CCosmology from confinement?
Mark Van Raamsdonk
Department of Physics and Astronomy, University of British Columbia,6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada.
E-mail: [email protected]
Abstract:
We describe a class of holographic models that may describe the physics of certainfour-dimensional big-bang / big-crunch cosmologies. The construction involves a pair of 3DEuclidean holographic CFTs each on a homogeneous and isotropic space M coupled at eitherend of an interval I to a Euclidean 4D CFT on M × I with many fewer local degrees offreedom. We argue that in some cases, when the size of M is much greater than the lengthof I , the theory flows to a gapped / confining three-dimensional field theory on M in theinfrared, and this is reflected in the dual description by the asymptotically AdS spacetimesdual to the two 3D CFTs joining up in the IR to give a Euclidean wormhole. The Euclideanconstruction can be reinterpreted as generating a state of Lorentzian 4D CFT on M × timewhose dual includes the physics of a big-bang / big-crunch cosmology. When M is R , wecan alternatively analytically continue one of the R directions to get an eternally traversablefour-dimensional planar wormhole. We suggest explicit microscopic examples where the 4DCFT is N = 4 SYM theory and the 3D CFTs are superconformal field theories with oppositeorientation. In this case, the two geometries dual to the pair of 3D SCFTs can be understoodas a geometrical version of a brane-antibrane pair, and the tendency of the geometries toconnect up is related to the standard instability of brane-antibrane systems. a r X i v : . [ h e p - t h ] F e b Introduction
In this note we describe specific holographic constructions through which the physics of 4Dbig-bang/big-crunch cosmologies might be encoded in the physics of certain non-gravitationalquantum field theories. We follow the construction of [1–3] (reviewed below; see also [4–6]for related low-dimensional constructions), which considers states of a 4D holographic CFTconstructed using a Euclidean BCFT path integral. These states were suggested to be dual toasymptotically AdS black hole spacetimes with a dynamical end-of-the-world (ETW) braneproviding an inner boundary for the spacetime behind the horizon of the black hole. In [1]was suggested that in favorable cases, gravity can localize to this ETW brane, so that theeffective description of the ETW brane physics is that of a four-dimensional big-bang/big-crunch cosmology.The present work refines and extends this picture in the following ways: • We point out that the physics of confinement and symmetry breaking plays a crucial rolein the relevant field theories. The construction relies on four-dimensional field theorieswith one compact direction flowing in the infrared to three-dimensional gapped/confinedtheories, with a particular pattern of global symmetry breaking. • We suggest specific microscopic examples constructed from N = 4 SYM theory andvarious 3D superconformal field theories that can be coupled to it at a boundary. Thedual gravity picture, including the ETW brane physics, is described in terms of typeIIB supergravity / string theory, and involves the physics of brane-antibrane systems. • We emphasize that the construction may continue to work when the 4D theory is nota conventional holographic theory (but the boundary theories involved in constructingthe state are). In this case, there can be a classical gravitational description for theETW brane (as a 4D theory of gravity, perhaps with a compact internal space) but noclassical bulk 5D spacetime. The encoding of a cosmological spacetime in the state ofa non-holographic CFT is similar to the encoding of black hole interiors in Hawkingradiation systems [8–10].While we don’t attempt to construct the relevant supergravity solutions in detail, weare able to describe the asymptotic behaviour explicitly. It remains to check (or argue indi-rectly) that the proposed solutions exist and have the conjectured properties. Alternatively,we can hope to understand better the 4D effective description of the ETW physics and verify Related comments were made in the context of Euclidean wormholes in [7]. – 2 – igure 1 . Basic field theory construction (the CFT sandwich): a pair of 3D holographic CFTs relatedby a reflection are coupled at either end of an interval I to a 4D CFT. that the desired solutions relevant to cosmology exist there. If the construction succeeds,an analytically-continued version gives four-dimensional eternally traversable wormholes pre-serving 2+1 dimensional Poincar´e symmetry in the effective description. It has been arguedthat the existence of these would require an unnaturally large amount of negative null energy[11, 12]. We review these arguments in Section 4 and identify a novel field theory effect thatgives a possible mechanism for achieving the large amount of negative null energy requiredto support the wormhole.These models have some interesting phenomonological/model-building aspects that wediscuss in Section 5. However, we emphasize that the immediate motivation here is not tocome up with a phenomenologically accurate model of cosmology, but rather to come up with some completely defined physical theory which encodes the cosmological physics of a four-dimensional homogeneous and isotropic universe with a big bang. If these constructionssucceed, they could shed light on the question of what are the well-defined observables incosmological spacetimes and allow a first principles calculation of these observables assumingthe holographic dictionary is understood well enough and the field theory calculations can bedone. For example, cosmological correlators could be computed from correlation functions ina four-dimensional Euclidean field theory with boundaries at some past and future Euclideantime.
Summary of the basic construction
We begin by describing the basic mechanism of the construction, considering the case wherewe wish to describe cosmology with spatial geometry R . The construction is essentiallythe same when the spatial geometry is spherical or hyperbolic. Following [18], we begin byconstructing a Euclidean wormhole.To start, consider a pair of three-dimensional Euclidean holographic CFTs each living on R . Each of these is dual to a separate four-dimensional Euclidean gravitational theory onAdS with boundary geometry R . We now introduce an interaction between the theories by For other approaches to cosmology using holography, see for example [13–17]. – 3 – igure 2 . Dual geometries for various field theory setups, showing end-of-the-world branes (red)with asymptotically AdS regions. (a) Dual of a single 3D CFT (b) Dual of a 3D CFT coupled tothe boundary of a 4D CFT. Gravity remains well-localized to the ETW brane when c D (cid:28) c D .(c) Possible dual of a pair of 3D CFTs coupled to a 4D CFT, where the IR physics is a conformal3D CFT (d) Possible dual of a pair of 3D CFTs coupled to a 4D CFT, where the IR physics is agapped/confining 3D theory. coupling them to an auxiliary four-dimensional quantum field theory on R times an interval[ − τ / , τ / We take the four-dimensional theory to have many fewer local degrees offreedom than the original 3D theories. In particular, the four-dimensional theory need notbe a conventional holographic theory. Since the fourth dimension is compact, the field theorywe have constructed will flow to some three-dimensional theory in the IR, which provides agood description of the physics at distance scales much larger than τ . This theory could bea non-trivial three-dimensional conformal field theory, but more generically, we expect thatit will be gapped/confining. We will argue that in some cases, the gravitational descriptionof this confinement is that the asymptotically AdS spacetimes associated with the two 3DCFTs join up in the IR so that the full spacetime is a Euclidean wormhole.To motivate this assertion, consider the case where the four-dimensional auxiliary theoryis also holographic. In this case, coupling one of the 3D holographic theories to the auxiliary4D system gives a holographic boundary conformal field theory. In the dual description, thefour-dimensional gravitational theory dual to the 3D CFT now describes the physics of anend-of-the-world brane in a five-dimensional geometry (Figure 2b). When the 4D theory hasmany fewer local degrees of freedom than the 3D theory ( c D (cid:28) c D ), gravity localizes tothis ETW by the Karch-Randall mechanism [19–21], and the 4D graviton gets a mass that In other words, we make a CFT sandwich. – 4 – igure 3 . Connection to cosmology. (a) State of the 4D CFT on R produced by the Euclidean pathintegral terminated by a 3D CFT b in the Euclidean past at τ = − τ . (b) τ < τ = 0 slice of the Euclidean solution servesas the initial data for Lorentzian evolution. (d) Full Lorentzian solution dual to | Ψ (cid:105) b,τ . can be made arbitrarily small by taking c D /c D small. In our construction with two 3Dtheories on either end of an interval, we have two ETW branes in the UV. But if the full fieldtheory is gapped in the IR, the dual geometry must be capped off somehow in the IR [22],and a natural mechanism for this is for the two ETW branes to join up (Figure 2d). Wewill provide evidence for this picture via a string theory construction, where the 4D auxiliarytheory is taken to be N = 4 SYM theory and the 3D theories are holographic superconformaltheories with opposite orientation. In this case, the ETW branes are related to a certainbrane-antibrane system in string theory, and the tendency for the ETW branes to join upis directly related to the instability of the brane-antibrane system. From the field theoryperspective, this situation is characterized by a spontaneous breaking G × G → G of globalsymmetry, where G is the global symmetry associated with each of the 3D CFTs and becomesa gauge group for gauge fields on the ETW brane.To connect with cosmology, we interpret the τ direction as a Euclidean time directionand interpret the Euclidean theory for τ < | Ψ b,τ (cid:105) of our auxiliary 4D theory living on a spatial R (Figure 3a). Here, b labels our choiceof 3D theory. Note that the degrees of freedom of this 3D theory are not physical degrees offreedom in the Lorentzian theory, but appear only in the Euclidean path integral generatingthe state | Ψ b,τ (cid:105) . This state (evolved with the usual Hamiltonian for the 4D theory) is dual toa Lorentzian geometry that is the analytic continuation of the Euclidean wormhole describedabove (see Figure 3b,c,d). This will generally be a flat FRW big-bang/big-crunch cosmology.In the case where the 4D field theory is holographic, the 4D cosmological physics is confinedto an ETW brane living at the IR end of a five-dimensional asymptotically AdS spacetime. As we discuss below, there are more general possibilities for which the effective description of the ETWbrane physics is not a single 4D Euclidean wormhole. This is similar to the Hartle-Hawking construction [23], but with a CFT path integral. In this context, the Euclidean wormhole is interpreted as a “bra-ket wormhole”. – 5 – igure 4 . CFTs on R , times S , with each CFT covering an interval on the S . A holographicmodel suggests that the negative Casimir energy of the CFT with larger central charge can becomemuch larger than that for this CFT on R , × S for special choices of the interface corresponding toa bulk interface tension close to a lower critical value in the holographic model. This lies behind the horizon of a (planar) black hole, emerging from the past singularity andending up in the future singularity (Figure 3d). But when the 4D theory is not a conventionalholographic theory, there is no geometrical 5D spacetime. In the language of [24–26], we canthink of the 4D cosmological spacetime is an “island” whose physics is encoded in the stateof a field theory that is not conventionally holographic. Eternally traversable wormholes
If they exist, the Euclidean wormholes we describe (for the case of spatial R ) could insteadbe analytically continued along one direction of the R to give a four-dimensional eternallytraversable wormhole (Figure 2d, but with one of the translationally invariant directionsanalytically continued to give a time direction). The existence of such solutions in the effectivedescription requires a substantial violation of the averaged null energy condition [28]. In the1+1-dimensional construction of Maldacena and Qi [29], this arises through a direct couplingof the CFTs associated with the asymptotic regions. It has been argued in [11] and [12] thatobtaining the required amount of negative null energy in a higher-dimensional construction isdifficult. We recall these arguments in detail in Section 4, and explain a possible mechanismto produce the required negative energy in the effective description. We argue that thematter in the effective description can be modeled as an interface theory as shown in Figure4; a holographic model suggests that such setups can lead to large negative Casimir energydensities for interfaces with specific properties. This will be discussed in more detail in [30]. This is similar to the ideas in [27], which argued that bubbles of a spacetime associated with someholographic CFT can be encoded in states of a different CFT, which might have a significantly smaller centralcharge. – 6 – igure 5 . (a) Probe brane solution dual to N = 4 SYM with parallel D5-brane defects. (b)Probebrane configuration for parallel defects with opposite orientation (D5- ¯ D Microscopic construction
The generalities of our construction are motivated and described more fully in section 2 below.In order to make everything more concrete, we discuss a possible specific realization of theconstruction within string theory in section 3. Here, the field theory arises as the low-energylimit of D3-branes stretched between a D5-brane/NS5-brane stack and a complementary D5-brane/NS5-brane stack, with extra D3-brane degrees of freedom added to the fivebrane stacksat either end (see Figure 8). The low-energy field theory description is U ( N ) N = 4 SYMtheory on R times an interval, coupled to 3D superconformal field theories at either end ofthe interval. These SCFTs are holographic with many more local degrees of freedom than the N = 4 theory. These SCFTs individually preserve half the supersymmetry when coupled tothe N = 4 theory. However, the full construction breaks supersymmetry. We argue that thetheory still has a gravitational dual well-described by type IIB supergravity, and we describethe asymptotic geometry explicitly using the work [31–34]. In these geometries, the ETWbranes are geometrical, characterized by an internal space which grows in size before pinchingoff smoothly (Figure 11). We can think of them as a geometrized stack of branes emergingfrom one boundary and a geometrized stack of the corresponding anti-branes emerging fromthe other boundary. The ETW branes connecting up would then be a non-perturbativegeometrized version of the joining of probe D5 and anti-D5 branes associated with paralleldefects in the N = 4 theory (Figure 5) [35–37].– 7 – General construction
In this section, we describe and motivate the general construction in more detail before turningto the specific microscopic construction in section 3.
Our goal is to construct models of big bang cosmology using the tools of AdS/CFT. Malda-cena and Maoz pointed out that certain big-bang / big-crunch spacetimes arise by analyticcontinuation from Euclidean AdS wormholes, with geometry of the form ds = dτ + f ( τ ) ds M (2.1)where M is a homogeneous isotropic space and the geometry is asymptotically AdS for τ →±∞ . The form of this Euclidean geometry suggests that it could be related holographicallyto a pair of Euclidean CFTs, each living on M . However, for a pair of decoupled CFTs, thepartition function and all correlation functions would factorize between the two CFTs, whileholographic calculations in the geometry (2.1) would give non-factorizing results.There is another puzzle with the geometries in (2.1) that applies to the flat case. Here,we could analytically continue one of the spatial directions in M = R to obtain a staticLorentzian geometry with two asymptotically AdS regions. Such a planar traversable worm-hole geometry cannot exist without violating the averaged null-energy condition (ANEC)[28]. To resolve these puzzles, it has been suggested that Euclidean AdS wormholes may corre-spond to ensemble-averaged products of CFT partition functions (see e.g. [18, 38–42]), orpartition functions for CFTs that are weakly interacting in some way (see e.g. [7]). Either ofthese can explain the non-factorization of correlators, and for the flat case, it is understoodthat introducing interactions between the CFTs associated with asymptotic regions can giveANEC-violating matter in the bulk that allows a traversable wormhole[29, 43, 44]. A specific approach that incorporates features of ensemble averages and interactions isto couple the original CFTs to some auxiliary degrees of freedom spread over an extra spatial Note, however that there has not been an explicit construction of an eternal traversable wormhole in fourdimensions with R spatial slices. – 8 –imension [3]. Specifically, we can consider a four-dimensional CFT on M times a spatialinterval [ − τ / , τ / τ = ± τ coupled to our original CFTs (Figure 1). Thepartition function for the full theory can be understood as a product of partition functionsof the original CFTs, averaged over an ensemble of sources [3]. Here, the sources are fieldsin the auxiliary theory and the probability distribution for the sources comes from the pathintegral over the auxiliary degrees of freedom.In order that the dual gravitational system associated with the coupled CFTs remainseffectively four-dimensional, we require that the number of local degrees of freedom in theauxiliary CFT is small compared to the number of local degrees of freedom in the originalCFTs. In this case, the addition of auxiliary degrees of freedom can be understood to be asmall perturbation of the original theory, at least in the UV. Localized gravity and Karch-Randall branes
To understand the effects on the gravitational physics from coupling to a 4D auxiliary CFT,it is helpful to consider the case where these auxiliary degrees of freedom are also holographic.Consider first the case where we have a single 3D holographic CFT which we couple to a four-dimensional CFT on a half-space. In this case, the full dual geometry has an asymptoticallyAdS region, and the original four-dimensional gravitational theory describes the physics ofan end-of-the-world brane, as shown in Figure 2b. Gravity is localized to this end-of-the-world brane via the Karch-Randall mechanism [19]. Specifically, the physics of the ETWbrane has an effective description as four-dimensional gravity, where the 4D graviton obtainsa tiny mass ( m ∼ c /c ), and we have a tower of massive fields coming from the modes of the5D-graviton [47, 48]. In a bottom-up description, we can think of the brane as a hypersurfaceliving at some angle θ in the Poincar´e coordinates of AdS × S . When this angle is close to − π/ See [45] for a related construction involving the coupling of two theories via an auxiliary system. Below, it will be important that such perturbations can significantly alter the IR physics. In microscopic examples, the full geometry can be understood as a warped product of AdS and an internalspace. When we have only the 3D CFT, the internal space is compact. Coupling to the 4D CFT on a halfspace modifies this compact space to include a narrow semi-infinite throat. . – 9 – equirement for strong IR correlations We have emphasized that the number of auxiliary degrees of freedom should be small in orderto maintain the four-dimensional character of the dual gravitational theory. However, endingup with a connected wormhole means that the interactions between the two CFTs inducedby the auxilary degrees of freedom lead to large correlations. In particular, we require thatln( Z/ ( Z Z )) ∼ c D . In the Lorentzian case where we have analytically continued one ofthe directions of M = R , the geometrical connection between the two sides implies that thevacuum entanglement between the original CFTs induced by the auxiliary degrees of freedomis large, with entanglement entropy of order c D . Thus, we wish to introduce an auxiliary theory whose number of degrees of freedom issmall, c D (cid:28) c D , but which leads to entanglement/correlations between the original CFTsthat are large, of order c D . In the next section, we will argue that the physics of RG flowsand confinement may help achieve this. Our general Euclidean field theory setup has two 3D CFTs on a homogeneous and isotropicspace M coupled to a 4D CFT on M times an interval I τ = [ − τ / , τ / τ , our field theory has a dual gravitational description as a 4DEuclidean AdS wormhole. We expect that the correlations between the original CFTs willbecome larger for smaller values of τ , so it is natural to consider the small τ limit and askwhether the wormhole exists here. Since the field theories we are dealing with are assumedto be scale-invariant, we can equivalently keep τ fixed and take the curvature length scaleof M large, so that our field theory geometry approaches R × I . If the wormhole exists inthis flat case, it should also exist in the spherical and hyperbolic cases for sufficiently smallspatial curvature.At length scales much larger than τ , the Euclidean field theory on R × I τ or the relatedLorentzian field theory on R , × I τ should be described by some three-dimensional fieldtheory. The IR limit of this theory could either be a non-trivial 3D CFT, or it could be agapped theory. We will now argue that the latter may correspond to some type of connectedETW brane wormhole geometry in the dual gravitational description. To regulate the entanglement entropy, we can consider a subsystem including a ball-shaped region of oneof the 3D theories and compare the entanglement entropy of this region to the entanglement entropy for thesame region in the case where there is not a second 3D CFT. – 10 – igure 6 . Two possibilities for the gravity dual of a theory that confines in the IR. These cansometimes be distinguished by a different pattern of global symmetry breaking. The ETW branegeometry in the left case is a Euclidean wormhole.
To see this, it is helpful again to consider the case where the auxiliary degrees of freedomare holographic. In this case, we have a dual geometry with an asymptotically AdS × S region whose boundary geometry is R × I τ , and we have ETW branes in the bulk anchoredto the ends of the interval.In the case where the IR theory is conformal, we have non-trivial physics at arbitrarilylong wavelengths, and the bulk picture is that the radial direction extends to infinite distancein the IR. Here, the ETW branes can remain separate and extend infinitely into the IR (Figure2c). Alternatively, they could join up somehow and extend into the IR. In neither case do weget the desired wormhole geometry.On the other hand, when the field theory is gapped in the IR, we expect that the radialdirection should terminate somehow in the IR at a finite distance from any interior point ofthe geometry. A natural way for this to occur is for the two ETW branes to join up intoa single brane (Figure 6, left). In this case, the ETW brane worldvolume geometry is thedesired asymptotically AdS Euclidean wormhole.The configuration of Figure 6 (left) is not the only way to end up with a confining theory.Indeed, adding a relevant deformation to the individual 3D theories, or to the 4D CFT couldlead to confinement. In this case, the ETW brane geometries and the bulk geometry couldindividually truncate in the IR, as shown in Figure 6, right. Here, the effective 4D descriptionof the ETW brane physics on the gravity side would have two disconnected asymptoticallyAdS spacetimes with an IR end.In order to ensure a single connected ETW brane, one strategy is to endow the ETWbrane with properties similar to those of a string theory brane, such that the pair of ETW– 11 – igure 7 . States created by Euclidean path integrals and gravity duals. (a) Thermofield double stateof two 3D CFTs, dual to a two-sided black hole. (b) State of a pair of 3D CFTs coupled by a 4D CFT.The two-sided black hole is now the geometry of an ETW brane. The black hole may be traversablefor some time. (c) Vacuum state of the 3D CFTs coupled by a 4D CFT, if the ETW brane remainsconnected and gives an eternally traversable wormhole in the effective description. branes acts like a probe brane-antibrane pair. For example, we can take the 3D CFTs to eachinclude some global symmetry G and the theories to be related to each other by reflectionthrough τ = 0. The G × G global symmetry in the UV is related to the presence of bulk gaugefields associated with the ETW branes. If the branes connect up in the bulk the G × G globalsymmetry is broken to a single diagonal copy. We will provide explicit examples below. Before turning to specific microscopic models, we motivate the existence of a connected worm-hole in a different way. Here, we focus on the Lorentzian picture, where we would have aneternally traversable wormhole after analytically continuing on one of the directions of the R . Consider first the pair of 3D CFTs on spatial R in the thermofield double state. This is– 12 –ual to the single connected geometry of a two-sided planar 4D black hole. This state may beconstructed using a Euclidean path integral with path integral geometry R × I that connectsthe two spatial R s, as shown in Figure 7a.Next, we can consider coupling the 3D CFTs via a 4D CFT as above. For the coupledtheory, we can consider the path-integral state shown in Figure 7b. We can choose to evolvethis state forward using the time independent Hamiltonian for the 4D theory on spatial R × I .When the 4D CFT has many fewer degrees of freedom than the 3D CFTs, we expect that thegravitational description of the new state is in some sense a small perturbation of the originaltwo-sided black hole geometry. In particular, we expect that the ETW brane geometry forthe t = 0 spatial slice should be almost the same as that of the two-sided 4D planar blackhole and the ETW branes from the two different 3D CFTs should still connect. When the4D theory is holographic, we can visualize the full geometry as having a 5D bulk such thatthe original 4D black hole becomes an ETW brane in this geometry. The presence of the4D CFT may alter the time-dependence of the ETW brane. Since it represents a couplingbetween the original 3D theories, it can have the effect of making the ETW brane geometrytraversable. However, the state is still time-dependent.By continuously modifying the path integral geometry to the strip geometry of figureFigure 7c, we end up with the vacuum state of the theory with the two 3D CFTs coupled bythe 4D auxiliary theory. In the case we are interested in, the ETW brane geometry wouldremain connected in the limit where we reach the vacuum state. Since the final dual geometryis static, the ETW brane geometry should be an eternally traversable wormhole, and afteranalytic continuations give a Euclidean AdS wormhole and a flat cosmological spacetime.Of course, for some theories, it could be that the ETW brane disconnects in the limitwhere the state approaches the vacuum state. Our goal is to find examples where the ETWbrane remains connected in this limit. In this section, we describe a family of specific microscopic constructions designed to realizethe picture we have described. In order to have the largest amount of control, we take asbuilding blocks quantum field theories with large amounts of supersymmetry, though thissupersymmetry will end up being broken in the final construction.For simplicity and maximal control over the gravity picture, we start by choosing the U ( N ) N = 4 SYM theory as the 4D CFT that gives our auxiliary degrees of freedom. Here,– 13 – igure 8 . Left: Brane construction for N = 4 SYM coupled to a 3D superconformal gauge the-ory. D3-branes in the 0123 directions (horizontal, red) are streched between D5-branes in the 0456directions (vertical, black) and NS5-branes in the 0789 directions (blue, angled). Right: quiver gaugetheory describing the low-energy physics. Circles represent circles represent gauge theory sectors cou-pled by bifundamental hypermultiplets (horizontal lines). Squares represent additional fundamentalhypermultiplets. N (= c D ) and the ’t Hooft coupling λ can both be taken large if we wish to have a 5Dgravity dual, but we can also consider the case where they are not large.Next, we introduce the 3D holographic CFTs. We take these to be superconformaltheories that can be coupled to the N = 4 theory at a boundary while preserving halfof the original supersymmetry of the N = 4 theory. Such theories preserve OSp (2 , | U ( N ) N = 4 theory describesthe low-energy physics of N semi-infinite D3-branes (in the 0123 directions) ending on stacksof D5-branes (in the 123456 directions) and NS5-branes (in the 123789 directions), withadditional D3-branes stretched in the 3 direction between the D5s and NS5s, as shown inFigure 8. The 3D theory on its own corresponds to the physics of these extra D3-branes.Since we are free to add an arbitrarily large number of these, we can take c D /c D as largeas we want.The 3D theories can also be understood as the IR limit of certain supersymmetric quivergauge theories of the type shown in Figure 8 (right). The parameters describing the quiver –the ranks of the gauge groups and the number of fundamental hypermultiplets – are relatedto the numbers of D5-branes, NS5-branes, and D3-branes in the string theory construction.The dual gravity solutions for both the 3D SCFTs and the BCFTs obtained by couplingthese to the N = 4 theory are known explicitly. These solutions of type IIB supergravitywere described in [31–34], based on the general OSp (2 , | N = 4 SYM theory to couple to a 3D CFT at each endof the Euclidean time interval. From the string theory perspective, we can obtain such a– 14 –heory by introducing additional stacks of D5-branes and NS5-branes so that the D3-branestack has two boundaries. The distance between the boundaries can be scaled so that weend up with a finite separation between the two boundaries in the low-energy limit. We canpreserve supersymmetry if the new D5-branes and NS5-branes have the same orientation asthe original ones. However, we instead want to take them to have the opposite orientation,i.e. to use anti-branes instead of branes. From the field theory perspective, what we want isto take the same Euclidean SCFT at either end of the Euclidean time interval, but coupledwith the opposite orientation to the N = 4 theory. The reason is that we want a constructionthat is symmetric under Euclidean time reversal. This is required for our interpretation ofthe Euclidean geometry as a bra-ket wormhole, and ensures that we obtain a real Lorentziangeometry under analytic continuation. Choosing the boundary theories to have the oppositeorientation breaks supersymmetry in the resulting low-energy field theory, but only nonlo-cally, due to boundary conditions in the N = 4 theory that are mutually incompatible withsupersymmetry.We will argue that the resulting non-supersymmetric theory should be gapped in theIR, and that the dual gravity interpretation has a connected ETW brane whose effectivedescription can be a four-dimensional Euclidean AdS wormhole. Before discussing the complete two-boundary construction, let us describe more explicitlythe dual gravitational physics of the single-boundary theories that preserve supersymmetry,following [31–33, 51].The bosonic symmetry of the full BCFT includes the SO (3 ,
2) 3D conformal symmetryplus an SO (3) × SO (3) subset of the original SO (6) R symmetry. Accordingly, the dualgeometries takes the form of AdS × S × S fibered over a two-dimensional space. In general,we can write the metric as f ( r, θ ) ds AdS + f ( r, θ ) d Ω + f ( r, θ ) d Ω + 4 ρ ( r, θ )( dr + r dθ ) (3.1)where r and θ are polar coordinates on the first quadrant of a plane. This is illustratedin Figure 9. The metric functions f , f , f , and ρ are determined by a pair of harmonicfunctions h , ( r, θ ) , h ( r, θ ), and these are determined by choosing the locations { l A } for a setof poles of h on the x axis and the locations { k B } for a set of poles of h on the y axis, wheremultiplicities are allowed. The explicit form of the metric for these solutions is reviewed inAppendix A; see the references [31–33, 51] for more details, including the expressions for theother supergravity fields. – 15 – igure 9 . Left: Geometries dual to N = 4 on a half-space. Each point in the quadrant has an AdS × S × S fiber. Curves connecting the axes are topologically AdS × S . Poles on the x and y axis correspond to D5 and NS5-brane throats. Right: Full geometry is well-approximated byPoincar´e-AdS away from the dark grey shaded region, where the internal space smoothly degenerates.This region can be understood as an end-of-the-world brane from the lower-dimensional perspective. The geometry is illustrated in Figure 9. At each point in the quadrant, we have an
AdS × S × S fiber, where the volumes of the three factors can vary independently. Thefirst and second S volumes go to zero for θ = 0 and θ = π/ D x -axis poles) or N S y -axis poles) in the geometry. The pair of S s fibred over a curve connectingthe two axes (e.g. a constant r curve) gives a geometry that is topologically S . For large r , the curves of constant r describe AdS × S slices of the local AdS × S geometry thatdescribes the asymptotic region. These are the slices of fixed Poincar´e angle, as shown on theright in Figure 9.In the region where the solution is well-described by Poincar´e AdS × S , the variable r is related to the angular coordinate Θ p in the τ − z plane in Poincar´e coordinates via rr = 1 − sin Θ p cos Θ p (large r ); . (3.2)For smaller values of r , the geometry deviates from AdS × S ; the r = 0 point correspondsto a smooth part of the geometry where the S contracts to zero size.The pole locations are constrained in the microscopic type IIB string theory by therequirement that the various fluxes originating from the fivebrane throats should be quantized.Specifically, we must have that L A = √ gl A + 2 π (cid:88) B arctan l A k B – 16 – B = k B √ g + 2 π (cid:88) A arctan k B l A (3.3)are integers for each A and B . These integers are related to the number of units of D3-braneflux per fivebrane in a given throat, and are directly related to integer parameters in thebrane or quiver pictures of Figure 8 specifying the gauge theory. Microscopic picture of the ETW brane
The geometry for a given microscopic 3D SCFT labeled by parameters { L A } , { K B } andcorresponding supergravity parameters { l A } , { k B } will contain a portion that is a good ap-proximation to the part of AdS × S with Poincar´e angle Θ p > Θ for some angle Θ thatis different for different parameter choices. . The remainder of the geometry (grey shadedregion in Figure 9) can be understood as a fat ETW brane in which the S contracts. Thispart of the geometry also includes the fivebrane throats. We can think of Θ as the ETWbrane angle.We will now argue that by choosing the boundary SCFT appropriately, we can findexamples with c D (cid:29) c D where Θ is arbitrarily close to − π/
2, so that our geometry includesan arbitrarily large portion of
AdS × S . In this case, the ETW brane is like a Planck branecutting off the asymptotic region on half the space, and we expect that gravity should be welllocalized on the brane.First, we note that the number and location of the poles determines the rank N of the N = 4 SYM theory gauge group, and the asymptotic AdS radius L by N = L π(cid:96) s = (cid:88) A l A + (cid:88) B k B . (3.4)For the solution specified by parameters { l A } and { k B } , we can expand the metric functionsasymptotically in r to verify that the solution asymptotes to AdS × S with these parametervalues. The same asymptotic behavior is obtained for many different choices of poles; thesechoices correspond to our choice of 3D SCFT.For fixed N , we find that the ETW brane angle Θ can be made to approach − π/ { l A } and { k B } small compared with r = √ N = (cid:115)(cid:88) A l A + (cid:88) B k B . (3.5) Here, Θ that depends on how closely we require the geometry to match with AdS × S – 17 –his requires taking a large number of poles. Since each pole corresponds to a fivebranein the brane construction, these cases correspond to having a complicated 3D SCFT with c D (cid:29) c D . Thus, we find that by choosing a boundary SCFT with many degrees of freedom, theeffective ETW brane tilts strongly outward so that it should behave like a Planck branecutting off half of the asyptotic region of Poincar´e-AdS. We expect gravity to localize on thebrane via the Karch-Randall mechanism. The resulting theory is expected to have an effectivedescription as the N = 4 theory on a half-space coupled to a theory on the other half-spacethat includes the gravitational theory dual to our 3D SCFT and a cutoff version of the N = 4theory.Even when Θ is not close to − π/
2, the ETW brane may be effectively described by 4Dgravity provided that c D (cid:29) c D . As argued in [46], this generally corresponds to a situationwhere the internal space volume in the ETW region becomes large before contracting; sucha geometry (shown in Figure 11 (left)) was described in [46] as a “bagpipe”. Here, the “bag”without the pipe is the internal space geometry for the dual of the 3D SCFT without the N = 4 theory. Coupling to the N = 4 theory adds the pipe, and this is very narrow comparedto the bag when c D (cid:29) c D . From the effective field theory perspective, the addition of thepipe gives the 4D graviton a small mass m ∼ c D /c D , and adds a tower of higher-massmodes coming from the 5D-graviton. But the physics is still a small perturbation to theoriginal 4D theory dual to our 3D SCFT. Next, we consider the case with two boundaries, where we have N = 4 SYM theory on R × I with 3D superconformal field theories of opposite orientation on either side of the interval. Inthis case, each SCFT preserves a different half of the supersymmetries of the N = 4 theory,so the full theory has no remaining supersymmetry. Before discussing this case, it will beuseful to understand also the case where the two SCFTs preserve the same supersymmetries. As a specific example (setting g = 1), we can take a pole with multiplicity N at location k = N/ (2 N )and a pole with multiplicity N at location l = N/ (2 N ). The flux quantization constraints (3.3) require that N/ (2 N ) + N / N (cid:29) N , the solution includes a region that is a goodapproximation to the portion of AdS × S with θ < π/ − (cid:15) , where (cid:15) = N / √ N . Note that since N can beas large as N , we can make (cid:15) parametrically small. – 18 – igure 10 . Left: geometry dual to 4D holographic CFT on R times a circle with antiperiodic bound-ary conditions for fermions (internal space is suppressed). Right: geometry dual to 4D holographicCFT on R times an interval, with SCFT of opposite orientation at either end. The blue ETW braneis described microscopically by a smooth degeneration of the internal space. Aside: wedge holography and a 3D dual for
AdS × S The two-boundary theories preserving supersymmetry case can be understood as arising froma string theory construction with D3-branes stretched between two separate stacks of D5-branes and NS5-branes, where we adjust the length of the D3-branes to remain finite inthe decoupling limit. In this case, the dual supergravity solution should include a wedge of
AdS × S with ETW branes on either side. In the IR limit, the field theory will flow to asingle 3D SCFT, namely the one associated with the string theory construction above whereall the fivebranes are taken to be coincident. Such SCFTs were discussed in [52], and provideexamples of the “wedge holography” discussed in [53], where a 3D CFT is dual to a wedge ofa 5D AdS space. It is interesting to note that, as for the single boundary case, by a judicious choice of theboundary SCFTs, our dual geometry can include an arbitrarily large wedge of
AdS × S ,i.e. a wedge − π/ (cid:15) ≤ Θ p ≤ π/ (cid:15) for arbitrarily small (cid:15) . Thus, we have in a sense a3D dual to AdS × S , though the full dual geometry also includes the ETW branes. From afield theory point of view, this suggests that the full physics of the N = 4 SYM theory maybe contained within an appropriately chosen 3D SCFT. This may be related to the idea ofdimensional deconstruction [54]. Supersymmetry breaking boundary conditions
We now return to the case of interest where the two boundary theories preserve non-intersectingsubsets of SUSY generators, so the full theory breaks all supersymmetry (as well as some of Note that because of the AdS geometry, the proper distance between the ETW branes actually remainsconstant as a function of the radial coordinate, so we can think of this as an example of ordinary holographywhere the internal space includes an interval. In the microscopic examples taking into account the spheres,the full internal space takes a dumbbell shape, with two “bags” connected by a narrow tube [52]. – 19 –he global symmetries present in the UV theory).Most well-controlled microscopic examples of AdS/CFT are supersymmetric, so one maybe concerned that the examples we have described cannot be studied holographically in acontrolled way. However, in our case, the supersymmetry is only broken by the fact that theboundary conditions at either end of the interval are incompatible with each other from aSUSY-perspective. A simpler example where we have SUSY broken by boundary conditions isthe N = 4 theory compactified on a circle with antiperiodic boundary conditions for fermions,introduced by Witten. Here, the theory has a well-controlled gravity dual in which the circlebecomes contractible in the bulk for the case where the noncompact directions of the fieldtheory are R . From the lower-dimensional perspective, this gives us a 3D confining gaugetheory [22] since the radial direction in the dual geoemtry has finite extent in the IR.Our situation is very similar to Witten’s example, except that the compact direction isan interval rather than a circle. Supersymmetry is broken by the boundary conditions, andwe expect that the interval contracts and pinches off in the bulk. This can happen smoothlyif the ETW branes originating from the two 3D SCFTs join up in the IR as shown in figure10. This implies that the full geometry is capped in the IR, and the IR physics of the fieldtheory is that of a confining/gapped 3D theory.It is also possible to get a gapped theory in the IR without the ETW branes connectingsmoothly. The two ETW branes and the bulk geometry between then could each terminateindependently in the IR (Figure 6). Thus, we want to further motivate the idea that theETW branes do connect in some cases.
It will be useful to consider a probe example. Instead of taking D3-branes that terminateon stacks of fivebranes, we can consider D3-branes that are intersecting parallel D5-branes.In this case, the field theory description is a theory with two parallel codimension 1 defects.The physics of these defects was described in [20, 21, 55]. We have a 3D hypermultiplet inthe the fundamental representation of U ( N ) coupled to the N = 4 fields at the defect [55].In the supergravity description, we have a probe D5-brane originating from each defect withworldvolume geometry AdS × S , where the S lives in S .If the defects arise from parallel D5-branes with the same orientation, the field theory For the case where we replace R with S , the circle is contractible in the bulk if its radius in field theoryis sufficiently small compared to the S radius. The resulting transition is the same one that appears in theHawking-Page transition (where the circle is taken to represent Euclidean time). – 20 –reserves supersymmetry. The scale L breaks conformal invariance, but we expect thatthe theory flows to a conformal defect theory in the infrared associated with the D3-branesintersecting two coincident D5-branes. In the dual description, we now have probe D5-branesliving on parallel AdS slices of the AdS (Figure 5a).For parallel D5-brane defects with the opposite orientation (associated with D3-branesintersecting a separated D − ¯ D D D
5) but this is now unstable, both nonperturbatively and perturbatively. To seethis, recall that open strings stretched between a D − ¯ D z > L AdS (cid:96)/α (cid:48) , where (cid:96) is the separation in the field theory.The endpoint of perturbative instability is another classical solution in which the branesare connected, as shown in Figure 5b. The explicit solutions were constructed in [35]; wereview them in Appendix B. The physics is qualitatively similar if the probe is a small number n of D5-branes and if we allow some small number k of D3-branes to end on these. In thiscase, the probe branes tilt outward as they enter the bulk, but they still connect providedthat k/n is not too large. The details are presented in appendix B.1. Starting with our probe brane setup, we can generalize to consider defects corresponding tolarger numbers D5-branes or combinations of D5s and NS5s, and finally our case of interestwhere the D3-branes all end on the fivebranes. In these situations, backreaction must betaken into account, and the probe brane is replaced by the geometrical ETW brane in somesolution of type IIB supergravity.It is plausible that the behavior of the ETW branes is similar to that of the probe branes.In the non-supersymmetric case where we have defects/interfaces/boundaries that are relatedby a reversal of orientation, we expect that the instability of the brane-antibrane configurationin the string theory picture should be reflected in a tendency for the branes to connect up inthe preferred solution. A schematic of the proposed geometry, emphasizing the geometricalnature of the ETW branes, is shown in Figure 11 (right). In Appendix B.1, we consider an intermediate situation with one boundary (associated with D3-branesending on stacks of D5-branes and NS5-branes) and one defect associated with an anti- D
5. This anti- D D – 21 – igure 11 . Left: Schematic of dual geometry for N = 4 SYM theory on M × R + coupled to a 3DSCFT with c D (cid:29) c D . Away from the ETW brane, the internal space is S . The ETW brane is aregion of the 10D geometry where this is deformed, growing and then pinching off smoothly. Right:the case with N = 4 on an interval coupled to 3D SCFTs with opposite orientation at the ends ofthe interval. Shown are the Euclidean time (horizontal), radial direction (into the page) and internalspace. To verify this picture, we would ideally want to look for solutions of type IIB supergravitywith the appropriate asymptotic behavior, showing that a connected solution exists and thatthis is the solution with least action.
The asymptotic behaviour of the dual geometries for our setup can be understood from theUV physics of the field theory. Correlators of bulk N = 4 SYM operators separated bydistances much smaller than their distance to the boundary should be well-approximatedby those of N = 4 SYM on R , so the asymptotic region of the dual geometry associatedwith points in the field theory away from the boundaries will be AdS × S . Short-distancecorrelators involving operators on one of the boundaries and nearby bulk operators will begoverned by the superconformal theory of N = 4 coupled to the 3D SCFT degrees of freedomat a single boundary. Thus, the asymptotic geometry near each of the boundaries shouldmatch with one of single-boundary solutions described above.In the field theory with two boundaries, conformal invariance is broken, but we preservetranslations and rotations in the three transverse directions. The UV theory also preserves SO (3) × SO (3) symmetry. If this is not broken spontaneously, the metric would take theform f ( (cid:126)x ) d(cid:126)y + f ( (cid:126)x ) d Ω + f ( (cid:126)x ) d Ω + g ij ( (cid:126)x ) dx i dx j (3.6)where the three coordinates x i on which the metric functions depend correspond to theEuclidean time direction, a radial direction, and one internal direction. The other fields oftype IIB supergravity will also generally be nonzero.– 22 –ince the asymptotic behavior is known, we (optimistically) expect that it is a tractablenumerical problem to find the desired solutions and investigate their properties. However,this lies beyond the scope of the present investigation. An alternative to searching for the full type IIB supergravity solutions would be to look forqualitatively similar solutions in a simpler theory of gravity. The simplest possibility withEinstein gravity and a constant tension ETW brane was considered in [1]. There, it wasfound that connected solutions exist and have least action provided that the tension of theETW brane is below some value T ∗ . But this value is below the critical tension T c where thePoincar´e angle of the ETW brane approaches − π/ T > T ∗ , the connected ETW brane solutions are self-intersecting and don’t makesense. It seems likely that the non-existence of solutions for
T > T ∗ reflects a failure of thesimple bottom-up model to properly capture the physics of the CFT setup. In the picturewhere the Euclidean path integral is preparing a state of the auxiliary degrees of freedomand we take the spatial geometry to be S , the model suggests that for all the boundarytheories corresponding to T > T ∗ , the state e − βH | b (cid:105) has energy of order N even in thelimit β →
0, in conflict with the expectation that boundary states | b (cid:105) should generally besingular. The resolution is likely that the bottom up model needs additional elements in orderto properly capture the physics. The full type IIB supergravity solutions involve a non-trivialdilaton, fluxes, and an internal space that becomes larger in the vicinity of the ETW brane.There are also light degrees of freedom localized near the fivebrane throats. Likely some ofthese additional elements are required in a bottom-up model to properly capture the physics.Below, we will suggest a particular resolution for the problem of self-intersecting ETW branesfor T > T ∗ . An alternative approach to understanding physics on the gravity side is to consider theeffective field theory of the ETW brane. Here, the simplest possible model is to take 4Dgravity coupled to a cutoff CFT (which takes the place of the 5D bulk). This should give thesame physics as the bottom up model with pure 5D gravity coupled to a constant tension This is in contrast to the case of a 2D CFT, where solutions of the simple model exist for all values up to T c . A possible alternative resolution was presented by Antonini and Swingle in [2]. These authors consideredadding a bulk gauge field and making the ETW brane charged under this field. In this case, connected branesolutions were found to exist all the way up to the critical tension. – 23 – igure 12 . Left: microscopic setup. Right: model for matter, in the Minkowski space conformalframe.
ETW brane. As in that description, we do not find the desired solutions in this setup[11, 12]. We review this analysis in the four-dimensional effective description in the nextsubsection. However, we expect that the correct 4D effective description should includeadditional elements. We note in particular that the microscopic models we consider arecharacterized by a global symmetry, with a symmetry breaking pattern G × G → G . Inthe effective field theory description, we then have a gauge field with gauge group G . Sincethe underlying theory is supersymmetric, this comes along with scalar fields and fermions.Since supersymmetry is broken by the combination of boundary conditions in our model,the vacuum energies of these fields do not cancel. Thus, it may be important to take intoaccount the physics of these extra fields in the effective description. This will be the case inthe analysis that we describe presently.As emphasized in [11, 12], the main challenge in these effective models is obtaininga sufficient amount of negative energy from the matter coupled to the 4D gravity (in thepicture where we are describing an eternally traversable wormhole). In the next subsection,we will review this effective field theory analysis and present a novel mechanism for achievingthe large negative energy. We begin with our basic setup of 3D holographic conformal field theories on R , coupledtogether by a 4D CFT on R , times an interval [ − z / , z /
2] as in Figure 12 (left).We would like to understand whether the dual description can include a connected trav- We thank Henry Lin and Juan Maldacena for emphasizing this. – 24 –sable wormhole ( a connected ETW brane with localized gravity in the case where the 4Dtheory is holographic). We will try to come up with an effective 4D description. Our analysisis similar to that in [11, 12]. This description should include: • The 4D gravitational theory dual to the 3D CFT, describing a spacetime with twoasymptotically AdS regions. This may include additional matter. • A cutoff version of the 4D CFT (accounting for the bulk physics). • A non-gravitational version of the 4D CFT on a strip, which couples the fields at thetwo AdS boundaries.The geometry of the ETW brane should be ds = a ( z )( dz + dx µ dx µ ) (4.1)where a ( z ) has simple poles at z = ± z / z = 0. Here, the parameter z is dynamical.The zz-component of Einstein’s equation gives3 (cid:18) a (cid:48) a (cid:19) − a L AdS = 8 πGT zz (4.2)The stress-energy tensor comes from a cutoff version of the 4D CFT plus additional matterfields that appear in the gravity dual of the 3D CFTs. We will model the whole matter systemas some 4D CFT, which we call CFT to distinguish it from CFT , the original 4D CFT.To understand the stress-energy tensor of CFT , we can perform a conformal transfor-mation to flat space R , × [ − z / , z / fields are coupled at eitherend of the interval [ − z / , z /
2] to the CFT fields on the ends of the interval [ − z / , z /
2] sothat the z direction is periodic, as shown in Figure 12 (right). We can model the connectionbetween CFT and CFT as some conformal interface.Using the 2+1 Poincar´e symmetry and conformal invariance, the stress tensor of CFT in this flat space picture must take the form T zz = − z F (cid:18) z z (cid:19) T µν = η µν z F (cid:18) z z (cid:19) (4.3)where we have used the conservation and tracelessness properties, and used dimensionalanalysis to determine the possible dependence on z and z , which are the only scales.– 25 –n the original conformal frame, the stress tensor becomes T zz = − a z F T µν = η µν a z F (4.4)where we are ignoring the conformal anomaly for now (we will show in Appendix D thatit does not qualitatively change the results, though it does lead to interesting effects in theLorentzian cosmology picture).Then the zz component of Einstein’s equation gives (cid:18) a (cid:48) a (cid:19) − a L AdS = − πGz F a , (4.5)or da (cid:113) a L − πGFz = dz (4.6)The minimum value of a occurs where a (cid:48) = 0, so we have a min = 1 z (8 πGF L ) . (4.7)Integrating from this minimum radius (which occurs at z = 0) to the asymptotically AdSboundary at z = z /
2, we get (cid:90) ∞ a min da (cid:113) a L − πGFz = z / I = (cid:90) ∞ dx √ x − (cid:0) (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) = √ K (cid:32) √ (cid:33) ≈ . , (4.9)and rewriting the integral in (4.8) in terms of this, we get finally that F (cid:18) z z (cid:19) = 2 I L πG ∼ c D (4.10)This gives us an equation for z . We see that in order for solutions to exist, the function F must be able to take on a large value for some z . Naively, the value of F should be oforder c (the number of degrees of freedom of the matter theory coupled to gravity), whichwe expect to be much less than c D . However, in the next subsection, we will investigate thebehavior of F in a holographic model and show that large values F (cid:29) c can be achieved incertain cases. – 26 – igure 13 . Left: Gravity dual of the interface theory: patches of the double analytically continuedAdS-Schwarzschild geometries are glued together along a constant tension domain wall. Right: whenCFT has smaller central charge and the interface tension approaches its minimal value, the regionassociated with CFT is part of a multiple cover of the original AdS-Schwarzschild geometry. In this section, we consider the CFT setup of the previous section in the case where thetwo CFTs are holographic. Here, we are just using holography as a tool to answer the CFTquestion of whether F can be large. We first recall the behavior of a single holographic 4DCFT on R , × S , where we take the S to have length L [22]. In the case where we haveantiperiodic boundary conditions for fermions so that the S is allowed to contract in the bulk,the relevant solution is a double-analytic continuation of the planar Schwarzschild geometry: ds = f ( r ) dz + f − ( r ) dr + r (cid:96) dx µ dx µ , (4.11)where f ( r ) = r (cid:96) − µr . (4.12)The periodicity of the z direction is fixed by smoothness at the horizon to be π(cid:96) µ , (4.13)This should equal the CFT periodicity L so we have that µ = π (cid:96) L . (4.14)Using the standard dictionary to read off the stress tensor, we find T zz = − π (cid:96) G L T µν = η µν π (cid:96) G L (4.15)We recall that (cid:96) /G gives a measure of the number of CFT degrees of freedom. In the languageof the previous section, we can think of this case as having a trivial pair of interfaces with– 27 –ero separation, identifying L with z . In this case, we have F = π (cid:96) G ≡ c (4.16)so the behavior of F is as expected.Now we consider the setup of the previous section. We will employ a holographic model[27] where the CFT interface corresponds to a constant tension domain wall between tworegions with different AdS length scales (cid:96) and (cid:96) associated with CFT and CFT . Asexplained in [27], the tension parameter κ = 8 πG T / | /(cid:96) − /(cid:96) | and 1 /(cid:96) + 1 /(cid:96) in order that it can reach the AdS boundary. Theparameter κ is related to properties of the CFT interface (we can think of it as an interfacecentral charge; this is conjectured to decrease under interface RG flows).The dual geometries correspond to a patch of the geometry (4.11) with parameters (cid:96) , µ connected across the domain wall to a patch of the geometry (4.11), as shown in Figure13. The trajectory of the domain wall may be determined by solving the Israel junctionconditions. The details of this analysis will be presented in [56], but are essentially the sameas in [27, 57]. The results for the interface trajectories are given in Appendix C. From thesesolutions, we can read off the behavior of the function F defined in the previous section.It will be convenient to describe the behavior of F/c as a function of the dimensionlessratio z /z . For generic choices of parameters, F/c is of order 1, with mild dependence on z /z . However, there is an interesting behavior when (cid:96) > (cid:96) and we take κ towards thecritical value 1 /(cid:96) − /(cid:96) . In this case, the Poincar´e angle of the domain wall where itintersects the AdS boundary approaches − π/ (cid:96) and π/ (cid:96) . In the resulting solutions, the domain wall in the (cid:96) regionwinds around the center point (Euclidean horizon) more than once (see Figure 13, right),though the solutions are still smooth, since the horizon is not included in the geometry. Forthese solutions, we have that F/c >
0. If we take κ = 1 (cid:96) − (cid:96) + (cid:15)(cid:96) − (cid:96) , (4.17)we find that Fc is approximately constant as a function of z /z , with the value Fc ≈ (cid:15) (cid:18) − (cid:96) (cid:96) (cid:19) I π , (4.18)where I is the same order one constant as before. Returning to the equation (4.10), we seethat a solution requires 1 (cid:15) c ∼ c D . (4.19) An interesting behavior was also noted recently in this limit for 2D CFTs in [57]. – 28 –hus, it appears that solutions may be possible if the interface between the CFTs correspondsto an interface tension close to the minimal value , where we take κ as in (4.17) with (cid:15) ∼ c c D . (4.20)It turns out that the critical value κ = (cid:96) − (cid:96) corresponds to the BPS bound for a domainwall in supergravity [58, 59]; thus, in modelling our setup with softly broken supersymmetry,it seems natural that the interface should be modelled holographically by a domain wall witha tension close to this BPS value.For a given choice of the tension parameter, F varies very little as a function of as afunction of z /z (only by a fractional amount of order (cid:15) ), so it would seem that having asolution requires some fine-tuning of the tension to lie within a narrow window. It wouldbe interesting to understand if this also occurs naturally in our supersymmetric setup. Itshould be noted that in the actual models, the matter in the gravitational sector is probablymore accurately described by a non-conformal quantum field theory, and this may lead toadditional dependence on z /z that may eliminate the need for fine-tuning. Given the fieldtheory and string theory motivations for the existence of solutions described earlier in thepaper, there is reason to believe that this may be the case. Finally, we point out that the mechanism that we have found for producing large negativeCasimir energies may also resolve the original puzzle in the 5D gravity description, wherethe desired solutions with ETW branes failed to exist because of self-intersections. In themodel depicted in Figure 14, we have both an ETW brane and an interface brane. Theextra geometrical region with AdS length (cid:96) > (cid:96) represents the fact that there is morematter in the effective gravity theory that in the original 4D CFT that connects the two 3Dtheories. Understanding in detail whether these solutions make sense may require additionalinput about the intersection between the interface brane and the ETW brane, but the setupappears to cure the basic pathology of a self-intersecting ETW brane. We have presented a class of field theory constructions that may give rise to four-dimensionalEuclidean wormholes, eternally traversable wormholes, and big-bang / big-crunch cosmolo- We recall that the tension parameter is related to an interface central charge (the “boundary F ” forthe folded theory) and that this is conjectured to decrease under RG flows, so these small values may arisenaturally. For example, in the limit of large z for fixed z , the supersymmetry in our setup would be restored, andin this case, the vacuum energies may be expected to cancel. – 29 – igure 14 . Left: Problem with simple 5D model of gravity plus ETW brane. The ETW brane self-intersects above a tension T ∗ , below the value necessary for localization of gravity. Right: A modelwith an additional interface brane, which takes into account the extra matter in the gravitationaltheory beyond the cutoff CFT dual to the bulk. The two solutions on the right are glued along theinterface brane I . The ETW brane and the interface are both multiply wound relative to a single copyof the Euclidean AdS/Schwarzschild geometry, so self-intersections are avoided. gies in the effective description. In all cases, the gravitational theory is not purely four-dimensional, but couples to a higher-dimensional bulk which may or may not have a descrip-tion in terms of classical gravity. Comments on the effective theory
While our immediate goal in this work is not to come up with a phenomenologically realisticmodel of cosmology, we mention a few interesting points related to the effective field theorydescription of the cosmological physics. This matter has two sectors, one coming from acutoff version of the 4D CFT that we choose, and the other coming with the gravitationaltheory dual to the 3D CFTs. The gauge group in the latter sector is directly related to theglobal symmetry of our chosen 3D CFT. Thus, we can control the matter that appears in theeffective 4D gravity theory by choosing the 4D CFT and the global symmetry of the 3D CFTsappropriately. In the specific microscopic examples we have discussed, this global symmetrycan be chosen as an arbitrary product of unitary groups.In these and other examples where the one-boundary theory preserves supersymmetry,the matter in the sector dual to the 3D CFT should be that of some 4D gauged supergravitytheory with the appropriate amount of supersymmetry (e.g.
OSp (2 , |
4) for the microscopictheories we discussed). In the two-boundary setup relevant to cosmology, supersymmetry isbroken. So we expect that some of these fields will become massive, and the effective fieldtheory relevant to low-energy physics will not be supersymmetric. In this case, we expectthat the vacuum energies of the fields would alter the cosmological constant, perhaps in away that depends on time (in which case, the “constant” should be modeled using a field).– 30 –n the Euclidean picture, the asymptotic value should be the negative cosmological constantassociated with ETW-brane theory with unbroken supersymmetry, but this could be modifiedin the interior region of the ETW brane. It is interesting to ask how the construction can avoidthe cosmological problem. Presumably it has to do with the very soft way that supersymmetryis broken, via incompatible boundary conditions at the boundaries in the Euclidean past andfuture.
Future directions
Moving forward, it will be important to verify the existence of the proposed solutions, eitherfrom a gravity point of view (e.g. finding solutions of type IIB supergravity with the specifiedasymptotics, or arguing they exist), or from a field theory point of view (e.g. understandingwhether the proposed microscopic theories have the suggested IR behavior and pattern ofsymmetry breaking).It would be interesting to understand better the mechanism for the enhancement ofnegative Casimir energies presented in Section 4.2. In the context of our holographic model,this can be made arbitrarily large for a CFT on a strip of fixed width by coupling the two sidesvia another strip CFT with smaller central charge and choosing the interface to correspond toa bulk domain wall close to a critical tension. But in microscopic examples, there is likely anupper bound on the energy, since the bulk tension corresponds to a central charge associatedwith the interface, and there should be some lower bound on this.Assuming that the setup we have described is viable, it will be interesting to understandbetter what are the well-defined observables in the cosmological theory and how to computethese from the CFT perspective. As discussed in [3], these calculations may be very difficultin the Lorentzian picture where we start with a state of the auxiliary 4D CFT, since thecosmological physics happens behind a black hole horizon. But at least some observables(e.g. cosmological correlators at the time-symmetric point), seem straightforward to obtaindirectly from the Euclidean picture.Finally, we discuss a connection to the physics of islands in black hole evaporation,also discussed in [3] and in the low-dimensional models of [4–6]. Following [1], we begin byasking about the entanglement wedge for subsystems of the 4D CFT, considering a three-dimensional ball in particular in the case where the spatial geometry is R . In the case whereour 4D theory is holographic and we have a 5D bulk spacetime, we recall that the ETW brane Here, we are talking about the picture where the Euclidean path integral is constructing some state of theLorentzian 4D CFT on M . – 31 –ies behind a planar black hole horizon. There are two possibilities for the Ryu-Takayanagisurface [60] of a ball-shaped region. We can have a surface that stays outside the black holehorizon, or we can have a surface that crosses the horizon and ends on the ETW brane. Inthe first case, the entropy will scale like the volume of the ball for large balls, while in thesecond case, the entropy will scale like the area of the ball for large balls. For large enoughballs, the latter case will have lower area, and the entanglement wedge will include a portionof the ETW brane. In the case where our 4D theory is not conventionally holographic, weexpect that the density matrix for a large enough ball still encodes the information about aball-shaped region of the cosmological spacetime. In this case, this region does not have ageometrical connection to the original CFT, so it is an island in the sense of [24–26]. Thus,we expect that any cosmological spacetime with an underlying description as in this papershould have islands. In a recent paper [26], Hartman et. al. analyzed the conditions underwhich islands can exist in cosmological spacetimes. They found that generically, sufficientlylarge ball-shaped regions of radiation dominated FRW big-bang / big-crunch universes withnegative cosmological constant contain ball-shaped regions satisfying the conditions to beislands. A possible explanation for the observations of [26] is that the underlying microscopicdescription of these big-bang / big-crunch cosmologies is always similar to the one describedin this paper. Acknowledgements
We are grateful to Juan Maldacena for questions and comments which prompted this workand guided several parts of this investigation. We also thank Nima Arkani-Hamed, CostasBachas, Ben Freivogel, Andreas Karch, Henry Lin, Emil Martinec, Mukund Rangamani,Brian Swingle and the string theory group at UBC for useful comments and discussion. Thiswork is supported by the Simons Foundation via the It From Qubit Collaboration and aSimons Investigator Award and by the Natural Sciences and Engineering Research Councilof Canada.
A Type IIB Supergravity solutions for N = 4 SYM theory coupled to a3D SCFT
In this appendix, we briefly recall the solutions of [31–33] corresponding to N = 4 SYMtheory with half-supersymmetric boundary conditions. The metric is given as ds = f ds + f ds S + f ds S + 4 ρ | dw | , (A.1)– 32 –here f , f , f , and ρ are real-valued functions of the complex coordinate w = x + iy = re iθ ,which we take to be restricted to the first quadrant 0 < θ < Π /
2. We also have non-trivialdilaton, NS-NS and R-R three-form fields, and five form fields.The explicit form of the metric and other fields may be expressed in terms of a pair h , h of real harmonic functions. In terms of these, the Einstein-frame metric functions may beexpressed as ρ = e − Φ2 √− N Wh h , f = 2 e Φ2 h (cid:114) − WN , f = 2 e − Φ2 h (cid:114) − WN , f = 2 e − Φ2 (cid:114) − N W , (A.2)where e = e φ = N N , (A.3)is the dilaton field and W ≡ ∂ w h ∂ ¯ w h + ∂ w h ∂ ¯ w h , X ≡ i ( ∂ w h ∂ ¯ w h − ∂ w h ∂ ¯ w h ) ,N ≡ h h | ∂ w h | − h W , N = 2 h h | ∂ w h | − h W . (A.4)Explicit expressions for the other fields may be found in the references.In general, we have a local solution for arbitrary harmonic functions h , h , but to obtaina global solution without singularities, we have additional constraints, e.g. that the polesmust lie on the axes.As an example, we can describe AdS × S with the choice h = L θ )( rr + r r ) , h = L θ )( rr + r r ) (A.5)Here, the codimension-one slices of the spacetime corresponding to a fixed r correspond to AdS × S slices of AdS × S (as in Figure 9); r = r corresponds to the vertical slice. The S arises from the angular coordinate θ and the two S s, which contract to zero on the x and y axes respectively.The general solutions corresponding to N = 4 SYM theory on a half-space correspondto the choice h = π x + 14 (cid:88) A ln (cid:18) ( x + l A ) + y ( x − l A ) + y (cid:19) h = π y + 14 (cid:88) A ln (cid:18) x + ( y + k A ) x + ( y − k A ) (cid:19) . For this choice, the asymptotic value of the dilaton field has been set to zero, but we can use the symmetry φ → φ + φ , B → e φ B , C → e − φ C to restore more general values. – 33 –here we are choosing units with (cid:96) s = 1 [33, 51]. As described in [33, 34], the singularities at x = l A , y = 0 corresponds to D5-brane throats, where the number of units N AD of D5-braneflux associated to a throat is the multiplicity of l A in the sum. Similarly, the singularities at y = k A , x = 0 corresponds to NS5-brane throats, where the multiplicity of the singularity k A in the sum gives the number of units N BNS of NS5-brane flux. From the five-form fluxes inthe solution, [33] found that the number of units of five-form flux (the flux associated withD3-branes) per fivebrane coming from the D5-branes in the A th stack and the NS5-branesthe B th stack are n AD = l A − π (cid:88) B arctan k B l A n BD = k B + 2 π (cid:88) A arctan k B l A (A.6)Thus, microscopic solutions with properly quantized fluxes are obtained by choosing positive k A and l A (including their multiplicities) so that n AD , and n BD are integers ( n AD can benegative). These integer parameters are directly related to the parameters which specifythe underlying field theory [33, 34]. For example, the total amount of D3-brane flux, whichcorresponds to the rank N of the U ( N ) N = 4 SYM Theory gauge group is N = (cid:88) A l A + (cid:88) B k B . (A.7)Note that the l A s and k A s appearing in these sums may appear with some multiplicity.Ignoring these quantization conditions, we note that in the limit where l A and k A are takento be small with (cid:80) A l A + (cid:80) B k B fixed, the harmonic functions approach those correspondingto AdS × S . Thus, the full 10D supergravity solutions also approach AdS × S . In theETW brane picture, we can say that the ETW brane angle goes to Π / AdS × S . However, in the microscopic theory, we cannot take l A and k A arbitrarily small and still satisfy the quantization conditions. B Probe D5-brane solutions
In this section, we review explicit solutions for probe D5-branes in an
AdS × S background.We use Poincar´e coordinates for the AdS, where the metric is ds = L z ( dz + dτ + d(cid:126)y ) (B.1)and describe the S by a metric ds = dψ + cos ψ ( dθ + sin θdφ ) + sin ψ ( dη + sin ηdχ ) . (B.2)– 34 –or a single D5-brane defect at x = 0 in the field theory, the D5-brane worldvolume isdescribed by the hypersurface τ = ψ = 0, filling the z, (cid:126)y, θ, and φ coordinates.For parallel D5-brane defects with the same orientation, we have two probe branes atpositions x , x . We note that the proper distance between the branes becomes small for large z . For D5-branes with the opposite orientation, we also have this solution, but it is unstable.The least energy solution is a connected brane solution, studied in [35], in which the D5-branehas some trajectory z ( τ ) (but still fills the (cid:126)y directions). On the sphere, the brane lives at ψ = 0 and fills the θ and φ directions.The induced metric on the brane is ds = L z (( z (cid:48) ) + 1) dτ + d(cid:126)y ) + L d Ω (B.3)so the brane action gives S ∼ (cid:90) dτ (cid:18) Lz (cid:19) (cid:112) z (cid:48) ) . (B.4)The action does not depend explicitly on τ , so we find1 z (cid:112) z (cid:48) ) = 1 z (B.5)where z is the maximum z coordinate at the turning point on the brane. Solving, we obtain τ ( z ) = z (cid:34) √ π Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) − (cid:18) zz (cid:19) F (cid:32) ,
58 ; 138 ; (cid:18) zz (cid:19) (cid:33)(cid:35) . (B.6)The parameter z is related to the defect separation τ by z = Γ (cid:0) (cid:1) √ π Γ (cid:0) (cid:1) . (B.7)This brane trajectory is plotted in Figure 15a.More generally, we can consider the case with a small number n of D5-branes where asmall number k of the D3-branes terminate on the D5s [36, 37, 61]. In this case, the fieldtheory description is an interface theory where we have gauge group U ( N ) between the defectsand U ( N − k ) outside the defects. The probe D5-branes now include k units of magneticflux on the S , and this induces k units of D3-brane charge from the (cid:82) C ∧ F term in theD5-brane action.Here, the relevant terms in the D5-brane action are S = − T (cid:90) d σ (cid:112) − det( g ab + 2 π(cid:96) s F ab ) − T (cid:90) π(cid:96) s F ∧ C . (B.8)– 35 – igure 15 . (a) Trajectory of D5-brane probe in Poincar´e coordinates. (b) Lorentzian trajectoryof D5-brane probe in Poincar´e coordinates. (c) Evolution of the scale factor for the worldvolumeD5-brane metric in FLRW coordiantes. With our ansatz, the action governing the trajectory z ( τ ) becomes [61] S ∼ (cid:90) dτ z − (cid:110)(cid:112) (1 + ( z (cid:48) ) )(1 + f ) ± f (cid:111) , (B.9)where f = πq/ ( √ λn ). In this case, the probe branes tilt outward as they enter the bulk, atan angle tan θ = f from the radial direction in Poincar´e coordinates, but we have a connectedsolution provided that f < C where C ≈ . Lorentzian probe brane solutions
As an example of the connection to cosmology, we can determine the FRW spacetime as-sociated to the probe brane trajectory (without D3 charge), though here, gravity does notlocalize to the brane. In the analytically continued case, the brane exists the Poincar´e horizon,reaches some minimum z value z and then falls back into the horizon.The trajectory satisfies 1 z (cid:112) − ( ˙ z ) = 1 z . (B.10)We can write the solution as t ( z ) = z F (cid:18) − ,
12 ; 78 ; (cid:16) z z (cid:17) (cid:19) − z √ π Γ (cid:18) (cid:19) Γ (cid:18) (cid:19) sin (cid:18) π (cid:19) . (B.11)This is displayed in Figure 15b. – 36 – igure 16 . Probe brane in the background dual to N = 4 SYM with one SUSY boundary. Left:a SUSY-preserving D5-brane probe remains separated from the ETW brane (left). Right: a SUSY-breaking ¯ D AdS × S . The worldvolume metric can be written most simply using z to parameterize the timedirection. This gives (for the metric in the t > ds = L z (cid:20) − z dz z − z + d(cid:126)y (cid:21) (B.12)In order to see the evolution of the scale factor, we can convert to standard flat FLRWcoordinates − dη + a ( η ) d(cid:126)y . (B.13)We have a ( η ) = Lz ( η ) ds = Lz z (cid:112) z − z dz (B.14)which gives a ( η ) = Lz cos (cid:18) ηL (cid:19) (B.15)This is plotted in Figure 15c. B.1 Probe ¯ D -brane in the background of a single-boundary solution. In this section, we consider a probe D5-brane in the general half-supersymmetric supergravitybackgrounds reviewed in the previous section. The brane action is the sum of Born-Infeldand Wess-Zumino terms, given in Einstein frame by S BI = − T (cid:90) d σe φ (cid:113) − det( g ab + e − φ ( B ab + 2 πα (cid:48) F ab )) (B.16) S W Z = − T (cid:90) e πα (cid:48) F + B ∧ (cid:88) C . (B.17)As for the
AdS × S solutions, we have that the D5-branes live at θ = 0, wrapping the first S . They are described by some trajectory r ( u ), where u is the Poincar´e radial direction in– 37 – dS and r is the radial coordinate on the quadrant. The brane is stretched in the otherthree directions of AdS .The worldvolume gauge field can be consistently set to zero. . Using the form of thesupergravity solution, we then find a Lagrangian density L = A ( r ) u (cid:115) B ( r ) u (cid:18) dudr (cid:19) + K ( r ) u (B.18)where A ( r ) = e φ f (cid:113) f + e − φ B B ( r ) = 4 ρ f K ( r ) = ± B C where the two possible signs in K correspond to a D5-brane probe or and anti D5-braneprobe. Redefining u = exp ( x ), we obtain equations of motion ABr (cid:48)(cid:48) − dKdr (1 + B ( r (cid:48) ) ) − dAdr − AB ( r (cid:48) ) + 12 ( r (cid:48) ) ( A dBdr − dAdr B ) − ABr (cid:48) = 0 (B.19)We want to consider a probe brane starting at r = ∞ at some fixed x . Solutions that returnto r = ∞ at some fixed x can be ignored since these correspond to having additional defects.We cannot have a solution that goes to r = ∞ for x → ∞ since the solution approaches AdS × S for large r and we do not have such solutions in this case. Thus, we have twopossibilities: the solution approaches some finite r for x → ∞ in the region away from theETW brane, or the brane is drawn toward the ETW brane (specifically to one of the D5-branesingularities).For the first type of solution, we have r (cid:48) → x , so (B.19) gives r (cid:48)(cid:48) = 1 AB (cid:18) dAdr + dKdr (cid:19) . (B.20)Thus, we can have solutions where the probe brane is not drawn into the ETW brane if andonly if A + K has a extremum. As an example, for AdS × S , we have A + K ∝ (cid:18) rr + r r (cid:19) (B.21)so we have a minimum at r = r which corresponds to the vertical slice in Poincar´e coordi-nates. It’s possible to consider solutions with world-volume flux, but these correspond to fivebranes carryingadditional D3-brane charge We find that C (6) vanishes for θ = 0, so the Wess-Zumino term only receives a contribution from B ∧ C (4) . – 38 –e have investigated the probe brane configurations for various parameter values. Witha SUSY-preserving D5 orientation, it appears that solutions of the first type always exist. Onthe other hand, replacing the D5 with an anti-D5, we find that such solutions do not exist inmany cases so the anti-D5 brane is necessarily drawn in to the ETW brane. This is a probeversion of the situation where two ETW branes from two boundaries reconnect in the bulk.On the other hand, we have solutions that are arbitrarily close to AdS × S . Since A + K hasa minimum for AdS × S it will continue to have a minimum in solutions that are very closeto AdS × S , for either sign of K , but these solutions correspond to boundary conditionsinvolving a very large number of D5-branes. So in this case, considering a single D5-braneprobe may not give us useful insight about the physics of adding a second boundary thatwould involve a very large number of anti-branes. C Holographic model for conformal interfaces
The details of our holographic model for conformal interface theories can be found in [27].We include here the formulae for z /β and z /β , the fraction of the boundary of Euclideanglobal AdS spacetime covered by the spacetime regions associated with CFT and CFT respectively. We have z β = − µ π(cid:96) (cid:90) ∞ r dr ( f − f + κ r )2 κrf √ Vz β = 1 − µ π(cid:96) (cid:90) ∞ r dr ( f − f + κ r )2 f κr √ V where f i = r (cid:96) i − µ i r V = f − (cid:18) f − f − κ r κr (cid:19) (C.1)and r is the largest value of r for which V ( r ) = 0. These formulae are valid in cases such asFigure 13b where the CFT side does not include the Euclidean horizon but the CFT sidedoes. We have the restriction that z /β < z /β can begreater than 1 in the case where we have multiple wound solutions as in Figure 13b. D Effective field theory description with conformal anomaly
In the analysis of section, we have ignored the conformal anomaly. However, if we assumethat the CFT matter theory is holographic with an Einstein gravity dual, it is possible toexplicitly take the conformal anomaly into account.– 39 –he contribution to the stress tensor from the conformal anomaly can be determined viaa gravity calculation to be T zz = − c a (cid:18) a (cid:48) a (cid:19) (D.1)where up to a numerical factor, c is the a or c type central charge of CFT (which are equalfor a holographic CFT with Einstein gravity dual).With this contribution, the zz component of Einstein’s equation gives (cid:18) a (cid:48) a (cid:19) − a L AdS = − πGz F a − πGc a (cid:18) a (cid:48) a (cid:19) , (D.2)or dadz = a √ πGc (cid:118)(cid:117)(cid:117)(cid:116)(cid:115) πGc L − π G F c z a − a is still a min = 1 z (8 πGF L ) . (D.4)Integrating from this minimum radius (which occurs at z = 0) to the asymptotically AdSboundary at z = z / F (cid:18) z z (cid:19) = c D Q (cid:18) c c D (cid:19) (D.5)where we have taken c D = L / (2 πG ) and Q ( (cid:15) ) = (cid:90) √ (cid:15)dy (cid:113)(cid:112) (cid:15) (1 − y ) − (D.6)We can check that in the limit c /c D → Q ( c /c D ) approaches a constant to give the sameequation as before. For small c /c D (as we expect), we get a slightly different number onthe right hand side of (D.5). D.1 Lorentzian solutions
Including the effects of the conformal anomaly does have some interesting consequences forthe Lorentzian solutions, namely the existence of a minimum scale factor and/or maximuminitial energy density.If we work in standard FRW coordinates, − dt + A ( t ) d x (D.7)– 40 –he Einstein equation leads to the Friedmann equation (cid:32) ˙ AA (cid:33) + 1 L AdS = πGc (cid:32) ˙ AA (cid:33) + 16 Fz c A . (D.8)Defining ˆ A = z (cid:16) c F (cid:17) √ πGc A ˆ t = 1 √ πGc ˆ L = L √ πGc ∼ c D c , (D.9)the equation simplifies to (cid:32) d ˆ Adt (cid:33) = ˆ A π (cid:115)(cid:18) L + 1 (cid:19) ˆ A − − sign) and the other giving rise to inflating spacetimes. We are inter-ested in the former.For either branch of solutions, we have a minimum possible value of ˆ A , where the expres-sion inside the square root vanishes,ˆ A min = (cid:18) L + 1 (cid:19) − . (D.11)Presumably, this is where the semiclassical approximation is breaking down. The energydensity at this point is T ∼ c G (D.12)dominated by the conformal anomaly contribution.The explicit solutions for a ( t ) can be obtained by integrating (D.10); the scale factorexpands from its minimum value to a maximumˆ A max = (cid:32) ˆ L (cid:33) (D.13)before contracting. We have ˆ A max ˆ A min = (cid:32) ˆ L (cid:33) ∼ (cid:18) c D c (cid:19) (D.14)so we have a large amount of expansion when c D (cid:29) c , i.e. the number of local degreesof freedom in the original holographic 3D CFT is much larger than the number of matterdegrees of freedom in the dual theory. This mechanism for inflation is closely related to Starobinsky’s original model for inflation [62]. – 41 – eferences [1] S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk, C. Waddell and D. Wakeham,
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