (n), N N S5 (n), L(n), K(n)) by this choice (α, β, γ, δ), taking f n to be any growing sequence. Since √ gN D5 l + 1 √ g N N S5 k = N implies that both l and k are at most O f −1 n , the equations (4.1) yield and thus It follows that as desired.
Thus, in the case that the string coupling g satisfies (D.1), we are able to identify a one-parameter family with scaling It is notable that such g are dense in R + , since the map tan π 2 (·) : (0, 1) → (0, ∞) is a continuous bijection, implying that the image of a dense set in this function is dense. We should therefore be able to extend the above result by considering sequences of suitable rational approximations.
Indeed, suppose that we fix arbitrary g and take as ansatz the linear scaling with z ∈ R + any fixed positive constant. In this case, requiring (B.2) to be satisfied implies 14) and given the relationship between linking numbers and SUGRA parameters (and the assumption that l, k will be suppressed), this would appear to require We would like to construct a sequence of parameters (N D5 (n), N N S5 (n), L(n), K(n)) exhibiting the scaling that we have suggested, subject to the requirement that these parameters must be positive integers. The most natural way to approach this is to take sequences of rationals an bn , pn qn in reduced form such that a n b n → 2 π arctan(z/g) , p n q n → z , (D. 16) and then define 12 L(n) = a n p n f n + γ n , γ n = o(a n p n f n ) where f n is left undetermined for the time being. Equation (C.1) then implies N D5 (n)L(n) + N N S5 (n)K(n) = (b n q n f n + α n ) (a n p n f n + γ n ) + ((a n − b n )p n α n + b n q n γ n − a n q n β n + b n p n δ n ) f n + α n γ n + β n δ n − α n β n .

(D.18)
For any fixed n, this is precisely the same as (D.3), which we found to be consistent with the requirement N D5 L + N N S5 K = N + N D5 N N S5 for suitably chosen (α, β, γ, δ). Consequently, we may here find (α n , β n , γ n , δ n ) which make our definitions of the parameters consistent with this equation for each n; once we have defined (a n , b n , p n , q n ) and (α n , β n , γ n , δ n ) in this way, we may then simply choose a sequence f n which scales sufficiently quickly such that we recover the necessary asymptotics The sequence of solutions that we have defined will then have the desired asymptotic suppression of max{l, k} and |cl − dk|, as can be shown in a manner identical to that discussed above.

E Nearby solutions with multiple poles
It is reasonable to expect that the precise form of our boundary condition, and in particular the linear quiver from which our boundary condition descends, can be relaxed somewhat, and indeed we expect that the broad geometrical features of the holographic description, including the existence of a large AdS 5 × S 5 region, should be robust to certain "small" deformations of this quiver. As a concrete example, we may consider a family of solutions (the simplest family constructed earlier in Appendix D) with parameters of the form here, α, β, γ, δ are constants chosen to satisfy N = αγ + βδ − αβ, and the constants a, b, z satisfy Each element of this sequence corresponds to a quiver of the form provided in Figure 5.
We will now consider deforming these quivers for each n by coupling an additional s(n) fundamental hypermultiplets to the (L + 1) th node of the quiver, where s(n) may scale with n but we require s(n) = o(n). In this deformation, we have two stacks of D5-branes and two stacks of NS5-branes with inequivalent linking numbers, described by the parameters and L 1 = azn + γ , L 2 = azn + γ + 1 , At leading order (namely at order O(n)), (2.4) gives the conditions π 2 a = a tan −1 (l 1 /k 1 ) from which we can infer and thus from (3.8) and Since max{l A , k B } and ∆ are again suppressed for large n, we find that we recover the desired geometrical features in this limit. In particular, while we now have two D5-brane throats and two NS5-brane throats, the total D5-brane and NS5-brane charges are approximately the same as before, and the separation between each pair of 5-brane throats in this case is subleading in n, It is straightforward to show that a similar argument can be applied to a more general version of this deformation, where we couple o(n) fundamental hypermultiplets at each of O(n 0 ) nodes in the quiver, where the location of these nodes relative to the left endpoint of the quiver scales proportionally to the overall size of the quiver with n.
Another deformation of interest involves coupling an additional small quiver to the left endpoint of our initial quiver, i.e. the endpoint opposite that which is coupled directly to the 4D theory upon imposing our field theory boundary condition. Here, "small quiver" refers to a quiver described by an O(1) number of parameters (N . We can couple the large and small quivers together via bifundamental matter coupled to an extra U (m) node, where m is also dominated by our initial parameters; the result will be a good quiver, provided that the small quiver is good. This procedure results in a boundary condition described by many distinct parameters, which we can denote by (N  N S5 for A < p and B < q, we see that the leading behaviour of l p , k q will be the same as before the deformation. Moreover, the remaining equations for the linking numbers imply for A < p and B < q. Consequently, the newly added parameters are suppressed compared to l, k, and contribute to ∆ at subleading order; we therefore arrive again at a solution with a large AdS 5 × S 5 region.

F Multi-wedge generalizations
Our goal in this section is to understand how to construct theories whose holographic description involves several wedges of AdS 5 × S 5 connected by interface branes; this applies to the BCFT case as well as the case involving 3D SCFTs which descend from linear or circular quiver gauge theories. The intuition behind our construction is illustrated in Figure 8.
Our construction in this section will begin with a list where the linking numbers are listed in increasing order. We will define these via the brane configuration depicted in Figure 8; we have "blocks" with large numbers of D5-branes and NS5-branes N N S5 , L A , K A ) which parametrize the A th block may be constructed in a completely identical manner to the construction of the one-parameter families we considered in Section 4.3 and Appendix D, with the simple replacement N → (m A − m A−1 ); in particular, the linking numbersL A ,K A that we would obtain from that construction will be related to the correct linking numbers L A , K A in the full quiver of the present construction by since we need to account for the fact that the linking numbers depend on the quantities of 5-branes present in previous blocks. Ultimately, we will take all of the m A (and the number of blocks p) to be O(1) in some large parameters which will determine the number of 5-branes and linking numbers in the A th block. and the linking numbers are increasing by construction, We will also require that f (A) n scales sufficiently quickly relative to f (A−1) n such that the parameters in block A scale at least as quickly as the parameters in block A − 1. We can now consider how the SUGRA parameters behave for each case. as well as the relations to SUGRA parameters (F. 13) Comparing these expressions at leading order, we see that consistency is achieved by requiring We therefore find as desired.
We have demonstrated that our construction thus far possesses a large AdS 5 × S 5 asymptotic region; to ensure that we recover a multi-wedge deep in the interior, we will actually consider a subset of the families defined so far for which the parameters (N N S5 ,L A ,K A ) of block A are all taken to scale with the same large parameter as the parameters of block A−1, as opposed to scaling strictly faster. Note that the "doubled" construction of Section 6.1 is an example of this choice. In this case, it suffices to note that for l A , k A r l A+1 , k A+1 , we find the leading behaviour of h 1 , h 2 to be N S5 and √ gl B ∼ z B √ g k B , the geometry in this region is approximately that of AdS 5 × S 5 , where the value of r 0 is proportional to the geometric mean of l A (or k A ) and l A+1 (or k A+1 ), and the AdS radius in this wedge scales relative to the AdS radius in the asymptotic region as L 4 wedge /L 4 ∼

F.2 Multi-wedge dual of SCFT: linear quiver
As at the end of last section, we will continue to restrict to the case where the linking numbers and charges for each block are all taken to scale with the same large parameter. The linking numbers L A , K A are related to parameters N It is immediate that we obtain the desired behaviour in this case, since this system of equations is identical to the system from the BCFT case up to sub-leading terms if we identify l A ↔ e −δ A and k B ↔ e −δ B , and the definitions of h 1 , h 2 will have the same leading behaviour in the regions of interest.
on a torus with modular parameter τ = it/π.
The linking numbers and supergravity parameters are now related by L. Again, in the region δ A Re(z) δ A+1 , the harmonic functions h 1 , h 2 agree with those from the linear quiver case at leading order, since the additional contributions coming from the n = 0 terms will be suppressed.