PPrepared for submission to JHEP
AdS Euclidean wormholes
Donald Marolf, a Jorge E. Santos b a Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106,U.S.A. b Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilber-force Road, Cambridge, CB3 0WA, UK
E-mail: [email protected] , [email protected] Abstract:
We explore the construction and stability of asymptotically anti-de Sitter Eu-clidean wormholes in a variety of models. In simple ad hoc low-energy models, it is not hardto construct two-boundary Euclidean wormholes that dominate over disconnected solutionsand which are stable (lacking negative modes) in the usual sense of Euclidean quantumgravity. Indeed, the structure of such solutions turns out to strongly resemble that of theHawking-Page phase transition for AdS-Schwarzschild black holes, in that for boundarysources above some threshold we find both a ‘large’ and a ‘small’ branch of wormhole solu-tions with the latter being stable and dominating over the disconnected solution for largeenough sources. We are also able to construct two-boundary Euclidean wormholes in avariety of string compactifications that dominate over the disconnected solutions we findand that are stable with respect to field-theoretic perturbations. However, as in classic ex-amples investigated by Maldacena and Maoz, the wormholes in these UV-complete settingsalways suffer from brane-nucleation instabilities (even when sources that one might hopewould stabilize such instabilities are tuned to large values). This indicates the existenceof additional disconnected solutions with lower action. We discuss the significance of suchresults for the factorization problem of AdS/CFT. a r X i v : . [ h e p - t h ] M a r ontents U (1) theory with S boundary 9 (cid:96) S = 0 sector 164.3.2 Scalar-derived perturbations: the (cid:96) S ≥ sector 194.3.3 Scalar-derived perturbations: the (cid:96) S = 1 sector 234.3.4 Vector-derived perturbations: the (cid:96) V ≥ sector 244.3.5 Vector-derived perturbations: the (cid:96) V = 1 sector 264.3.6 Tensor-derived perturbations: the (cid:96) ≥ sector 26 S boundary 27 Ansatz for the wormhole solutions 295.2
Ansatz for the disconnected solution 325.3 Results 325.4 Negative Modes 345.4.1 The scalar homogeneous mode : (cid:96) S = 0 (cid:96) S ≥ (cid:96) S = 1 (cid:96) V ≥ (cid:96) V = 1 mode 435.4.6 The tensor modes with (cid:96) T ≥ U (1) wormholes in 11-dimensional supergravity 44 U (1) − theory 446.2 The disconnected phase 456.3 The wormhole phase 466.4 M branes on wormhole backgrounds 476.5 Negative modes 486.6 Results 49– i – Mass deformation of ABJM 50 probe branes 537.4 Results 53 U (1) wormholes 62 A.1 The symmetric Matrix V for the scalars 62A.2 The symmetric Matrix V for the tensors 64 B Symmetric matrices for the scalars 64C Wormholes with toroidal boundaries in simple low-energy theories 70
C.1 U (1) with a toroidal boundary 71C.2 Wormholes sourced by scalar fields with a toroidal boundary 71C.2.1 Negative modes with toroidal boundaries 75 D Table with longer list of models studied 83
It has long been understood that S-matrices, boundary correlators, or boundary partitionfunctions defined by bulk gravitational path integrals may fail to display familiar factor-ization properties due to contributions from spacetime wormholes [1–6]. For our purposes,it is convenient to define spacetime wormholes as connected geometries whose boundarieshave more than one compact connected component. This definition includes real geome-tries of any signature as well as those that are intrinsically complex. Using an AdS/CFTlanguage, the point is that a boundary partition function Z is naively represented by a bulkpath integral over configurations with a single compact boundary. Similarly, a product ofboundary partition functions (say, Z ) is naively represented by a bulk path integral over– 1 – igure 1 . An example showing failure of factorization due to spacetime wormholes. The top linerepresents a path integral (cid:104) Z (cid:105) . Although we have drawn the configuration as connected, it mayinclude contributions from disconnected spacetimes as well. In any case, the natural path integral (cid:104) Z (cid:105) associated with a pair of boundaries yields all terms generated by squaring (cid:104) Z (cid:105) , but alsocontains additional contributions connecting the two boundaries as indicated by the second termin the bottom line. configurations with two disconnected boundaries. But contributions from spacetime worm-holes suggest that the latter path integral (which we call (cid:104) Z (cid:105) ) is not necessarily the squareof the former path integral (which we call (cid:104) Z (cid:105) ); see figure 1.Such failures of factorization would clearly require the standard picture (see e.g. [7–9])of AdS/CFT duality to be modified; see e.g. [4–6, 10, 11] for related discussions. A possibleresolution is that the bulk path integral is in fact dual to an ensemble of boundary theories,and our notation (cid:104) Z (cid:105) , (cid:104) Z (cid:105) is chosen to reflect this idea. A non-zero “connected correlator” (cid:104) Z (cid:105)−(cid:104) Z (cid:105) is then interpreted as describing δZ for fluctuations δZ that allow the partitionfunction Z in any particular element of the ensemble to differ from the ensemble-mean (cid:104) Z (cid:105) . Such an effective description was derived in [4–6] under certain locality assumptions,though this assumption can be dropped by using the argument of [12]. In addition, dualitiesof this kind have been explicitly constructed between (appropriate completions of) variousversions of Jackiw-Teitelboim (JT) gravity and corresponding double-scaled random matrixensembles [13–15]. See also [16] for discussion of related issues for c = 1 matrix models, and[17] for discussion of ensembles of Sachdev-Ye-Kitaev models [18, 19] and a proposed relationto wormholes in Jackiw-Teitelboim gravity [20] [21] coupled to matter fields. Discussions ofoff-shell wormholes in both JT gravity and pure gravity in AdS can be found in [22, 23];see also [24, 25] for related discussions of averaging 2d conformal field theories.However, it is far from clear that this ensemble interpretation will hold for the mostfamiliar examples of AdS/CFT. In particular, such cases involve bulk theories with largeamounts of supersymmetry, and this supersymmetry should be reflected in each memberof the boundary ensemble . However, in more than two boundary dimensions d the set While these works pre-date the discovery of AdS/CFT, their arguments apply immediately to thatcontext. This property holds in the matrix model examples of [15]. A general argument follows from the factthat the full bulk system admits an algebra of asymptotic SUSY charges, and that these charges must act – 2 –f local boundary theories with large amounts of supersymmetry is expected to be verylimited; e.g., for d = 4 and N = 4 SUSY, super Yang-Mills theory is known to be theunique local maximally-supersymmetric theory that admits a weakly-coupled limit , andmay thus be the unique local theory. One might thus expect that – at least when thefull physics associated with the UV-completion of bulk gravity is taken into account –the ensemble of dual boundary theories degenerates in this case so as to contain onlya single physically-distinct theory (here d = 4 , N = 4 SYM); see related discussions in[10, 12, 27–29]. On the other hand, this would require strong departures from the low-energy semi-classical description of the bulk and would thus render unclear the status ofrecent apparent successes [30–33] in using semi-classical bulk physics to resolve issues inblack hole information.We thus return to the question of whether partition functions defined by higher-dimensional bulk path integrals should factorize across disconnected boundaries; i.e., whetherin such cases we should in fact find (cid:104) Z (cid:105) = (cid:104) Z (cid:105) . Such factorization would require eithersome rule or effect to forbid spacetime wormholes from appearing in the path integral, oralternatively that in all computations one finds precise cancellations when one sums overall connected spacetimes with given disconnected boundaries. We note that, while it mayat first appear somewhat contrived, the latter option is precisely what occurs in one of thebaby-universe superselection sectors described in [4–6, 12] and in the limit of the eigen-branes described in [34] where one fixes all of the eigenvalues of the relevant matrices. Ofcourse, in both of those cases one has carefully tuned some extra ingredient (the baby uni-verse state or the eigenbrane source) to make such cancellations occur. Furthermore, thisoption is difficult to see explicitly unless one can solve the theory in detail.It will come as no surprise that a complete study of the higher dimensional gravitationalbulk path integral is far beyond the scope of this work. We will therefore resort to theusual crutch of studying bulk saddle points. Assuming that the contour of integration canbe deformed to pass through our saddles, they should dominate over contributions fromnon-saddle configurations in the limit of small bulk Newton constant G .Before proceeding, it is useful to review the literature concerning asymptotically AdSspacetime-wormhole saddle-points. The simplest context in which one might imagine suchsaddles to arise would be Euclidean pure AdS-Einstein-Maxwell theory with two sphericalboundaries (each S d for a d + 1 -dimensional bulk). However, such solutions are forbiddenby the results of [35], which showed that spacetime-wormhole saddle-points cannot arise inEuclidean pure AdS-Einstein-Maxwell theory [35] when the scalar curvature of the boundarymetric is everywhere positive. On the other hand, although none of these solutions haveall of the properties that one might desire, a variety of Euclidean spacetime-wormholesaddles have been constructed by allowing the boundary metric to be negative or by addingcertain types of matter [10, 27, 28] (with the latter based on the zero cosmological constantconstructions of [36]); see also [17] in the context of JT gravity with matter. We also trivially on the ‘baby universe sector’ of the theory (i.e., on the H BU of [12, 26]). Thus the SUSY algebraacts within each bulk superselection sector. The boundary dual interpretation is then that each member ofthe associated ensemble has a well-defined SUSY algebra. This follows from classifying the supersymmetric marginal deformations of free field theory. – 3 –efer the reader to the very interesting story of constrained instantons (off-shell wormholes)described in [37], though at least as of now their stability has been analyzed only in theoriesof pure gravity (and JT gravity for d = 2 ).In particular, the spacetime-wormhole solutions of [10, 27, 28] may be divided into fourcategories. The first are the Euclidean solutions of [28] which have spherical (and thuspositive-curvature) boundaries but follow [36] in using axionic matter. The axion kineticterm can become negative in Euclidean signature, though in this case it is positive definitebut has a surprising zero at a radius that depends on angular momentum. This unusualkinetic term and a potential that is negative in the region where the kinetic term is smallcombine to allow saddles to suffer from bulk negative modes associated with non-trivialbulk angular momentum [38]. It is thus hard to argue that the dominate over non-saddleseven at small G . Other spacetime wormholes with field-theoretic bulk negative modes weredescribed in section 4.2 of [10].The second are Euclidean wormholes with negative-curvature boundaries (perhaps com-pact hyperbolic manifolds). As discussed in [35], in the simplest AdS/CFT examples suchsolutions have string-theoretic negative modes associated with the nucleation of D-branes.Indeed, these negative modes render the entire theory unstable in the UV. While the UVissues can be stabilized by appropriately deforming the CFT [10] (and, in particular, break-ing conformal invariance), and while this will forbid any brane nucleation instability closeto the AdS boundary, it was found in [27] that – at least in the model studied there – thewormholes remain unstable to the nucleation of D-branes at finite locations in the interior.The third class consists of Euclidean wormholes that have no known negative modes,and which can even have lower action than disconnected solutions, but which have no knownembedding in string theory. Examples include the large α solutions in section 4.2 of [10].Finally, the fourth category (section 5 of [10]) contains Euclidean wormholes with knownstring-theoretic embeddings and no known negative modes, but where the correspondingdisconnected solution is not yet known so that it is unknown which saddle dominates atlarge G .We should also mention that the so-called ‘double-cone’ solutions of [13] define a 5thclass of spacetime wormhole solutions. These solutions are constructed by starting with e.g.the complexification of a two-sided AdS-Schwarzschild black hole and taking the quotientby a discrete Lorentz-signature time translation. The real Lorentz-signature section ofthis quotient is connected and has two compact boundaries, though it is also singular atthe bifurcation surface. However, there are non-singular complex sections (on which thequotient acts freely) that can be used to connect the same two real boundaries. As a result,we prefer to think of the double-cone as an inherently complex solution. This is in no way afundamental problem, but we will instead discuss real Euclidean solutions below. We hopeto return later to a more detailed analysis of stability in the (complex) double cones. The situation is even worse if one makes a further analytic continuation of the axion according to φ → iφ , as the kinetic term then becomes negative definite. One might expect the same to be true in areformulation writing the axion in terms of the Hodge-dual 2-form potential, but we have not carried outa detailed analysis. – 4 –ur goal here is thus to expand the class of known Euclidean spacetime wormholeswith simple disconnected boundary metrics, which we will take to consist either of twocopies of a (possible squashed) sphere ( S d ) or two copies of the torus ( T d ). In particular,we wish to identify saddles without field-theoretic negative modes, where the saddles can beembedded in simple compactifications of string theory, and where the Euclidean wormholesdominate over disconnected saddles. We will also explore negative modes associated withbrane nucleation, and we will analyze the structure of any the phase transition associatedwith the exchange of dominance between Euclidean wormholes and disconnected saddles.We begin in section 2 by describing how a general class of potential Euclidean worm-holes may be interpreted as homogeneous isotropic Euclidean cosmologies; i.e., as EuclideanFriedmann–Lemaitre–Robertson–Walker (FLRW) solutions. This point of view providesuseful intuition for why Euclidean wormholes with positive curvature boundaries are forbid-den in pure AdS-Einstein-Hilbert gravity, and also for what sort of ingredients are requiredto overcome this obstacle.Sections 4 and 5 then study toy models in four bulk dimensions with simple bulkmatter content that concretely illustrate the construction suggested by the FLRW analysisof section 2 with S boundaries. As examples of bulk matter we consider both U (1) gaugefields (section 4) and non-axionic scalar fields (section 5). These models do not directlyembed in string theory, but turn out to be similar to some that do. In the toy models weidentify connected (wormhole) solutions that are free of bulk field-theoretic negative modesand which dominate over disconnected (non-wormhole) solutions in appropriate regimes. Asa function of the matter sources, we find the associated space of solutions to have structuremuch like that of the well-known Hawking-Page transition, including in particular a first-order phase transition associated with exchanging dominance between the connected anddisconnected saddles. This is precisely the structure that was found in studies of wormholesin JT gravity coupled to matter [17] and in studies of constrained wormholes [37] in theoriesof pure gravity. Because these are only ad hoc low-energy theories or JT models, they do notcontain fundamental branes. Thus there can be no notion of brane-nucleation instabilityto explore in such models.We therefore turn in sections 6, 7, and 8 to studying truncations of string-theory or M -theory. In particular, section 6 considers a truncation of 11-dimensional supergravity thatreduces to U (1) Maxwell-Einstein theory on AdS , section 7 examines a mass-deformedversion of the ABJM theory [39], and section 8 investigates a truncation of type IIB stringtheory to an Einstein-scalar system on AdS . In each of these cases we require spheri-cal boundaries and construct wormhole solutions. The structure of the space of solutionsis generally much as in the toy models, with wormholes dominating over the most obvi-ous disconnected solution at large values of the relevant boundary source, and with suchwormholes being free of field-theoretic negative modes. However, in all cases we find suchwould-be dominant wormholes to suffer from brane-nucleation instabilities. In sections 6and 8 the brane-nucleation occurs only in a finite region of the bulk and does not occur inthe deep UV. Thus the theory itself remains stable with these boundary conditions and onlythe particular wormhole solution is destabilized. In contrast, section 7 studies a contextwith hyperbolic boundaries of the form reviewed above, but with a deformation parameter– 5 – that is expected to stabilize the theory at large enough µ . While it does in fact appearto do so, we find wormholes only in the small- µ regime where the theory remains unstable.The results of section 7 are thus similar to those found in [27] for mass deformations of N = 4 SYM.We close with some interpretation and discussion of open issues in section 9. The maintext is supplemented by various appendices with additional technical details. This includesappendix C, which describes studies of two UV-complete models with torus boundaries(with results broadly similar to those associated with spherical boundaries). It also includesappendix D, which lists results for a larger set of 14 low energy models and 22 string/M-theory compactifications in a variety of dimensions that we have studied at least briefly butwhose investigation we may not have chosen to describe in detail. While the explorationsof those models were not always as complete as the ones described in the main text (seeappendix D for details), they suggest that the results of sections 6, 7, 8 are typical.
Consider any Euclidean wormhole whose boundary consists of two copies of a maximally-symetric Euclidean geometry Σ d ; i.e., where Σ d is a sphere S d , a Euclidean plane R d , or ahyperbolic plane H d . If the metric of the full bulk solution preserves this symmetry, thenthe wormhole admits a preferred slicing which again preserves the symmetry. We may thusdescribe the wormhole as a d + 1 -dimensional homogeneous isotropic Euclidean cosmologywith S d , R d , or H d slices and with Euclidean time running transverse to each slice.We may thus write the metric in the (Euclidean) FLRW form d s = d τ + a ( τ )dΣ d , (2.1)with dΣ d the standard metric on S d , R d , or H d , where in the case of S d or H d we take thespaces to have unit radius for simplicity. Labelling the 3 cases by k = 1 , , − as usual, itis well known that the Einstein equation R ab − g ab R = 8 πGT ab (2.2)with T ab the stress energy tensor for bulk matter, reduces to the Friedman equation (cid:18) ˙ aa (cid:19) = − πGd ( d − ρ + 1 L + ka . (2.3)Here ˙ a = d a d τ , L is the bulk AdS scale and the term L encodes the explicit effects of thenegative cosmological constant, and ρ is the standard (Lorentz-signature) energy density ofany matter fields ρ ≡ − T ττ . (2.4)Due to the Euclidean signature this equation differs from its familiar Lorentz-signaturecounterpart by an overall sign. Of course, we can also take quotients of the above solutions The so-called second Friedman equation follows from the time derivative of (2.3) and conservation ofstress-energy. – 6 –nd thus use this formalism when Σ d is a torus T d or a compact hyperbolic manifold H d / Γ for some properly discontinuous isometry group Γ of H d .Let us now briefly investigate what (2.3) implies for the existence of Euclidean worm-holes. For τ → ±∞ , we ask that a ( τ ) → ∞ to satisfy asymptotically AdS boundaryconditions. As a result, on any wormhole solution a ( τ ) must have some minimum a where ˙ a = 0 . This clearly requires the right-hand side of (2.3) to vanish. Without matter we have ρ = 0 , so this condition can be satisfied only when k = − ; i.e., when the boundary metrichas negative scalar curvature.However, we immediately notice two encouraging features. First, for k = 0 the abovefailure is only marginal. Multiplying (2.3) by a and setting k = 0 , we see that ˙ a can vanishat a = 0 . Indeed, there is a k = 0 vacuum solution a ( τ ) = L e τ/L . One may think of twocopies of this solution as describing a degenerate limit of Euclidean wormholes where theneck of the wormhole has been stretched to become both infinitely long and infinitely thin.Second, for any k it is clear that the ˙ a = 0 constraint can be satisfied by adding matterwith positive Lorentz-signature energy ρ . For k = 0 an arbitrarily small amount of suchmatter will do, but for k = 1 we require ρ to exceed a critical threshold. We thus expect k = 0 wormholes to appear with arbitrarily small matter sources on the boundaries, whilefor k = 1 wormholes will arise only when the scalar sources are sufficiently large.Many familiar kinds of matter yield positive Lorentz-signature energy densities ρ . How-ever, especially since the matter energy density ρ will be large for k = 1 , it is useful to choosematter which is gravitationally attractive in Lorentz signature. Wick rotation to Euclideansignature then gives gravitational repulsion, which helps to make ¨ a positive at a = a andalso helps to satisfy the asymptotically AdS boundary conditions. In particular, it would not be useful to use the potential energy of a scalar field, where the condition ρ > wouldeffectively require adding a new positive cosmological constant to cancel the old (negative)one. Furthermore, if the energy at a = a comes from time-derivatives, then it is naturallypositive if the Lorentz-signature field is real. But that will make τ -derivatives imaginaryat a , and thus tend to give imaginary (or complex) fields in Euclidean signature. Thekinetic terms of such fields then tend to give negative contributions to the Euclidean action,and are thus a likely source of negative modes. This is the essential reason why the axionsolutions of [28, 36] have many negative modes [38]. We thus wish to take all τ -derivativesto vanish at a . Assuming that the scalar sources are identical on the two boundaries, thisis equivalent to requiring the entire wormhole to be invariant under a corresponding Z symmetry.As a result, we study solutions below with a surface a = a invariant under sucha Z symmetry and where ρ > at this surface due to spatial gradients of the matterfields. Such kinetic energy is indeed gravitationally attractive in Lorentz signature, andthus gravitationally repulsive in Euclidean signature, but is consistent with real Euclideansolutions. Creating such gradients requires similar gradients in the scalar sources we choose As discussed above, the kinetic term of [38] does not become negative. But it does have a surprisingzero. – 7 –t the two boundaries. For k = 0 , we expect Euclidean wormhole solutions with arbitrarilysmall such sources. For k = 1 , we expect Euclidean wormhole solutions to appear once theboundary sources exceed some critical threshold.Note that the above analysis and statement of expectations applies only when thesurfaces of constant Euclidean time are homogeneous and isotropic. Since the matter fieldshave spatial gradients, we will need to choose a finely-tuned matter Ansatz to achieve this.We may expect similar behavior for more general solutions that violate homogeneity , butfinding solutions would then require the solution of partial differential equations. We thussave such analyses for future work. Let us consider a general Euclidean partition function Z associated with a Euclidean action S E (cid:104) (cid:126)φ ; (cid:126)φ ∂ M (cid:105) with some collection of fields (cid:126)φ and corresponding boundary conditions (cid:126)φ ∂ M .For the case of an Einstein-Scalar theory, (cid:126)φ would contain all ( d + 1)( d + 2) / independentmetric components and the scalar field. Such a partition function can be schematicallyrepresented as Z [ (cid:126)φ ∂ M ] = (cid:90) D (cid:126)φ e − S E [ (cid:126)φ ; (cid:126)φ ∂ M ] . (3.1)In the saddle-point approximation, we can expand Z as Z [ (cid:126)φ ∂ M ] ≈ e − S E [ (cid:126)φ ; (cid:126)φ ∂ M ] × (cid:90) D (cid:126)δφ e − S (2) E [ (cid:126)δφ ;0 ] + . . . , (3.2)where (cid:126)φ are classical solutions of the equations of motion derived from S E (cid:104) (cid:126)φ ; (cid:126)φ ∂ M (cid:105) , andwe obtain S (2) E (cid:104) (cid:126)δφ ; 0 (cid:105) by expanding the fields as (cid:126)φ = (cid:126)φ + (cid:126)δφ and keeping all terms up tosecond order in (cid:126)δφ . A first order term is absent in the expansion above, because (cid:126)φ satisfiesthe classical equations of motion derived from S E (cid:104) (cid:126)φ ; (cid:126)φ ∂ M (cid:105) .Let us imagine for a moment that S (2) E (cid:104) (cid:126)δφ ; 0 (cid:105) is not positive definite. In that casethe saddle (cid:126)φ is not a local minimum of the Euclidean action and does not dominate overintegration over nearby configurations if the integral is performed along the real Euclideancontour. In this case we say that (cid:126)φ has field-theoretic negative modes.For pure gravity S (2) E (cid:104) (cid:126)δφ ; 0 (cid:105) is infamously not positive definite [40, 41]. In fact onecan show that the conformal factor of the metric always has the wrong sign for the kineticterm. This is the conformal factor problem of Euclidean quantum gravity. One way aroundthis is to Wick rotate the conformal factor, leading to a convergent Gaussian integral. Thisprocedure, although slightly ad hoc in [40, 41], was justified at the level of linearized gravityin [42] and has been recently backed up by detailed dual field theory calculations [43–45] in For k = 0 , spacetime wormholes should require non-zero gradients along each leg of the torus. Otherwiseone could Wick-rotate along a leg with translational symmetry and find a Lorentz-signature solution withtwo disconnected non-interacting boundaries linked by a traversable wormhole (and thus violating boundarycausality). We thanks Douglas Stanford for discussions on this point. – 8 –he context of gauge/gravity duality. It was also shown in [46] that a version of this Wickrotation can be performed at the non-linear level.The case of gravity coupled to matter is more delicate. In particular, it is no longerobvious that the conformal factor is the right variable to Wick rotate [47] since perturbationsof the conformal factor, i.e. trace-type metric perturbations, will generically couple to otherscalar matter perturbations. In addition, if matter is present, the trace free part of themetric can also couple with the trace itself.Here we follow the procedure outlined in [48] which was used in [47] to investigate thenegative mode of an asymptotically flat Reissner-Nodström black hole. It turns out that inall the cases we studied, the action can be decomposed as S (2) E (cid:104) (cid:126)δφ ; 0 (cid:105) = ˆ S (2) E (cid:104) (cid:126)δ ˆ φ ; 0 (cid:105) + ˜ S (2) E (cid:104) (cid:126)δ ˜ φ ; 0 (cid:105) (3.3)where together the perturbations (cid:126)δ ˆ φ and (cid:126)δ ˜ φ span the space of the original perturbations (cid:126)δφ and both ˆ S (2) E (cid:104) (cid:126)δ ˆ φ ; 0 (cid:105) and ˜ S (2) E (cid:104) (cid:126)δ ˜ φ ; 0 (cid:105) have been written in first order form. The variables (cid:126)δ ˆ φ turn out to be non-dynamical, i.e. the action ˆ S (2) E (cid:104) (cid:126)δ ˆ φ ; 0 (cid:105) contains no derivatives of (cid:126)δ ˆ φ . It isin this sector that we find a mode with a non-positive action. Furthermore, the fact that thevariables (cid:126)δ ˆ φ enter the action algebraically is a consequence of the Bianchi identities, and inan appropriate canonical formalism these variables would become Lagrange multipliers thatenforce constraints. Let us denote by { (cid:126)δ ˆ φ } the problematic mode. This is the mode thatwe Wick rotate as { (cid:126)δ ˆ φ } → i { (cid:126)δ ˆ φ } . The Gaussian integral over (cid:126)δ ˆ φ can now be performedand we reabsorb it in the measure. We are then left to study the positivity properties of ˜ S (2) E (cid:104) (cid:126)δ ˜ φ ; 0 (cid:105) . At this stage we introduce gauge invariant variables (cid:126) ˇ q which can be used towrite ˜ S (2) E (cid:104) (cid:126)δ ˜ φ ; 0 (cid:105) solely as a function of (cid:126) ˇ q and their first derivatives. It is then ˜ S (2) E (cid:2) (cid:126) ˇ q ; 0 (cid:3) whose positivity properties we investigate. Note that the dimensionality of (cid:126) ˇ q is necessarilysmaller than that of (cid:126)δ ˜ φ because of gauge invariance.The procedure outlined above is consistent with the studies performed in [49–51] whichwere used in [38] to show the existence of multiple negative modes on wormholes sourcedby axions; see also [52]. U (1) theory with S boundary We now proceed to study a simple AdS Einstein-Maxwell model that illustrates both keyelements of the physics and our main techniques. We assume spherical symmetry, and inparticular a spherical boundary metric. We begin with an overview of our model and thendiscuss disconnected solutions in section 4.1, and connected wormhole solutions in section4.2. In particular, we will see that connected wormholes can have lower action than thedisconnect solution. Finally, we show in section 4.3 that these low-action wormholes arestable in the sense that they have no Euclidean negative modes. Since this is merely an adhoc low-energy model not derived from string theory (or any other UV-complete theory),the discussion in section 4.3 concerns only field-theoretic negative modes. There is no– 9 –ossible notion of a brane-nucleation negative modes as the theory does not contain branes(nor does it contain non-singular magnetic monopoles).As described below, choosing the model to contain three distinct Maxwell fields willhelp us to arrange a cohomogeneity-1 solution; i.e., a solution that is homogeneous at each‘Euclidean time’ in the FLRW sense described above. We will thus search for wormholeand disconnected solutions to the equations of motion derived from the following action: S U (1) = − (cid:90) M d x √ g (cid:32) R + 6 L − (cid:88) I =1 F ( I ) ab F ( I ) ab (cid:33) − (cid:90) ∂ M d x √ h K + S B , (4.1)where L is the AdS length scale, K is the trace of the extrinsic curvature associated withan outward-pointing normal to ∂ M , h the determinant of the induced metric h µν on ∂ M and F ( I ) = d A ( I ) . Here and throughout this paper will take units in which πG = 1 .The second term in (4.1) is the so-called Gibbons-Hawking-York term. The final term S B includes a number of counterterms that render the Euclidean on-shell action finite andare functions of the intrinsic geometry on ∂ M only and are dimension dependent. For theabove theory in four bulk spacetime dimensions these turn out to be given by S B = 4 L (cid:90) ∂ M d x √ h + L (cid:90) ∂ M d x √ h R , (4.2)where R is the intrinsic Ricci scalar on ∂ M . One might wonder whether we need additionalboundary terms associated with F ( I ) such as the ones reported in [53]. However, as notedin [53], no such terms are needed if we are interested in fixing the leading value of A ( I ) aswe approach the conformal boundary, i.e. work in the grand-canonical ensemble. These areprecisely the boundary conditions we choose. The equations of motion derived from (4.1)read R ab + 3 L g ab = 2 (cid:88) I =1 (cid:16) F ( I ) ac F ( I ) cb − g ab F ( I ) cd F ( I ) cd (cid:17) , (4.3a) ∇ a F ( I ) ab = 0 . (4.3b)subject to the boundary conditions that on ∂M , the induced metric h is fixed as well as A ( I ) .We are interested in finding solutions for which the metric has the same isometriesas a round three-sphere, i.e. spherical symmetry, and in particular SO (4) , but where theMaxwell fields explicitly break such symmetry. An easy way to do so is to write the 3-spherein terms of left-invariant 1-forms { ˆ σ , ˆ σ , ˆ σ } such that the metric on the unit round threesphere reads dΩ = 14 (cid:0) ˆ σ + ˆ σ + ˆ σ (cid:1) , (4.4)with dˆ σ I = 12 ε IJK ˆ σ J ∧ ˆ σ K . (4.5)– 10 –n terms of standard Euler angles, we can choose ˆ σ = − sin ψ d θ + cos ψ sin θ d ˆ ϕ (4.6a) ˆ σ = cos ψ d θ + sin ψ sin θ d ˆ ϕ (4.6b) ˆ σ = d ψ + cos θ d ˆ ϕ (4.6c)with ψ ∈ (0 , π ) , θ ∈ (0 , π ) and ˆ ϕ ∈ (0 , π ) .We then search for solutions of the form d s = d r f ( r ) + g S ( r )dΩ , (4.7)For the Maxwell fields, we choose A ( I ) = L ˆ σ I r ) , for I ∈ { , , } . (4.8) A primary question will be whether connected wormhole solutions dominate over discon-nected solutions. We thus begin here by constructing the simpler disconnected solution forcomparison, deferring discussion of wormhole solutions to section 4.2 below.As is often the case, it is convenient to fix the gauge in Eq. (4.7) by choosing g S ( r ) = r . (4.9)We take r ∈ (0 , + ∞ ) , with r = 0 describing the smooth center where the round S shrinkssmoothly to zero size and r = + ∞ the location of the asymptotic conformal boundary.Regularity of F ( I ) at the origin demands that dΦ( r )d r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = 0 , (4.10)with Φ(0) being a constant, whereas regularity of the metric at the point r = 0 demands f (0) = 1 . (4.11)Note that regularity of A ( I ) seen as a 1-form demands that Φ( r ) = O ( r ) , which is strongerthan the condition expressed by Eq. (4.10).With our choice of boundary conditions, we find a unique solution given by f ( r ) = 1 + r L and Φ( r ) = Φ √ L + r − L √ L + r + L . (4.12)It turns out that each member of this one-parameter family of solutions is self-dual, in thesense that (cid:63) F ( I ) = F ( I ) , (4.13)where (cid:63) is the Hodge dual operation in four spacetime dimensions. Recall that for self-dualsolutions the stress energy tensor induced by F ( I ) is identically zero, which is why the– 11 –etric is identically Euclidean AdS for any value of Φ . Note also that the above solutionhas Φ( r ) = O ( r ) near r = 0 so that our boundary condition is satisfied.It is straightforward to evaluate the Euclidean on-shell action on these solutions. Itmust of course be a function of Φ only, and we find the particular form S DU (1) = 8 π L (cid:0) (cid:1) . (4.14)Here the upper-script D on the left-hand-side denotes the on-shell action of the disconnectedsolution. Having found our disconnected solution, we now turn to the study of smooth connectedwormholes. Since such solutions will have a minimal sphere of some non-zero area πr , wenow choose to fix the gauge in Eq. (4.7) by writing g S ( r ) = r + r . (4.15)Now r ∈ R , with the two asymptotic boundaries located at r → ±∞ . We also imposea global Z symmetry that relates the two spheres of given r (cid:54) = r and which leaves theminimal sphere fixed. We shall see that the parameter r will be a function of Φ only.Without loss of generality we will take r > .Again, we can integrate our equations of motion to find the full space of such solutions.Note that our Z symmetry requires dΦ( r )d r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = 0 . (4.16)From the equations for the Maxwell fields we find f ( r ) = C + 4Φ( r ) ( r + r )Φ (cid:48) ( r ) , (4.17)where C is a constant to be determined later and (cid:48) denotes differentiation with respect to r . Consistency of the rr and S S components of the Einstein equation then demands Φ (cid:48) ( r ) − L r (cid:2) C + 4Φ( r ) (cid:3)(cid:0) r + r (cid:1) (cid:2) CL + (cid:0) r + r (cid:1) (cid:0) L + r + r (cid:1)(cid:3) = 0 . (4.18)Since this gives an explicit result for [ C + 4Φ( r ) ] / Φ (cid:48) it allows us to write f ( r ) in the form f ( r ) = CL + (cid:0) r + r (cid:1) (cid:0) L + r + r (cid:1) L r . (4.19)Thus, we find a singularity at r = 0 , unless we set C = − L r + r L , (4.20)With this choice for C one obtains f ( r ) = L + r + 2 r L , (4.21)– 12 –hich is smooth at r = 0 as desired. Using our choice of the constant C in (4.18) yields Φ (cid:48) ( r ) − L Φ( r ) − r (cid:0) L + r (cid:1) L (cid:0) r + r (cid:1) (cid:0) L + r + 2 r (cid:1) = 0 . (4.22)Since we require Φ (cid:48) (0) = 0 at r = 0 we want to impose, the above fixes r in terms of Φ(0) ≡ Φ (cid:63) to be r = b L with a ≡ (1 + 16Φ (cid:63) ) / and b ≡ (cid:114) a − . (4.23)With these choices, the equation for Φ can be readily solved to give Φ( r ) = Φ (cid:63) cosh (cid:20) b F (cid:18) arctan (cid:16) rL a (cid:17) (cid:12)(cid:12)(cid:12) − a b (cid:19)(cid:21) , (4.24)where F ( φ | m ) is the elliptic integral of the first kind.To determine the source Φ in terms of Φ (cid:63) we simply expand the above Φ( r ) at large r , to find Φ (Φ (cid:63) ) = Φ (cid:63) cosh (cid:20) b K (cid:18) − a b (cid:19)(cid:21) , (4.25)where K ( m ) is the complete elliptic integral of the first kind. We stress that Φ is the actualchemical potential for A ( I ) , but that we find it more convenient to parameterize the solutionsin terms of Φ (cid:63) . The reason for this is that there can be more than one solution for a givenvalue of Φ , but that solutions are uniquely determined by their value of Φ ∗ . This is bestillustrated by looking a plot of Φ (Φ (cid:63) ) (see Fig. 2). From this plot it is clear that wormholesolutions can only exist if Φ ≥ Φ min0 ≈ . , for which Φ (cid:63) = Φ min (cid:63) ≈ . . � � � � � ���������� Figure 2 . The source for the Maxwell fields A ( I ) Φ as a function of Φ (cid:63) . There is a minimumvalue of Φ , Φ min0 ≈ . , above which two types of wormhole solutions exist. But what distinguishes the two wormholes with a given value of Φ ? Perhaps the bestway to see the answer is to study the radius r of the wormhole throat as a function of Φ shown in Fig. 3. For any fixed value of Φ > Φ min0 ≈ . , two wormhole solutions exist:– 13 – large wormhole and a small wormhole. For Φ = Φ min0 we have r = r min0 ≈ . L ,and both the large and small branches merge. We shall see that these solutions behave justlike small and large Euclidean Schwarzschild black holes in global AdS. In particular, wewill show that the small wormhole branch has a negative mode, and the large wormholebranch does not. � � �� �� ������� Figure 3 . Radius of the wormhole solutions as a function of the Maxwell source Φ . For fixedvalue of Φ > Φ min0 two wormhole solutions exist. One can also evaluate the Euclidean on-shell associated with our wormhole solutions,which can be written in terms of complete elliptic integrals of the first and second kind inthe form S CU (1) = 8 L π ( X − / (cid:34) X − E ( − X ) − ( X − K ( − X )+ 3 X √ X − (cid:16) √ X − K ( − X ) (cid:17) (cid:35) . (4.26)Here we defined X ≡ L r , (4.27)and E ( m ) is the complete elliptic integral fo the second kind.We can finally plot one of our figures of merit, namely ∆ S U (1) ≡ S DU (1) − S CU (1) . (4.28)If ∆ S U (1) is positive, the wormhole solution has lower Euclidean action than the discon-nected solution with the same value of Φ . If, on the other hand, ∆ S U (1) < , it must bethat the wormhole solution is subdominant. We find that the large wormhole solutions aredominant for Φ > Φ HP ≈ . , and subdominant otherwise. We denote the transitionvalue by Φ HP due to the similarity to the familiar Hawking-Page transition. The smallwormhole solutions are always subdominant. These two behaviours are displayed in Fig. 4.– 14 – � � � � �� - ��� - �������� Figure 4 . The difference in Euclidean action between the disconnected and connected solutions.At any Φ , the larger value of ∆ S U (1) corresponds to the large wormhole and the smaller valuecorresponds to the small wormhole. Due to the similarity to the familiar Hawking-Page transition,we use Φ HP to denote the value of Φ at which ∆ S U (1) = 0 for the large wormhole. For Φ > Φ HP , the large wormhole solution becomes dominant while the small wormhole solution is alwayssubdominant. A similar structure was found previously for wormholes in JT gravity coupled to matter[17]. Having found dominant saddles, we now proceed to determine their stability.We now note that all the solutions we found, either connected or disconnected in thebulk, satisfy a Euclidean version of the first lawinvolving the Euclidean action S E . Accord-ing to standard lore in AdS/ CFT, one can find the expectation value of the operators dualto A I by simply taking a functional derivative of the action with respect to the correspond-ing boundary value of A I (cid:104)J Iµ (cid:105) = δS E δA I µ . (4.29)where Greek indices run over boundary coordinates. These can be easily evaluated onarbitrary on shell solution and it turns out that (cid:104)J Iµ (cid:105) = 8 π L ˜ J Iµ . (4.30)with ˜ J Iµ being given by A I = A Ia d x a = Φ Iµ d x µ − z ˜ J Iµ d x µ + O ( z ) (4.31)where z is a Fefferman-Graham coordinate [54]. This in turn implies that d S E = ε J Iµ dΦ I µ , (4.32)where ε = 1 for disconnected solutions and ε = 2 for wormholes with two boundaries. Wehave checked that our solutions satisfy this relation. Perhaps more importantly, appropriategeneralisation of this type of first law also arise when studying scalar wormhole solutions,which we were only able to study numerically. We have checked that all our numericalsolutions satisfy the above relations to better than − % accuracy.– 15 – .3 Negative modes We now discuss perturbations around our wormholes, and in particular, the possible ex-istence of negative modes. We will take advantage of the SO (4) symmetry of the S to decompose the perturbations into spherical harmonics. Perturbations then will comeinto three different classes: scalar-derived perturbations, vector-derived perturbations andtensor-derived perturbations. These are built from scalar, vector and tensor harmonicson the S . We shall label each of these structure functions by S (cid:96) S , S (cid:96) V i and S (cid:96) T ij , with (cid:96) S = 0 , , , . . . , (cid:96) V = 1 , , . . . and (cid:96) T = 2 , , . . . and i, j running over the sphere directions.These structure functions are chosen so that they are orthogonal to each other in theabsence of background fields that might break the SO (4) symmetry. Unfortunately, theMaxwell fields do break SO (4) , so we will need more structure. Nevertheless, we willbe able to use these building blocks to study the negative modes. When there is SO (4) background symmetry of the background, orthogonality only occurs if we take S (cid:96) V i to bedivergence free and S (cid:96) T ij to be traceless-transverse. All these operations are, of course, donewith respect to the metric on the round three-sphere. The scalars in addition satisfy (cid:3) S S (cid:96) S + λ S S (cid:96) S = 0 , (4.33a)with λ S = (cid:96) S ( (cid:96) S + 2) , the vectors (cid:3) S S (cid:96) V i + λ V S (cid:96) V i = 0 , (4.33b)with λ V = (cid:96) V ( (cid:96) V + 2) − and the tensors (cid:3) S S (cid:96) T ij + λ T S (cid:96) T ij = 0 , (4.33c)with λ T = (cid:96) T ( (cid:96) T + 2) − . (cid:96) S = 0 sector This is the only sector where we find a negative mode, and it occurs only for the smallwormhole branch. In fact, the threshold for the existence of this mode coincides preciselywith r = r min0 . This is akin of what happens with Schwarschild-AdS, where a negativemode exists for small black holes, but not for large black holes [55]. The threshold can befound analytically by inspecting when the Schwarzschild-AdS black holes become locallythermodynamically stable, i.e. when the specific heat becomes positive. Note that inSchwarzschild-AdS this transition occurs before the Hawking-Page transition, so that whenthe large black hole branch dominates over pure thermal AdS, the negative mode is nolonger presence. We shall see a similar behaviour with the spherical wormholes.We start with an Ansatz for the (cid:96) = 0 sector. Since there are no vector or tensorperturbations on the S with (cid:96) = 0 , our Ansatz preserves the same symmetries as thebackground solution. We thus search for negative modes which take the same form asEqs. (4.7)-(4.8) with g ( r ) = g + δg ( r ) , f ( r ) = f + δf ( r ) and Φ( r ) = Φ( r ) + δ Φ( r ) , (4.34)– 16 –here Φ is given in Eq. (4.24) and g = r + r , and f = L + r + 2 r L . (4.35)We expand the action (4.1) to second order in δg , δf and δ Φ . The terms linear in δg , δf and δ Φ vanish by virtue of the background equations of motion. It is then a simple exerciseto recast the action as a function of δg , δf and δ Φ and their first derivatives only. Doingso involves integrating by parts, and the resulting boundary term precisely cancels theGibbons-Hawking-York term. We shall denote this quadratic action by S (2) . Furthermore,we find that δf enters the action algebraically as it should by virtue of the Bianchi identities.This makes it straightforward to integrate out δf (since the action is quadratic in δf andthus the path integral is Gaussian). The result of this procedure yields an action that isquadratic in δ Φ and δg and their first derivatives. We have also checked that the pathintegral in δf has the correct sign for a meaningful integration, i.e. the coefficient of theterm proportional to δf is negative definite. We denote the resulting action by ˜ S (2) .We wish to work with gauge invariant perturbations. In order to do this, we mustfirst understand how δg , δf and δ Φ transform under an infinitesimal gauge transformation ξ = ξ r ( r ) ∂/∂r . This is easily seen by recalling that a metric perturbation h and gauge fieldperturbation a , transforms under infinitesimal gauge transformations ξ as ∆ h = £ ξ g , ∆ a = £ ξ A (4.36)where £ ξ is the Lie derivative along ξ , and ( g, A ) are the metric and gauge potentialbackground fields.We then find ∆ δf = ξ r f (cid:48) − f ξ (cid:48) r , ∆ δg = 2 rξ r and ∆ δ Φ = ξ r Φ (cid:48) . (4.37)As a result, we can then build the gauge invariant quantity q = δ Φ − Φ (cid:48) δg r . (4.38)It is a trivial exercise to show that ∆ q = 0 . We can use this definition to write the quadraticaction ˜ S (2) for δg and δ Φ as a function of q . This is done by effectively solving Eq. (4.38)for δ Φ and substituting the resulting expression for δ Φ in the quadratic action ˜ S (2) . Thedependence in δg completely cancels out, as it should, due to gauge invariance. We are arethus left with ˜ S (2) written in terms of q , q and its first derivative only: ˜ S (2) = 2 π (cid:90) ∞−∞ d r (cid:115) gf (cid:34) f K q (cid:48) + V q (cid:35) , (4.39a)where K = 6 L r r − gL Φ (cid:48) and V = 4 Kg gL f r L (cid:16) r − L Φ Φ (cid:48) (cid:17) + g (cid:16) r − L Φ Φ (cid:48) (cid:17) r − gL Φ (cid:48) − . (4.39b)– 17 –wo comments regarding boundary terms are now in order. First, to show that δg drops out one needs to integrate by parts twice. This generates two boundary terms. Thesetwo terms precisely cancel the counterterms in Eq. (4.1). Second, in order to ensure thatterms proportional to qq (cid:48) do not show up in the final form of the action, we again had tointegrate by parts. It is easy to show that the resulting boundary terms vanish so long as q ∼ o ( r ) near the conformal boundary. As we shall see below, our boundary conditions for q will require that q vanishes at this rate near the boundary, so these terms can be safelyneglected.To search for negative modes, we integrate the first term in Eq. (4.39a) by parts towrite ˜ S (2) = 2 π (cid:90) ∞−∞ d r (cid:115) gf q (cid:40) − (cid:115) fg (cid:20)(cid:113) f g K q (cid:48) (cid:21) (cid:48) + V q (cid:41) . (4.40)The resulting boundary term can be neglected so long as q ∼ o ( r − / ) , and we will verifybelow that such boundary conditions may be imposed. The negative mode equation simplybecomes − (cid:115) fg (cid:20)(cid:113) f g K q (cid:48) (cid:21) (cid:48) + V q = λ q . (4.41)If we can find values of λ < for which this equation admits non-trivial solutions, thenfluctuations about the saddle will make large contributions and the Euclidean solution islocally unstable. Finally, we still need to check whether the possible behaviours that thisequation admits near the conformal boundary are consistent with imposing a boundarycondition that requires q ∼ o ( r − / ) , so that we can indeed neglect the above boundaryterms. A Frobenius analysis close to the conformal boundary reveals that q ∼ r − ∆ ± with ∆ ± = 12 ± (cid:114) − λ , . (4.42)We see that for λ < the ∆ + branch satisfies q ∼ o ( r − / ) . It is thus consistent to requirethis as a boundary condition and to then neglect the above-mentioned boundary terms.Since V ( r ) is symmetric around r = 0 , we can decompose our modes into modes with q (0) = 0 or q (cid:48) (0) = 0 . For numerical convenience we also define q = L ∆+ ( r + r ) ∆ + / ˆ q and r = r y − y (4.43)so that the conformal boundary is located at y = 1 and the origin at y = 0 . SolvingEq. (4.42) off the conformal boundary yields ˆ q (cid:48) (1) = 0 for the choice q ∼ r − ∆ + .Using the numerical methods first outlined in [56] and reviewed in [57] we search fornegative modes with the above boundary conditions. For ˆ q (0) = 0 we find no negativemodes for any value of r /L . On the other hand, for ˆ q (cid:48) (0) = 0 we find exactly one negativemode, which becomes positive when r = r min0 (see Fig. 5). This is precisely when thetransition between small and large wormholes occurs, in complete analogy with sphericalSchwarzschild-AdS black holes. – 18 – ��� ���� ���� ���� - �� - �� Figure 5 . The homogeneous negative mode with (cid:96) = 0 as a function of the black hole radius,measured in units of r min0 . At r = r min0 the negative mode vanishes, and becomes positive thereafter. (cid:96) S ≥ sector Here we partially follow the pioneering work of Kodama and Ishibashi in [58]. For themetric we take our perturbations to have the form δ d s (cid:96) S = h (cid:96) S rr ( r ) S (cid:96) S d r + 2 h (cid:96) S r ( r ) ∇∇ i S (cid:96) S d r d y i + H (cid:96) S T ( r ) S (cid:96) S ij d y i d y j + H (cid:96) S L ( r ) S (cid:96) S g ij d y i d y j (4.44)where ∇∇ is the covariant derivative on S , y are coordinates on the S , lower case Latinindices live on S , g is the round metric on S and S (cid:96) S ij = ∇∇ i ∇∇ j S (cid:96) S − g ij ∇∇ k ∇∇ k S (cid:96) S . (4.45)Here S (cid:96) S ij is by construction trace free. The perturbation of the gauge fields is more intricate.Since the background Maxwell fields break the full SO (4) , we expect that their perturba-tions will strongly depend on the background field. Note that S (cid:96) S ij vanishes identically for (cid:96) S = 1 so that this mode must be treated separately; see section 4.3.3.It is our objective to list all independent vector fields on S that are linear in S (cid:96) S andcan be built with the background fields A I and metric g . These turn out to be δA (cid:96) S I = A (cid:96) S ( r ) S (cid:96) S A I + B (cid:96) S ( r ) ˜ £ Ξ (cid:96)S A I + C (cid:96) S ( r ) A iI ∇∇ i S (cid:96) S d r + D (cid:96) S I ( r )d r + E (cid:96) S I ( r ) ∇∇ i S (cid:96) S d y i , (4.46)where ˜ £ is the Lie derivative acting on S and Ξ (cid:96) S = ∇∇ i S (cid:96) S ∂∂y i . (4.47)One might think that one would need to include additional terms of the form ι Ξ (cid:96)S d A I , (4.48)but these can be reabsorbed into the coefficients A (cid:96) S , B (cid:96) S , C (cid:96) S with a U (1) gauge transfor-mation on δA (cid:96) S I . – 19 –n order to check that our Ansatz is nontrivial, we have linearized the correspondingEinstein-Maxwell equations and found that there are three dynamical second order gaugeinvariant equations that govern such perturbations. Since we want to work with gaugeinvariant perturbations, we must find how these perturbations transform under infinitesimalgauge transformations (both U (1) and diffeomorphisms). In order to do this, we need tosort out how to write infinitesimal diffeomorphisms in terms of the scalar harmonics S (cid:96) S .Let ξ (cid:96) S be the infinitesimal diffeomorphism associated with S (cid:96) S . Following [58] we write ξ (cid:96) S as ξ (cid:96) S = ξ (cid:96) S r ( r ) S (cid:96) S d r + ξ (cid:96) S V ( r ) ∇∇ i S (cid:96) S d y i . (4.49)Our perturbations then transform as δh (cid:96) S rr = ξ (cid:96) S r f (cid:48) f + 2 ξ (cid:96) S (cid:48) r , δh (cid:96) S r = ξ (cid:96) S r − ξ (cid:96) S V g (cid:48) g + ξ (cid:96) S (cid:48) V ,δH (cid:96) S T = 2 ξ (cid:96) S V , δH (cid:96) S L = f g (cid:48) ξ (cid:96) S r − (cid:96) S ( (cid:96) S + 2) ξ (cid:96) S V ,δA (cid:96) S = f Φ (cid:48) Φ ξ (cid:96) S r , δB (cid:96) S = ξ (cid:96) S V gδC (cid:96) S = ξ (cid:96) S (cid:48) V − g (cid:48) g ξ (cid:96) S V , δD (cid:96) S I = 0 , and δE (cid:96) S I = 0 . (4.50)Similarly, under U (1) transformations δA (cid:96) S I → δA (cid:96) S I + d χ (cid:96) S I with gauge parameter χ (cid:96) S I = ˆ χ (cid:96) S I ( r ) S (cid:96) S we find δh (cid:96) S rr = δh (cid:96) S r = δH (cid:96) S T = δH (cid:96) S L = δA (cid:96) S = δB (cid:96) S = δC (cid:96) S = 0 ,δD (cid:96) S I = ˆ χ (cid:96) S (cid:48) I , and δE (cid:96) S I = ˆ χ (cid:96) S I . (4.51)Just as we did for the (cid:96) S = 0 mode, we now substitute our Ansatz into the Einstein-Maxwellaction (4.1) and expand to second order in the perturbations ˆ q ≡ { h (cid:96) S rr , h (cid:96) S r , H (cid:96) S L , H (cid:96) S T , A (cid:96) S , B (cid:96) S , C (cid:96) S , D (cid:96) S I , F (cid:96) S I } . (4.52)The resulting action, S (2) [ˆ q , ˆ q (cid:48) ] depends on both ˆ q and its first derivative ˆ q (cid:48) . Crucially,the dependence in the angular coordinates drops out, as it should. Upon further inspection,one notes that by performing further integrations by parts we can in fact write S (2) in aform that depends only algebraically on h (cid:96) S rr (i.e., it does not depend on derivatives of h (cid:96) S rr ).The associated boundary terms again either cancel with the Gibbons-Hawking-York termor with one of the boundary counter-terms. This means we can easily integrate out h (cid:96) S rr from the action (though this requires the Wick-rotation described in section 3) and find areduced action ˜ S (2) that does not depend on h (cid:96) S rr . Upon further scrutiny, one finds that,upon a couple of integration by parts, ˜ S (2) does not depend on derivatives of h (cid:96) S r so again itcan be integrated out. At this stage we are left with a quadratic action ˆ S (2) that dependsonly on { H (cid:96) S L , H (cid:96) S T , A (cid:96) S , B (cid:96) S , C (cid:96) S , D (cid:96) S I , F (cid:96) S I } . (4.53) To bring the action to this form, we integrated by parts term proportional to H (cid:96) S (cid:48)(cid:48) T and H (cid:96) S (cid:48)(cid:48) L , withthe boundary terms cancelling with the Gibbons-Hawking-York boundary term in (4.1) . – 20 –nd their first derivatives. At this stage, we introduce gauge-invariant variables with respectto the U (1) . Under such gauge transformations the variables { H (cid:96) S L , H (cid:96) S T , A (cid:96) S , B (cid:96) S , C (cid:96) S } are already invariant. However D (cid:96) S I and F (cid:96) S I transform non-trivially, so we define theinvariant combination f (cid:96) S I ≡ D (cid:96) S I − E (cid:96) S (cid:48) I ⇒ D (cid:96) S I = f (cid:96) S I + E (cid:96) S (cid:48) I . (4.54)Using this definition, we find that ˆ S (2) depends only on { H (cid:96) S L , H (cid:96) S T , A (cid:96) S , B (cid:96) S , C (cid:96) S , f (cid:96) S I } (4.55)and their first derivatives (as one would expect from the U (1) gauge invariance). Further-more, f (cid:96) S I decouples from all other variables and appears in ˆ S (2) as ˆ S (2) = ˇ S (2) + 4 π (cid:96) S ( (cid:96) S + 2) (cid:96) S + 1 (cid:88) I =1 (cid:90) + ∞−∞ d r (cid:112) f g ( f (cid:96) S I ) (4.56)where ˇ S (2) depends only on { H (cid:96) S L , H (cid:96) S T , A (cid:96) S , B (cid:96) S , C (cid:96) S } . (4.57)This means that f (cid:96) S I are non-dynamical in the sense that it enters the action algebraicallyand contributes positively to ˆ S (2) . We may thus focus our attention on ˇ S (2) .At this stage we introduce gauge invariant variables with respect to the diffeomorphisms(4.50). Since ξ (cid:96) S r and ξ (cid:96) S V enter the gauge transformations for H (cid:96) S L and H (cid:96) S T algebraically,we can use H (cid:96) S L and H (cid:96) S T to easily construct gauge invariant variables as follows. Define Q (cid:96) S = A (cid:96) S − Φ (cid:48) r Φ (cid:104) H (cid:96) S L + (cid:96) S ( (cid:96) S + 2) H (cid:96) S T (cid:105) , (4.58a) Q (cid:96) S = B (cid:96) S − H (cid:96) S T g , (4.58b) Q (cid:96) S = C (cid:96) S − H (cid:96) S (cid:48) T g (cid:48) g H (cid:96) S T . (4.58c)Using the transformations 4.50 it is relatively simple to see that δQ (cid:96) S = δQ (cid:96) S = δQ (cid:96) S = 0 .Using this relations, we can solve for A (cid:96) S , B (cid:96) S and C (cid:96) S and insert those expressions into ˇ S (2) . After some integrations by parts, the terms with H (cid:96) S L and H (cid:96) S T drop out, as theyshould due to gauge invariance. At this stage, ˇ S (2) is a function of { Q (cid:96) S , Q (cid:96) S , Q (cid:96) S , Q (cid:96) S (cid:48) , Q (cid:96) S (cid:48) } . (4.59)Remarkably, and for reasons we don’t fully understand, Q (cid:96) S only enters the action alge-braically (and with positive coefficient for the quadratic term). This allows us to performthe Gaussian integral over Q (cid:96) S and be left with an effective action for { Q (cid:96) S , Q (cid:96) S } which wedenote by S (2) F . – 21 –t turns out to be beneficial to perform one final change of variable and write Q (cid:96) S = 2 L Φ (cid:34) q (cid:96) S − gλ S (cid:0) r + 4 L ΦΦ (cid:48) (cid:1) r ( gλ S + 24 L Φ ) q (cid:96) S (cid:35) , and Q (cid:96) S = − L Φ q (cid:96) S . (4.60)The final action then takes the following form S (2) F = (cid:90) + ∞−∞ d r g / √ f (cid:104) f ( D q (cid:96) S ) I K IJ ( D q (cid:96) S ) J + q (cid:96) S I V IJ q (cid:96) S J (cid:105) (4.61)where I, J ∈ { , } , ( D q (cid:96) S ) I = q (cid:96) S (cid:48) I + ε JI q (cid:96) S J , ε JI = ε IJ K IJ , ε IJ = (cid:34) λ (cid:35) , (4.62)and λ = ( (cid:96) S + 1)32 π ( λ S − L rf ( gλ S + 24 L Φ ) (cid:104) L r Φ f Φ (cid:48) ( g − L Φ ) − L Φ (3 g + gλ S L + 3 gL + 3 L r + 3 r ) + gλ S r ( L + r ) + 96 L Φ (cid:105) . (4.63)Finally, the symmetric matrix V is given in appendix A.1 and the symmetric matrix K ismore easily expressed in terms of its inverse ( K − ) = g ( (cid:96) S + 1)256 π r ( λ S − gλ S + 24 L Φ ) (cid:34) g L λ S (cid:18) Φ (cid:48) − r Φ g (cid:19) ++ gr ( λ S − λ S + 16 g Φ ( g + L )( λ S − f (cid:18) − L Φ g ( g + L ) (cid:19) (cid:35) , (4.64a) ( K − ) = ( (cid:96) S + 1)64 π λ S ( λ S − (cid:0) λ S g + 24 L Φ (cid:1) , (4.64b) ( K − ) = 0 . (4.64c)It is not hard to show that K is positive definite for r /L > / / / ≈ . . First wenote that ( K − ) is positive definite, second we note that is is only the last term in ( K − ) that is not positive definite. However, it is a simple exercise to show that − L Φ g ( g + L ) > for r /L > / / / . Since / / / < r min0 /L , all large wormholes have positive definite K . The non-existence of negative modes can then be investigated by looking at the prop-erties of V . As the reader can see in appendix A.1, V is a rather complicated matrix, whoseeigenvalues we can only compute numerically as a function of r , for particular values of (cid:96) and r /L . This allows us to exclude portions of the ( r /L, (cid:96) S ) plane as potential regionswith negative modes. In Fig. (6) we plot the regions of parameter space where V is positivedefinite. It appears that if V is positive definite for given ( r /L, (cid:96) S ) , then it remains positive– 22 –efinite if we increase (cid:96) S while holding r /L fixed. Since we are looking at large wormholes,we took r /L > .We see, perhaps counter-intuitively, that the larger the value of r /L , the larger valueof (cid:96) S we have to achieve to see V being positive definite. This might at first look worrying,but in fact it is natural to expect angular momentum to have less effect at larger r (sincethe gradients associated with given (cid:96) S are smaller at large r ). Indeed, as r /L increaseswe find that the most negative eigenvalue of V moves toward zero. Thus V becomes lessand less negative at large r . � � �� �� �� �� ��������� Figure 6 . Disks represent regions of moduli space in the ( r /L, (cid:96) S ) plane where V is positivedefinite, and thus no negative mode exists. Note that if V is not positive definite, it does not necessarily mean that negativemodes exist. Indeed, for the complementary region we resort to solving for the spectrumnumerically. After doing so, we find no evidence for the existence of negative modes for r > L and for (cid:96) S ≥ . We have performed a thorough search in parameter space byscanning the large wormhole branch to about r ∼ and up to (cid:96) S ∼ . (cid:96) S = 1 sector This mode is special, because H (cid:96)T no longer appears in the metric perturbation. Apart fromthat, the construction is similar to what we have done for (cid:96) S ≥ . Perhaps the end resultis somehow surprising. Again, we find that the second order action for the perturbationscan be brought to a form similar to (4.56) with again f I contributing positively to theaction. However, we find that the second order action for the remaining variables vanishesidentically. Thus in the linearized theory these additional variables are pure gauge. Thisin turn means that they are the linearization of a pure-gauge mode in the full theory, asthe method applied by [59] to showing that Einstein-Maxwell theory admits a symmetric-hyperbolic formulation will also apply to our system. This in turn means that no gauge– 23 –egrees of freedom can remain in the linearized theory once the gauge symmetries of thefull theory have been fixed. We thus find that our wormhole is stable with respect togauge-invariant perturbations in this sector. (cid:96) V ≥ sector Things get more complicated, perhaps unexpectedly, when we move on to study vector-derived or tensor-derived perturbations. Note that when A I = 0 , these sectors are reallyeasy to study! The issue is that one can contract the fundamental vector harmonics S i with A Ii and build a scalar harmonic. So, in general, the vectors harmonics couple toscalar-derived perturbations. Thus their treatment will require all of the complicationsdiscussed above in the context of scalar-derived perturbations as well as treatment of thevector harmonics.An explicit discussion is thus extremely tedious but can be performed using preciselythe same techniques as in section 4.3.2. We suppress the details, but provide the followingremarks. It turns out that the vector derived perturbations only excite a few scalar-derivedperturbations and that they do not excite tensors-derived perturbations. For a given vectorharmonic S (cid:96) V i with (cid:96) V ≥ we find that we need to consider a sum of three scalar harmonicsof the form S (cid:96) V − and three harmonics of the form S (cid:96) V +1 . In each of these sectors, one of theharmonics is proportional to cos ψ , sin ψ or has no ψ dependence. It is also possible to findthe exact differential map between these harmonics. It is then an incredibly tedious exerciseto find the resulting action, and diagonalise accordingly using appropriate numerics. Wehave done so and find that the action is again positive definite.However, there is a sector in which this unpleasant coupling does not occur and whereone can work purely with vector harmonics. We will describe the calculations for thissimpler case in detail to illustrate the mechanics of working with the vector harmonicsthemselves. The simple sector involves vector-harmonics of the form S (cid:96) V i d y i = [ − sin( mψ ) d θ + cos( mψ ) sin θ d ˆ ϕ ] sin | m − | θ . (4.65)Regularity at θ = 0 and θ = π demands that m ∈ Z . The case with m = 1 is special,and because the background is invariant under ψ → − ψ , φ → − φ we can take m ≥ without loss of generality. It is a simple exercise to show that (cid:96) V = 2 m − .For the metric perturbation we take δ d s (cid:96) V = 2 h (cid:96) V r ( r ) S (cid:96) V i d r d y i + H (cid:96) V T ( r ) S (cid:96) V ij d y i d y j (4.66)with S (cid:96) T ij = ∇∇ i S (cid:96) V j + ∇∇ j S (cid:96) V i . (4.67)While for the gauge field perturbations we take δA (cid:96) V I = A (cid:96) V ( r ) ˜ £ Ξ (cid:96)V A I + B (cid:96) S ( r ) A iI S (cid:96) V i d r + C (cid:96) V I ( r ) S (cid:96) V i d y i (4.68)where ˜ £ is the Lie derivative acting on S and Ξ (cid:96) V = S i (cid:96) V ∂∂y i . (4.69)– 24 –ust as before we can ask how these perturbations behave under infinitesimal coordinatetransformations. Here, the infinitesimal diffeomorphisms are parameterized via ξ (cid:96) V = L (cid:96) V ( r ) S i (cid:96) V ∂∂y i , (4.70)and it turns out this induces the following transformations δh (cid:96) V r = L (cid:96) V (cid:48) − L (cid:96) V g (cid:48) g , δH (cid:96) V T = L (cid:96) V , δA (cid:96) V = L (cid:96) V g ,δB (cid:96) V = L (cid:96) V (cid:48) − L (cid:96) V g (cid:48) g and δC (cid:96) V = 0 . (4.71)From here onwards, the procedures are very similar to what we have seen for the scalars.First, we compute the second order action S (2) which is a function of h (cid:96) V r , h (cid:96) V T , A (cid:96) V , B (cid:96) V r , C (cid:96) V r and their first derivatives. Furthermore, it also depends on the second derivativesof h (cid:96) V T . One can integrate by parts the term proportional to the second derivative of h (cid:96) V T ,reducing S (2) to a function of first derivatives only. The resulting boundary term is cancelledby the Gibbons-Hawking-York boundary action, as it should.One then notes that S (2) only depends on h (cid:96) V r , but not on its first derivative. Thismeans we can perform the (here already with the convergent sign) Gaussian integral over h (cid:96) V r and find an action ˜ S (2) depending on h (cid:96) V T , A (cid:96) V , B (cid:96) V r , C (cid:96) V r and their first derivatives.At this point we further notice that C (cid:96) V r completely decouples from the remaining action,and furthermore that its contribution to ˜ S (2) is manifestly positive definite. We are thusleft to study an effective action ˆ S (2) for h (cid:96) V T , A (cid:96) V , B (cid:96) V r and their first derivatives. Finally,we make use of the gauge transformations (4.71) and introduce gauge invariant variables ofthe form q (cid:96) V = A (cid:96) V − h (cid:96) V T g and q (cid:96) V = B (cid:96) V + g (cid:48) g h (cid:96) V T − h (cid:96) V (cid:48) T . (4.72)It is a simple exercise to check that δq (cid:96) V = δq (cid:96) V . After some integration by parts, the de-pendence in h (cid:96) V T drops out completely (as it should by virtue of diffeomorphism invariance),and ˆ S (2) is solely written in terms of q (cid:96) V , its first derivative and of q (cid:96) V . Since q (cid:96) V onlyenters algebraically, we can perform the Gaussian path integral and find an action S (2) F for q (cid:96) V only. It is convenient to perform one last change of variable and write q (cid:96) V = Q (cid:96) V Φ . (4.73)The final action for Q (cid:96) V reads S (2) F = 32 π / L ( m − ( m + 1)Γ( m )Γ (cid:0) m + (cid:1) (cid:90) + ∞−∞ d r g / √ f (cid:40) fg ( m − m + 4 L Φ Q (cid:96) V (cid:48) + V (cid:96) V ( r )[ g ( m − m + 4 L Φ ] Q (cid:96) V (cid:41) , (4.74)– 25 –ith V (cid:96) V ( r ) = 4 g L ( m − (cid:40) f gL m Φ r Φ (cid:48) + 4 L Φ (cid:2) L (cid:0) gm − gm + r (cid:1) + r (cid:3) + gmL (cid:2) m (cid:0) gm − gm + r (cid:1) + r (cid:3) + m ( m + 1) gr (cid:41) . (4.75)For m ≥ all terms appearing in V (cid:96) V ( r ) are positive definite (note that r Φ (cid:48) is positivedefinite for r ∈ R ), and thus no negative mode exists in this sector as well. (cid:96) V = 1 sector This mode is again special because S (cid:96) V ij vanishes identically, and thus h (cid:96) V T does not enter thecalculation. Again, we find that C contributes positively to the action, but the remaininggauge invariant variables have a vanishing action (and thus as in section 4.3.3 are in factpure gauge under some special residual gauge transformations). We thus find that ourwormhole is stable with respect to gauge-invariant perturbations in this sector. (cid:96) ≥ sector The analysis of general tensor-derived perturbations is even more complicated than in thevector-derived case. The tenor-derived perturbations in general couple to both scalar-derived (with (cid:96) S = (cid:96) T ± ) and vector-derived perturbations (with (cid:96) V = (cid:96) T ± ). Onceagain, the general calculation can be performed by combining an analysis of the tensor-derived harmonics with the techniques used above, and doing so yields an action which(with appropriate numerics) can be verified to be positive definite. As in the vector-derivedcase we suppress the details of this extremely tedious general study and limit explicitdiscussion to a special type of tensor-derived perturbation that does not source eithervector- or scalar-derived harmonics. This allows us to illustrate the treatment of the purelytensor-derived part. Combining such a treatment with the methods used above for scalar-and vector-derived perturbations then suffices to treat the general case.The simple tensor-derived sector is descreibed by metric perturbations of the form aδ d s = g √ L h T ( r ) (cid:110) ( σ − σ ) cos [( m − ψ ] + 2 σ σ sin [( m − ψ ] (cid:111) sin | m − | θ ≡ g √ L h T ( r ) S ij d y i d y j , (4.76)with m ∈ Z and without loss of generality we take m ≥ . For the gauge field perturbationwe take δA I = g L a T ( r ) S ij d y i A I j . (4.77)Both h T ( r ) and a T ( r ) are automatically gauge invariant with respect to both infinitesimaldiffeomorphisms and gauge perturbations. To show this, recall that S ij is transverse andtrace free. One can readily compute (cid:96) T by using Eq. (4.33c) and it turns out (cid:96) T = 2( m − .Everything is much simpler now because these quantities are gauge invariant. In particular,in order to cast the quadratic action in an adequate form we only need to remove a term– 26 –roportional to the second derivative of h T . The boundary term readily cancels off theusual Gibbons-Hawking-York term. It turns out we can write the second order action inthe form S (2) F = π / Γ( m − L Γ (cid:0) m − (cid:1) (cid:90) + ∞−∞ d r g / f / (cid:104) f ( D q (cid:96) S ) I K IJ ( D q (cid:96) S ) J + q (cid:96) S I V IJ q (cid:96) S J (cid:105) , (4.78)where I, J ∈ { , } , ( D q (cid:96) S ) I = q (cid:96) S (cid:48) I + ε JI q (cid:96) S J , ε JI = ε IJ K IJ , ε IJ = (cid:34) β Φ (cid:48) β Φ (cid:48) (cid:35) . (4.79)where β , β , β and β are constants, K IJ = L g ( β + β Φ) L g ( β + β Φ) L g ( β + β Φ) L g (4.80)and h T = q and a T = ( β + β Φ) q + q . (4.81)Note that det K > and Tr K > so that K is a positive definite symmetric matrix. Inaddition, we must take β = 2 √ − β + β so that terms of the form q (cid:48) I q J for I (cid:54) = J donot feature the action. The symmetric matrix V is incredibly cumbersome to write downexplicitly, and not very illuminating. What is important is that we can choose β , β and β so that V is also positive. A good choice turns out to be β = − √ m − , β = − √ m + 2) and β = 17( m − √ m − . (4.82)For the above choice, we present V in appendix A.2. S boundary Let us now consider a different class of simple asymptotically AdS models in which thewormhole is sourced by the stress-energy of scalar fields. We again assume spherical sym-metry, and in particular a spherical boundary metric. After giving an overview of themodel, we briefly describe the ansätze we use to study the connected wormhole and thedisconnected solution. Computing the actions require more numerics than in the Einstein-Maxwell model of section 4, so we devote a separate subsection to discussing the results ofsuch computations. A final subsection considers potential negative modes in direct parallelwith the discussion of section 4.3. The final results also mirror those of section 4, as weagain find a Hawking-Page-like structure with two branches of wormholes (large and small).Moreover, the large wormholes are once again free of negative modes and dominate overthe disconnected solution when they are sufficiently large. However, this model also has nopossible notion of a brane-nucleation negative modes as the theory does not contain branes.– 27 –n particular, while we will later discuss its relation to certain string-theoretic setups, thecurrent model is not UV-complete.We will study both conformally coupled scalars and massless scalars, though for themoment we include an arbitrary mass parameter µ . Our action is given by S = − (cid:90) M d x √ g (cid:20) R + 6 L − ∇ a (cid:126) Π) · ( ∇ a (cid:126) Π) ∗ − µ (cid:126) Π · (cid:126) Π ∗ (cid:21) − (cid:90) ∂ M d x √ h K + S µ B , (5.1)where L is the four-dimensional AdS length scale, (cid:126) Π describes a doublet of complex scalarfields, and ∗ denotes complex conjugation. The second term in (5.1) is the usual Gibbons-Hawking term and S µ B the boundary counter-term to make the action finite and the vari-ational problem well defined. The precise form of S µ B will explicitly depend on µ .The Einstein equation and scalar field equation derived from this action read R ab − R g ab − L g ab = 2 ∇ ( a (cid:126) Π · ∇ b ) (cid:126) Π ∗ − g ab ∇ c (cid:126) Π · ∇ c (cid:126) Π ∗ − µ g ab (cid:126) Π · (cid:126) Π ∗ , (5.2a) (cid:3) (cid:126) Π = µ (cid:126) Π . (5.2b)We now note that if we take the trace of the Einstein equation, we find R = − L + 2( ∇ a (cid:126) Π) · ( ∇ a (cid:126) Π) + 4 µ (cid:126) Π · (cid:126) Π ∗ . (5.3)As such, the on-shell Euclidean action can be computed by evaluating the following bulkintegral S on − shell = (cid:90) M d x √ g (cid:20) L − µ (cid:126) Π · (cid:126) Π ∗ (cid:21) − (cid:90) ∂ M d x √ h K + S µ B . (5.4)The precise form of S B will be important to evaluate this on-shell action and depends onthe boundary conditions that we impose on scalar doublet (cid:126) Π . We will take µ to be zeroor to take the conformal value µ L = − .In standard Fefferman-Graham coordinates [54] the metric takes schematic form d s = L z (cid:2) d z + ˆ g µν ( x, z )d x µ d x ν (cid:3) , (5.5)where z = 0 marks the location of the conformal boundary and recall that Greek indices runover boundary directions only. One can then show that ˆ g µν ( x, z ) admits a simple expansionin terms of a power series in z , possibly with log z terms depending on the scalar field mass µ . For µ L = − and µ = 0 the log z terms can be shown to be absent and ˆ g µν ( x, z ) canbe expanded as ˆ g µν ( x, z ) = g µν ( x ) + z g (2) µν ( x ) + z g (3) µν ( x ) + o ( z ) , (5.6)where g µν ( x ) is interpreted as the boundary metric. We can now explain a little bit betterwith what we mean by ∂ M in Eq. (5.1). The surface ∂ M is defined as the ε → limit ofhypersurfaces ∂ M ε on which z = ε . Furthermore, h µν ( ε ) is the induced metric on ∂ M ε .In this sense, lim ε → ε h µν /L = g µν ( x ) . Note that in the wormhole case ∂ M has twoconnected components, one at each end of the wormhole.– 28 –n FG coordinates the scalar doublet (cid:126) Π can be expanded as (cid:126) Π = (cid:126) Π − ( x ) z ∆ − [1 + o (1)] + (cid:126) Π + ( x ) z ∆ + [1 + o (1)] , (5.7)with ∆ ± = 32 ± (cid:114)
94 + µ L . (5.8)Throughout this section fix (cid:126) Π + as a boundary condition (associated with “standard quan-tization” in the language of [60]), so that in AdS/CFT (cid:126) Π − ( x ) becomes the expectationvalue of the operator dual to (cid:126) Π . For the massless and conformal cases we have ∆ − = 0 and ∆ − = 1 respectively. This choice (partially) dictates what S B should be to make thevariational problem well defined. That is to say, when deriving the equations of motion wewant to ensure that we keep (cid:126) Π − ( x ) fixed and that our boundary terms are consistent withsuch choice. The remaining freedom in choosing S B is fixed so that the action (5.1) is finite.In the massless case we take S µ =0 B = − L (cid:90) ∂ M d x √ h − L (cid:90) ∂ M d x √ hR h + 2 L (cid:90) ∂ M d x √ hh µν ∇ hµ (cid:126) Φ · ∇ hν (cid:126) Φ (cid:63) , (5.9)where R h is the Ricci scalar associated with h and ∇ h its metric preserving connection.For the conformal case we take S µ L = − B = − L (cid:90) ∂ M d x √ h − L (cid:90) ∂ M d x √ hR h − L (cid:90) ∂ M d x √ h (cid:126) Φ · (cid:126) Φ (cid:63) . (5.10)We are interested in finding solutions where the metric enjoys spherical symmetry butthe scalars are chosen in such a way that these symmetries are broken. In particular, wemight consider d s = g rr ( r )d r + g S ( r )dΩ , (5.11)where dΩ is the unit round three-sphere, g rr ( r ) , g S ( r ) are to be determined later and r is an arbitrary bulk coordinate.For the scalar fields, we take (cid:126) Π = (cid:126)X Π( r ) (5.12)with Π( r ) ∈ R and (cid:126)X a two dimensional complex unit vector on S with d (cid:126)X (cid:54) = 0 and (cid:126)X · (cid:126)X ∗ = 1 .In fact, the coordinates of (5.11) turn out to be inconvenient for constructing our solu-tions. We thus present two new Ansätze below corresponding to connected or disconnectedsolutions and which specify our gauge choice slightly differently in each case.
Ansatz for the wormhole solutions
When searching for wormhole solutions, we will take d s = L (1 − ˜ y ) (cid:26) f (˜ y )d˜ y − ˜ y + y dΩ (cid:27) . (5.13)Clearly, Eq. (5.13) falls into the same symmetry class as Eq. (5.11), and the factors of ˜ y ∈ ( − , where chosen to that asymptotically (as ˜ y → ± ) f → . From the form of– 29 –he Ansatz above it is clear that y is the minimal radius of the S in the interior, which isattained at ˜ y = 0 . The value of this radius for given boundary sources will be determinednumerically. For the scalar field we take Π = (1 − ˜ y ) ∆ + q . (5.14)Let us describe the boundary conditions at ˜ y = 0 in detail. We wish to impose thereflection symmetry ˜ y → − ˜ y which leaves the locus ˜ y = 0 invariant. As a result, f (˜ y ) = f ( − ˜ y ) and q (˜ y ) = q ( − ˜ y ) . This implies d f d˜ y (cid:12)(cid:12)(cid:12)(cid:12) ˜ y =0 = d q d˜ y (cid:12)(cid:12)(cid:12)(cid:12) ˜ y =0 = 0 . (5.15)The requirement (5.15) in fact motivates us to choose a different coordinate that automat-ically enforces these conditions. In particular, we use y := ˜ y with y ∈ (0 , . The lineelement now reads d s = L (1 − y ) (cid:26) f ( y )d y (2 − y ) y + y dΩ (cid:27) , (5.16)and the boundary conditions (5.15) become just the statement that f and q are regular at y = 0 in the sense that (using further input from the 2nd order equations of motion) bothmust admit a Taylor series expansion about this locus. The equations in the y coordinatesand in this gauge are sufficiently compact to to present here: (1 − y ) √ − y √ y √ f dd y (cid:20) √ − y √ y √ f (1 − y ) dΠd y (cid:21) − − y ) y Π − L µ Π = 0 , (5.17a) f = (2 − y ) y y (cid:2) (1 − y ) + y (cid:3) − Π (cid:2) − y ) + µ L y (cid:3) (cid:34) − (1 − y ) (cid:18) dΠd y (cid:19) (cid:35) . (5.17b)A non-singular wormhole must have f being non-zero and finite everywhere, and in partic-ular at y = 0 . Looking at the expression above for f , it follows that at y = 0 we must have Π(0) = √ (cid:112) y (cid:112) µ L y , (5.18a)where without loss of generality we took q (0) > . Assuming a regular Taylor series for q around y = 0 , it also follows that dΠd y (cid:12)(cid:12)(cid:12)(cid:12) y =0 = √ (cid:112) y (cid:0) L y µ (cid:1) / y (3 − L µ ) and f (0) = 3 + L y µ − L µ . (5.18b)Note that the four-dimensional Breitenlohner-Freedman bound ensures that both quantitiesabove are positive definite.At the conformal boundary we find q = V − y V (1 − y ) + λ (1 − y ) − y V (1 − y ) + O (cid:2) (1 − y ) (cid:3) for µ = 0 , Vy + κ (1 − y ) + V ( V ) y (1 − y ) + O (cid:2) (1 − y ) (cid:3) for µ L = − (5.19)– 30 –here λ and κ are constants. The metric (5.16) is not yet in FG coordinates, so we are notyet ready to read off the values of boundary sources. In order to achieve this we need toperform a change of variables from y to z . We only require this change asymptotically, y = − y z − (cid:0) − V (cid:1) y z + O ( z ) for µ = 0 , − y z − (cid:0) V (cid:1) y z − κV y z + O ( z ) for µL = − . (5.20)We can now compare the expansion for Π in powers of z with the general expansion (5.7)which gives (cid:126) Π + = (cid:126)X V (5.21)for both µ = 0 and µ L = − . We thus see that V is to be interpreted as the source ofthe operator dual to (cid:126) Π .The numerical procedure to find these solutions is clear. We take a value for y , and usethe boundary conditions (5.18) to numerically integrate the equations outwards to y = 1 .Once this is done, we read off V , thus finding what source was needed to source a wormholewith minimal size y . To do this, we write the equations in first order form, and use aChebyshev collocation grid on Gauss-Lobatto points to perform the numerical integration.Because in our gauge the expansion of q can be shown to be analytic at the singular points y = 0 , we obtain exponential convergence in the number of grid points as we approachthe continuum limit.Our integration also determines the values of λ and κ in (5.19). These are related tothe expectation value (cid:104)O (cid:126) Π (cid:105) of the operator dual to (cid:126) Π via (cid:104) (cid:126) O (cid:126) Π (cid:105) = δSδ(cid:126) Φ + = (cid:126)X − π λL y for µ = 0 , π κL y for µL = − . (5.22)Finally, we give a few words on how to evaluate (5.4) in a numerically stable manner.We first look at how the first term in (5.4) diverges in inverse powers of (1 − y ) and √ y .We then add and subtract a regulator with precisely the same singularity structure, butone that we can integrate analytically. More precisely we use S π L = (cid:90) + ∞ d y √ f y (1 − y ) √ − y √ y (cid:0) − L µ Π (cid:1) − (cid:90) ∂ M d x √ h K + S µ B = (cid:90) + ∞ d y (cid:20) √ f y (1 − y ) √ − y √ y (cid:0) − L µ Π (cid:1) − G (cid:21) + (cid:90) + ∞ d y G − (cid:90) ∂ M d x √ h K + S µ B , (5.23)where G takes the form G = G ( − (1 − y ) + G ( − (1 − y ) + G (0) + G (1) (1 − y ) √ y . (5.24)Here all G ( i ) are constant and are chosen in such a way that the first integrand on thesecond line of (5.23) vanishes as y → and as y → . These constants can be found– 31 –nalytically, because we know the expansion for all functions via (5.19). Once this is done,we can analytically perform the integrations on the last line of (5.23) and check that theresult is finite as desired. The numerical integral that remains is then manifestly finite,with all the complicated cancelations having been implemented analytically. Ansatz for the disconnected solution
The procedure for the disconnected solutions is similar to what we have just described forthe wormhole, so we will be more brief here. The
Ansatz reads d s = L (1 − ˜ y ) (cid:26) f (˜ y )d˜ y − ˜ y + ˜ y (2 − ˜ y )dΩ (cid:27) , (5.25)with the regular centre located at ˜ y = 0 and the conformal boundary at ˜ y = 1 . Regularityat ˜ y = 0 now demands that f ( y ) = 1 + O (˜ y ) and Π( y ) = Q (0) ˜ y [1 + O (˜ y )] , (5.26)with Q (0) constant. These regularity conditions suggest introducing a coordinate ˜ y = y just as before, and they also suggest redefining Π = √ y (cid:112) − y (1 − y ) ∆ + q . (5.27)At the conformal boundary y = 1 we demand Π(1) = q (1) = V . For both µ = 0 and µ L = − this coincides with fixing the source for the expectation value dual to (cid:126) Π .Just as before, the Einstein equation and scalar field equation yield a single equationfor q which we can solve numerically given the boundary conditions above. The proceduresto regulate the action and to identify the source are very similar to the ones used for thewormhole geometry, so we will suppress this discussion here and pass directly to resultsbelow. Let us define ∆ S = 2 S D − S W , (5.28)where S D is the Euclidean on-shell action for the disconnected solution and S W the on-shellaction for the wormhole solution with the same values of the boundary sources (cid:126) Π + . Notethat both of these actions are finite due to the counter-terms.We first focus on the massless case µ = 0 . As shown in Fig. 7, our numerical resultsindicate that ∆ S crosses zero and becomes positive for V > V HP ≈ . . In addition,for any V ≥ V min ≈ . , there are two wormholes for a given value of V . At V min wefind y = y min0 ≈ . . The solution with larger ∆ S we will call the large wormhole,and the one with lower ∆ S we call the small wormhole. These phases are depicted as bluedisks (large wormhole) and orange squares (small wormhole), respectively, in Fig. 7. Forthe small wormhole phase we find ∆ S < in all cases. To backup our nomenclature, weplot the minimum radius y of the S as a function of V in Fig. 8 using the same colorcoding as in Fig. 7. – 32 – � � � � - ��� - �������������������� Figure 7 . Action difference ∆ S as a function of the source V for µ = 0 . There is a Hawking-Pagetransition around V > V HP ≈ . . The large wormholes are depicted as blue disks and thesmall wormholes as orange squares. � � � �� ��������� Figure 8 . Minimum value of the S radius as a function of V for µ = 0 . The blue disks representthe large wormholes and the orange disks the small wormholes. Wormholes only exist for V >V min ≈ . . Although the conformally coupled case is qualitatively similar, it turns out to be morechallenging numerically. This is because in that case V needs to be very large for thewormholes to exist, and even larger for the large wormholes to dominate (i.e., to see theHawking-Page-like transition). We find the transition to occur at S D ∼ × L and S W ∼ L , while the difference is of order ∆ S ∼ L . This means we have to accuratelyextract the first digits when evaluating the regulated integrals to determine the transitionwith good accuracy. At this point the use of high-precision arithmetics was essential. We– 33 – ��� ��� ��� ��� ���� - �� - �� - ������� Figure 9 . Action difference ∆ S as a function of the source V for µ L = − . There is a Hawking-Page transition around V > V HP ≈ . . The large wormholes are depicted as blue disksand the small wormholes as orange squares. used octuple precision throughout, keeping track of at least the first digits.The resulting phase diagram is similar in structure to what we found in massless case.For V ≥ V min ≈ . we find two wormhole solutions for a given value of V . At thispoint, y = y min0 ≈ . . The Hawking-Page transition occurs on the large wormholebranch for V = V HP ≈ . ; see Fig. 9. The two wormhole solutions for each V can again be distinguished by the different values of y ; see Fig. 10 where we used the samecolor coding as in Fig. 7. We now discuss perturbations around our scalar wormholes, and in particular, the potentialexistence of negative modes. Just as we did for the U (1) − Maxwell wormholes, we willtake advantage of the SO (4) symmetry of the S to decompose the perturbations into thespherical harmonics (4.33). (cid:96) S = 0 The metric perturbations are given by δ d s = L (1 − y ) (cid:20) δf ( y )d y (2 − y ) y + δg ( y )dΩ (cid:21) , (5.29a)and δ(cid:126) Π = (cid:126)X δ Π( y ) , (5.29b)Throughout, we shall work with gauge invariant variables. The most general infinitesimaldiffeomorphism compatible with SO (4) takes the form ξ = L ξ y ( y )d y . This in turn induces– 34 – ��� ��� ��� ��� ������������������� Figure 10 . Minimum value of the S radius as a function of V for µ L = − . The blue disksrepresent the large wormholes and the orange disks the small wormholes. Wormholes only exist for V > V min ≈ . . the following gauge transformations on the different metric and scalar perturbations δf = (cid:20) − y ) (cid:0) y − y + 1 (cid:1) − (1 − y ) (2 − y ) yf (cid:48) f (cid:21) ξ y + 2(2 − y ) y (1 − y ) ξ (cid:48) y , (5.30a) δg = 2(1 − y )(2 − y ) yf ξ y , (5.30b) δ Π = (1 − y ) (2 − y ) y Π (cid:48) f ξ y . (5.30c)where (cid:48) denotes differentiation with respect to y .Our procedure will be similar to the one we used for the U (1) − Maxwell wormholein section 4. We first evaluate the action (5.1) to second order in { δf, δg, δ Π } and theirderivatives. Let us denote this quadratic action by S (2) . Second derivatives of δg withrespect to y appear in the action, which we integrate by parts to write the action in first orderform. This procedure generates a boundary term which cancels the perturbed Gibbons-Hawking-York boundary term. The resulting action S (2) is written in terms of { δf, δg, δ Π } and their first derivatives only. We then note that, after some integration by parts whosesurface term vanishes or cancels with the boundary counterterms, δf only enters the actionalgebraically. That is to say, no derivatives act on δf . This means we can formally performthe Gaussian integral over δf , and find a new action ˜ S (2) that depends only on { δg, δ Π } and their first derivatives.At this stage we introduce gauge invariant quantities. Looking at the transformations(5.30), we introduce a gauge invariant variable Q through the relation δ Π = Q + 12 (1 − y ) q (cid:48) δg . (5.31)– 35 –ubstituting δ Π into ˜ S (2) yields an action for Q and its first derivative Q (cid:48) . To reducethe action to this form, more integrations by parts have to be performed and, remarkably,every non-vanishing term at the wormhole boundaries cancel with contributions from theperturbed boundary counter-terms. It turns out that the action takes a simpler form if wefurther redefine Q = (cid:112) − (1 − y ) Π (cid:48) √ Q . (5.32)The second order action ˜ S (2) then takes the following rather explicit form ˜ S (2) = 4 π L (cid:90) + ∞ d y √ f y (1 − y ) √ − y √ y (cid:20) (1 − y ) (2 − y ) yf ˜ Q (cid:48) + V ˜ Q (cid:21) , (5.33)where V ( y ) = 1Π (cid:40) f (cid:2) Π (cid:0) − Π (cid:1) m y q (cid:48) + 3 (cid:0) m y + y (cid:1)(cid:3) (2 − y ) yy z y + 3(2 − y ) yf (cid:104)(cid:0) m y Π Π (cid:48) (cid:1) + 1 (cid:105) − (cid:0) − q (cid:1) m y y z y (cid:34) (cid:0) − Π (cid:1)
8Π (1 − Π ) m y + Π (cid:48) (cid:35) − m y Π Π (cid:48) z y + 9 (cid:0) − Π (cid:1) m y − Π ) y z y − (cid:0) m y + y (cid:1) y z y (cid:41) , (5.34)with m y = 1 − y and z y = 3 − (1 − y ) Π (cid:48) . (5.35)The change of variable (5.32) only makes sense so long as the argument inside the squareroot is positive. One can show analytically that, so long as y > y c ≈ . for themassless case and y > y c ≈ . for the conformally coupled case, the argumentinside the square root is indeed positive definite. In both cases y c < y min0 , meaning thatthis transformation makes sense for all large wormholes, and for a range y ∈ ( y c , y min0 ) ofsmall wormholes. Recall that we want to establish that the large wormhole branch has nonegative modes so, for our purposes, it suffices to study wormholes in the range y > y c .To search for negative modes we simply study the eigenvalue equation − (1 − y ) √ − y √ y √ f (cid:20) √ − y √ y (1 − y ) √ f ˜ Q (cid:48) (cid:21) (cid:48) + V ˜ Q = λ ˜ Q. (5.36)Note that in the ˜ y coordinates the potential is an even function of ˜ y as expected from the Z symmetry. This means that we can study separately those eigenfunctions which areeven and odd in ˜ y . In terms of the y coordinates, this maps to functions that scale as ˜ Q ∼ √ y ( a + b y ) near y = 0 for the odd case while for the even case we have ˜ Q ∼ ˜ a + ˜ b y (where a , b , ˜ a and ˜ b are constants).Before proceeding, we must specify the boundary conditions at infinity. A Frobeniusanalysis near the conformal boundary reveals two possible near boundary behaviors, ˜ Q ∼ (1 − y ) ± (cid:113) + µ L − λ . (5.37)– 36 – �� ��� ��� ��� - � - � - ��� ���� ���� ���� ���� - � - � - � - �� Figure 11 . We find a negative mode on the small wormhole branch, which vanishes at y = y min0 On the left panel we plot the negative mode for the massless case and on the right the negativemode of the conformally coupled case. In both cases, the horizontal axis is given by y /y min0 . The integrations by parts we have performed throughout our analysis yield boundary termsthat vanish only consistent if we choose the + root. This motivates performing one lastchange of variable, ˜ Q = (1 − y ) + (cid:113) + µ L − λ y ε/ (2 − y ) ε/ ˜ q , (5.38)where we set ε = 0 for the even sector and ε = 1 for the odd sector of perturbations.We find no odd negative modes, and a single even negative mode exists in the regime y ∈ ( y c , y min0 ) but not for other values of y . In particular we find no negative modes forthe large wormhole branch. Perhaps even more interesting, the negative mode of the smallwormhole branch disappears precisely at y = y min0 . This can be seen in Fig. 11 where weplot the negative eigenvalue λ as a function of y /y min for the massless (left panel) andconformally coupled scalars (right panel). (cid:96) S ≥ We now study scalar-derived perturbations in detail for (cid:96) S ≥ . This turns out to be amuch easier task than studying perturbations of the U (1) Maxwell theory with a sphericalboundary metric. For the metric perturbations we take the same
Ansatz as in Eq. (4.44) δ d s (cid:96) S = h (cid:96) S yy ( y ) S (cid:96) S d y + 2 h (cid:96) S y ( y ) ∇∇ i S (cid:96) S d y d y i + H (cid:96) S T ( y ) S (cid:96) S ij d y i d y j + H (cid:96) S L ( y ) S (cid:96) S g ij d y i d y j , (5.39a)while for the scalar perturbation we choose δ(cid:126) Π (cid:96) S = (cid:126)X B (cid:96) S ( y ) S (cid:96) S + ( ∇∇ i S (cid:96) S ∇∇ i (cid:126)X ) A (cid:96) S ( y ) . (5.39b)Under an infinitesimal diffeomorphism of the form ξ (cid:96) S = ξ (cid:96) S y ( y ) d y + L (cid:96) S y ( y ) ∇∇ i S (cid:96) S d y i , (5.40)– 37 –he metric and scalar perturbations transform as δh (cid:96) S yy = (cid:18) y − − y − − y − f (cid:48) f (cid:19) ξ (cid:96) S y + 2 ξ (cid:96) S y (cid:48) , (5.41a) δh (cid:96) S y = − L (cid:96) S y − y + ξ (cid:96) S y + L (cid:96) S y (cid:48) , (5.41b) δH (cid:96) S T = 2 L (cid:96) S y , (5.41c) δH (cid:96) S L = − λ S L (cid:96) S y + 2 (2 − y ) y y (1 − y ) f ξ (cid:96) S y , (5.41d) δA (cid:96) S = (1 − y ) Π L y L (cid:96) S y , (5.41e) δB (cid:96) S = (2 − y )(1 − y ) y Π (cid:48) L f ξ (cid:96) S y . (5.41f)Our procedure will again be very similar to the one we used for the U (1) Maxwell theory.We first expand the action (5.1) to quadratic order in the perturbations, using S (2) todenote the result. This quadratic action involves second derivatives of H (cid:96) S T and H (cid:96) S L , whichwe readily remove via an integration by parts. The resulting boundary term cancels theperturbed Gibbons-Hawking-York boundary term. At this stage, S (2) is a function of h (cid:96) S yy , h (cid:96) S y , H (cid:96) S T , H (cid:96) S L , A (cid:96) S , B (cid:96) S and their first derivatives with respect to y . However, after a fewintegration by parts, whose boundary terms partially cancel some of the counter-terms, wecan write S (2) in a way where h (cid:96) S yy only enters the action algebraically. As such, it is easy toperform the (correctly-signed) Gaussian integral over this variable and obtain a new action ˜ S (2) which is a function of h (cid:96) S y , H (cid:96) S T , H (cid:96) S L , A (cid:96) S , B (cid:96) S and their first derivatives. Upon a fewmore integration by parts, we can rewrite ˜ S (2) in a manner where h (cid:96) S y again only entersalgebraically. Since it also has the correct sign, we can again perform the Gaussian integralto finally obtain an action ˇ S (2) which is a function of H (cid:96) S T , H (cid:96) S L , A (cid:96) S , B (cid:96) S and their firstderivatives.At this stage we introduce two gauge invariant quantities built using H (cid:96) S T , H (cid:96) S L , A (cid:96) S , B (cid:96) S . These are Q (cid:96) S = A (cid:96) S − (1 − y ) Π2 L y H (cid:96) S T , and (5.42a) Q (cid:96) S = B (cid:96) S − (1 − y ) Π (cid:48) L y (cid:18) H (cid:96) S L + 13 H (cid:96) S T (cid:19) . (5.42b)We then solve for A (cid:96) S and B (cid:96) S in terms of Q (cid:96) S , Q (cid:96) S , H (cid:96) S L , H (cid:96) S T , and substitute the resultingexpression into ˇ S (2) . After some integration by parts (which generate boundary terms thatagain cancel some of the perturbed counter-terms) we find that all dependence on H (cid:96) S L and H (cid:96) S T disappears (as it should due to gauge invariance). At this stage ˇ S (2) is a function of Q (cid:96) S , Q (cid:96) S and their first derivatives only.To proceed, we now treat the massless and conformal cases separately. Strictly speak-ing, the procedure we will apply to the conformally coupled case also works for the masslesscase, but as we shall see it is much more cumbersome so we will be more schematic there.– 38 –or the massless case we take Q (cid:96) S ≡ (1 + λ S ) / √ π √ λ S Π q (cid:96) S ,Q (cid:96) S ≡ (1 + λ S ) / √ π √ λ S (1 − y )Π (cid:48) q (cid:96) S . (5.43)The reason why this change of variable is only adequate for the massless case is the presenceof the multiplying factor Π (cid:48) in the definition of q (cid:96) S . It is a relatively simple exercise to showthat Π must have an extremum for y ∈ (0 , for any µ L < . This means that thisredefinition will necessarily include a singularity in those cases and is thus not appropriateto use. For the massless case, however, Π is monotonic and as such Π (cid:48) does not vanish in y ∈ (0 , .For the massless case, the resulting quadratic action takes the form ˇ S (2) = L (cid:90) + ∞ d y √ f y (1 − y ) √ − y √ y (cid:20) (1 − y ) (2 − y ) yf q (cid:96) S I (cid:48) K IJ q (cid:96) S J (cid:48) + q (cid:96) S I V IJ q (cid:96) S J (cid:21) , (5.44)where K − = 1 λ S − (cid:34) + λ S − λ S Π λ S Π λ S Π [ λ S − − y ) Π (cid:48) ] (1 − y ) Π (cid:48) (cid:35) . (5.45)Note that λ S = (cid:96) S ( (cid:96) S + 2) , and we are focusing on (cid:96) S ≥ , so that λ S ≥ . From the aboveexpression it is clear that Tr( K − ) > . Furthermore, we have det( K − ) = λ S Π (1 − y ) ( λ S −
3) Π (cid:48) (cid:2) λ S − + (1 − y ) (cid:0) − Π (cid:1) Π (cid:48) (cid:3) . (5.46)It turns out that det( K − ) is not positive definite for all values of y , but one can checknumerically that for (cid:96) S = 2 it is positive so long as y (cid:38) . < y min0 . We thus concludethat det( K − ) is positive for all large wormholes, and thus that K is positive definite for alllarge wormholes. Note that for larger values of (cid:96) S , these values will become even smaller.We now turn out attention to V , which has a rather complicated expression. However,it turns out that the combination V IJ = ( K − ) IK ( K − ) JL V KL (5.47a)has a more manageable form, namely V = m y y ( λ S −
3) Π (cid:0) λ S − (cid:1) (cid:0) λ S − (cid:1) , (5.47b) V = − m y y ( λ S − (cid:48) λ S (cid:2) λ S − − m y Π (cid:0) λ S − (cid:1) Π (cid:48) (cid:3) , (5.47c) V = λ S (2 − y ) m y yy ( λ S −
3) Π (cid:48) (cid:110) (2 − y ) m y yy λ S (cid:0) λ S − (cid:1) Π (cid:48) + 6 y ( λ S −
3) Π f + m y f (cid:2) y ( λ S −
2) + m y λ S − (cid:0) y λ S − y + 5 m y λ S (cid:1) Π + 4 m y λ S Π (cid:3) Π (cid:48) (cid:111) , (5.47d)– 39 –here one should recall that m y = 1 − y .It is now a simple exercise to compute the two real eigenvalues λ − ≤ λ + of V as afunction of y . We then take the minimum value of λ − in the interval y ∈ (0 , , and plotit as a function of y . It turns out a critical value of y exists above which λ − is positivedefinite for all λ S ≥ . For the massless case this occurs for y (cid:38) . < y min0 .Thisestablishes that no negative modes exist in the scalar sector with (cid:96) S ≥ for large wormholesgenerated by massless scalar sources.The conformally coupled case is more complicated because we cannot apply (5.43).Instead, we have to use a procedure more similar to the one we used for the U (1) theory.Here we consider a generic value of µ L < parametrized by the conformal dimension ∆ satisfying µ L = ∆(∆ − . (5.48)Instead of (5.43) we consider Q (cid:96) S ≡ (1 + λ S ) / √ π √ λ S (1 − y ) ∆ (cid:20) q (cid:96) S + (1 − y )Π Π (cid:48) λ S − − y ) Π (cid:48) q (cid:96) S (cid:21) ,Q (cid:96) S ≡ (1 + λ S ) / √ π √ λ S (1 − y ) ∆ q (cid:96) S . (5.49)The second order action now reads ˇ S (2) = L (cid:90) + ∞ d y √ f y (1 − y ) √ − y √ y (cid:20) (1 − y ) (2 − y ) yf ( D q (cid:96) S ) I K IJ ( D q (cid:96) S ) J + q (cid:96) S I V IJ q (cid:96) S J (cid:21) , (5.50)with K − = 1(1 − y ) Π [ − (1 − y ) Π (cid:48) ] λ S − − y ) Π (cid:48) λ S (cid:104) (1 − y ) Π (cid:48) λ S − (cid:105) . (5.51)This result is positive so long as y (cid:38) . < y min0 , thus again implying that it ispositive definite on the large wormhole branch. Furthermore, ( D q (cid:96) S ) I = q (cid:96) S I (cid:48) + ε JI q (cid:96) S J , (5.52)where ε JI = ε IK K KJ and ε IK = (cid:8)(cid:0) m y + L y µ (cid:1) Π − m y (cid:2) m y + 3 y − (cid:0) m y + L y µ (cid:1) Π (cid:3) Π (cid:48) (cid:9) × m − − y λ S Π fy y (2 − y ) (cid:0) λ S − m y Π (cid:48) (cid:1) (cid:34) − (cid:35) . (5.53)The matrix V turns out to be positive definite so long as y (cid:38) . < y min0 , thusrendering the large wormholes branch stable. The expression for V can be found in appendixB. The second formalism described above can also be used to study the massless case, butthe expressions are considerably more complicated than in the approach described for m = 0 above. – 40 – .4.3 Scalar-derived perturbations with (cid:96) S = 1 As we have seen, scalar modes with (cid:96) S = 1 are excluded from the previous analysis due tothe fact that S ij = 0 for this special mode. Our metric perturbation becomes simpler inthis case where it takes the form δ d s (cid:96) S =1 = h yy ( y ) S d y + 2 h y ( y ) ∇∇ i S d y d y i + H L ( y ) S g ij d y i d y j , (5.54a)while the scalar perturbation is essentially unchanged and yields δ(cid:126) Π = (cid:126)X B ( y ) S + ( ∇∇ i S ∇∇ i (cid:126)X ) A ( y ) . (5.54b)Since scalar derived infinitesimal diffeomorphisms still have two degrees of freedom, ξ (cid:96) S =1 = ξ y ( y ) d y + L y ( y ) ∇∇ i S d y i , (5.55)we only expect a single gauge-invariant master function. We shall see that this is indeed thecase. Under such diffeomorphisms the metric and scalar perturbation functions transformas δh yy = (cid:18) y − − y − − y − f (cid:48) f (cid:19) ξ y + 2 ξ y (cid:48) , (5.56a) δh y = − L (cid:96) S y − y + ξ y + L y (cid:48) , (5.56b) δH L = − L y + 2 (2 − y ) y y (1 − y ) f ξ y , (5.56c) δA = (1 − y ) Π L y L y , (5.56d) δB = (2 − y )(1 − y ) y Π (cid:48) L f ξ y . (5.56e)Just as for the case with (cid:96) S ≥ , both h (cid:96) S yy and h (cid:96) S y can be integrated out, leaving an actionwhich depends only on H L , A L and B L . At this stage we introduce a gauge invariantvariable Q = Π B − (1 − y )Π (cid:48) A − (1 − y ) Π Π (cid:48) L y H L . (5.57)The resulting quadratic action ˇ S (2) can be written entirely in terms of Q and its firstderivatives and takes the form ˇ S (2) π = (cid:90) + ∞ d y √ f y (1 − y ) √ − y √ y (cid:34) (2 − y ) yf m y Π + m y (1 − Π ) Π (cid:48) Q (cid:48) + V Q (cid:35) , (5.58)– 41 –ith V = Π (cid:2) + m y (1 − Π ) Π (cid:48) (cid:3) y (cid:40) (cid:0) m y + y (cid:1) − Π (cid:104) (cid:0) m y + y (cid:1) Π − m y Π − m y (cid:0) m y + 6 y − m y Π (cid:1) Π (cid:48) − m y Π (cid:0) m y + 2 y − m y Π (cid:1) Π (cid:48) − m y (cid:0) − Π (cid:1) Π (cid:48) (cid:105) − − y ) yy f (cid:104) − Π(9Π − m y Π (cid:48) − m y ΠΠ (cid:48) +3 m y Π Π (cid:48) +3 m y ΠΠ (cid:48) +2 m y Π (cid:48) − m y Π Π (cid:48) ) (cid:105) − (cid:0) m y + y (cid:1) f (2 − y ) yy (cid:0) − Π (cid:1) (cid:2) (cid:0) m y + y (cid:1) + m y Π (cid:0) − Π (cid:1) Π (cid:48) (cid:3) (cid:41) (5.59)Once more, one can easily verify that V and + m y (cid:0) − Π (cid:1) Π (cid:48) are positive definitealong large wormhole branch. This thus establishes that (cid:96) S = 1 scalar derived perturbationsyield no negative modes on the large wormhole branch. (cid:96) V ≥ This sector of perturbations turns out to be much easier to study than in the U (1) − Maxwelltheory. This is because in the current case the vector-derived perturbations do not sourcescalar-derived perturbations. Our
Ansatz for the metric perturbations reads δ d s (cid:96) V = 2 h (cid:96) V y ( y ) S (cid:96) V i d y d y i + H (cid:96) V T ( y ) S (cid:96) V ij d y i d y j , (5.60a)while for the scalar perturbation we choose δ(cid:126) Π (cid:96) V = ( S (cid:96) V i ∇∇ i (cid:126)X ) A (cid:96) V ( y ) . (5.60b)The most general vector-derived infinitesimal diffeomorphism can be written ξ (cid:96) V = L (cid:96) V y S (cid:96) V i d y i . (5.61)This infinitesimal diffeomorphism induces the gauge transformation δh (cid:96) V y = − L (cid:96) V y − y + L (cid:96) V y (cid:48) , (5.62a) δH (cid:96) V T = L (cid:96) V y , (5.62b) δA (cid:96) V = (1 − y ) Π L y L (cid:96) V y . (5.62c)The procedure now is very similar to what we have have seen for the scalar-derived per-turbations. We first derive the second order action S (2) and note that H (cid:96) V T appears in theaction with terms that involve second derivatives with respect to y . We integrate these byparts, with the non-vanishing boundary terms cancelling the perturbed Gibbons-Hawking-York term. Furthermore, after some further integration by parts, h (cid:96) V y only enters the actionalgebraically. This means we can perform the Gaussian path integral and find a new second– 42 –rder action ˇ S (2) which depends on H (cid:96) V T , A (cid:96) V and their first derivatives with respect to y .At this point, we introduce the gauge invariant variable Q (cid:96) V = A (cid:96) V − (1 − y ) Π L y H (cid:96) V T . (5.63)Solving this expression with respect to A (cid:96) V and substituting it back into ˇ S (2) gives an actionfor Q (cid:96) V and its first derivative with respect to y . The dependence in H (cid:96) V T completely dropsout from the calculations, as it should due to diffeomorphism invariance. For conveniencewe further define Q (cid:96) V = ( λ V + 2) / √ √ λ V − Q (cid:96) V . (5.64)The second order action for ˜ Q (cid:96) V reads ˇ S (2) = 4 π L (cid:90) ∞ d y y √ f Π √ − ym y √ y (cid:34) (2 − y ) y ( λ V − ) f ˜ Q (cid:48) (cid:96) V + ˜ Q (cid:96) V y (cid:35) . (5.65)Since the above quadratic action is manifestly positive for any value of y , there are nonegative modes in the vector-derived sector with (cid:96) V ≥ . (cid:96) V = 1 mode The situation here is completely analogous to that of the U (1) Maxwell theory case. Thismode turns out to give a zero-mode when written in gauge-invariant variables. So sincethe theory admits a symmetric hyperbolic formulation, it must be the linearization of apure-gauge mode and can be ignored. (cid:96) T ≥ Finally, we come to the easiest sector, which is the one defined by tensor-derived perturba-tions. The reason why this sector is easiest is twofold: the scalar perturbations are zero inthis sector, and the metric perturbation reduces to a single gauge-invariant variable. Unlikein the U (1) case, the tensors to not excite vectors or scalar derived perturbations. Themetric perturbation takes the rather simple form δ d s (cid:96) T = 2 √ y √ (cid:96) T L (1 − y ) H (cid:96) T ( y ) S (cid:96) T ij , (5.66)where the factors multiplying H (cid:96) V T are only there for later convenience in the presentationof the second order action. The second order action reads ˇ S (2) = 4 π L (cid:90) ∞ d y y √ f (1 − y ) √ − y √ y (cid:20) (2 − y ) yf H (cid:48) (cid:96) T + 1 y (cid:0) λ T + 2 + 4Π (cid:1) H (cid:96) T (cid:21) , (5.67)which is manifestly positive for all wormhole solutions.– 43 – Einstein- U (1) wormholes in 11-dimensional supergravity Having explored wormholes in simple but ad hoc low-energy theories in sections 4 and 5,we now wish to understand the behavior of at-first-sight similar wormholes in various UV-complete theories. This will be explored in the next few sections, and will in particularraise the important issue of possible brane instabilities.We begin in the current section by describing a truncation of 11-dimensional super-gravity that leads to a four-dimensional action similar to that studied in section 4, but withonly two Maxwell fields instead of three. In later sections we will also consider an examplein a mass-deformed ABJM setup and a type IIB compactification that leads to a theory ofscalar fields in AdS .To begin our discussion, recall that bosonic fields of 11-dimensional supergravity arejust a metric (11) g and a four-form F (4) = d A (3) . The Euclidean action reads S = − (cid:90) (cid:18) (11) R (cid:63) − F (4) ∧ (cid:63) F (4) − i F (4) ∧ F (4) ∧ A (3) (cid:19) , (6.1)with (11) R ≡ (11) R AB G AB being the − dimensional Ricci scalar associated with (11) g and (11) R AB its Ricci tensor. The equations derived from (6.1) are (11) R AB − (11) g AB (11) R = 112 (cid:20) F (4) ACDE F CDE (4) b − (11) g AB F (4) CDEF F CDEF (4) (cid:21) , (6.2a) d (cid:63) F (4) = i F (4) ∧ F (4) . (6.2b)Here upper case Latin indices are eleven-dimensional and (cid:63) is the eleven-dimensionalHodge dual operation. U (1) − theory We consider an
Ansatz where the eleven-dimensional fields take the form d s d ≡ (11) g AB d X A d X B = g ab d x a d x b +1 g (cid:40) d ξ + cos ξ (cid:20) d θ + sin θ d φ + (cid:16) d ψ + cos θ d φ − gA (1) (cid:17) (cid:21) + sin ξ (cid:20) d θ + sin θ d φ + (cid:16) d ψ + cos θ d φ − gA (2) (cid:17) (cid:21) (cid:41) , (6.3a)with ξ ∈ (0 , π/ , θ i ∈ (0 , π ) , φ i ∈ (0 , π ) and ψ i ∈ (0 , π ) , and F (4) = − g i Vol + i ˜ F (4) (6.3b)with ˜ F (4) = cos ξ g (cid:20) sin ξ d ξ ∧ (d ψ + cos θ d φ − gA (1) ) + 12 cos ξ sin θ d θ ∧ d φ (cid:21) ∧ (cid:63)F (1) − sin ξ g (cid:20) cos ξ d ξ ∧ (d ψ + cos θ d φ − gA (2) ) −
12 sin ξ sin θ d θ ∧ d φ (cid:21) ∧ (cid:63)F (2) , (6.3c)– 44 –here (cid:63) is the Hodge dual with respect to the four-dimensional metric g , F ( I ) = d A ( I ) with I = 1 , and Vol the volume form of g . Here, lower case Latin indices are four-dimensional.We also restrict to configurations where F (1) ∧ (cid:63)F (1) = F (2) ∧ (cid:63)F (2) . (6.4)Inserting the Ansätze for the eleven-dimensional metric (11) g and four-form field F (4) into the equations of motion (6.2) induces a set of equations for the four-dimensional metric g and gauge fields F ( I ) which can be derived from the following somehow familiar four-dimensional action S = − (cid:90) M d x √ g (cid:32) R + 6 L − (cid:88) I =1 F ( I ) ab F ( I ) ab (cid:33) − (cid:90) ∂ M d x √ h K + S B , (6.5)where the boundary terms are exactly as in Eq. (4.1) and g = L . This looks remarkablysimilar to the Einstein- U (1) theory, but with only two Maxwell fields. Any solution of theequations of motion induced by (6.5) can be uplifted to a solution of eleven-dimensionalsupergravity via Eqs. (6.3). This truncation is a sub-truncation of a more general truncationthat (to our knowledge) first appeared in [61]. We will later consider in section 7 yet anothersub-truncation of [61].We are interested in finding solutions whose boundary metric is a round S . However,we are going to relax the assumption that the SO (4) symmetry is preserved in the bulk.In fact, in the bulk, we will only require our geometry to enjoy U (2) symmetry. The bestway to visualise how we are going to do this in a simple manner is to again introduce theleft-invariant one forms ˆ σ i described in section 4 and to consider a line element of the form d s = d r f ( r ) + g ( r )4 (cid:2) h ( r ) ˆ σ + ˆ σ + ˆ σ (cid:3) , (6.6)where the U (2) = U (1) × SU (2) symmetry is manifest (with the U (1) parametrising theangle that rotates ˆ σ into ˆ σ ). The functions f , g and h are functions of r only. For thegauge fields we take A (1) = L r ) ˆ σ and A (2) = L r ) ˆ σ . (6.7)We want to construct solutions where the dual operator to A ( I ) has a non-vanishingsource. This is obtained by searching for solutions for which lim r → + ∞ Φ = Φ . (6.8)The objective of the sections below is to construct the phase diagram of the wormhole anddisconnected solutions as a function of Φ . The disconnected phase is easy to find analytically. Just as for the Einstein- U (1) theory,it satisfies F ( I ) = ± (cid:63) F ( I ) , (6.9)– 45 –ith the lower sign yielding a singular solution. We thus take the upper sign .The stress energy tensor is then identically zero, and g is just the usual metric onEuclidean AdS with a boundary S for which g ( r ) = r , f ( r ) = r L + 1 , h ( r ) = 1 and Φ( r ) = Φ √ r + L − L √ r + L + L . (6.10)We were also able to find solutions where h (cid:54) = 1 , but it turns out they all lead to boundarymetrics that are not round spheres; i.e., the squashing does not disappear on the boundary.We will comment further on this more general case later. It is a simple exercise to evaluatethe on-shell action for which we find ∆ S U (1) = 8 π L (cid:0) (cid:1) . (6.11) Despite our best efforts we were not able to find an analytic solution for the wormholephase. We thus proceed numerically. Our
Ansatz takes the form d s = L (1 − y ) (cid:20) f d y y (2 − y ) + y (cid:0) g ˆ σ + ˆ σ + ˆ σ (cid:1)(cid:21) , (6.12a)with f and g to be determined numerically and depending only on y . Just as for theEinstein- U (1) theory, y measures the minimal size of the wormhole at the neck.For the Maxwell fields we again take A (1) = L y ) ˆ σ and A (2) = L y ) ˆ σ . (6.12b)The equations of motion read √ g √ − y √ y √ f (cid:20) √ g √ − y √ y √ f Φ (cid:48) (cid:21) (cid:48) − y Φ = 0 , (6.13a) f gn y yy + 8 f m y Φ n y yy g + g (cid:48) gm y − n y yy m y (cid:2)(cid:0) m y + 3 y (cid:1) f − n y y (cid:0) y − m y Φ (cid:48) (cid:1)(cid:3) = 0 , (6.13b) f gm y n y − f Φ m y gy n y − f (cid:0) m y + 3 y (cid:1) n y − ym y y f (cid:48) f − m y yy n y + (1 + 2 y ) y + 2 m y y Φ (cid:48) = 0 . (6.13c)with n y = 2 − y and m y = 1 − y .We now discuss the boundary conditions at y = 0 , the wormhole Z plane of symmetry.Demanding that f and g have a regular Taylor expansion around y = 0 gives the following– 46 –et of Dirichlet conditions at y = 0Φ(0) = (cid:112) g (0) y (cid:112) y + 4 − g (0)2 √ , Φ (cid:48) (0) = y (cid:112) y + 8 − g (0) (cid:112) g (0) (cid:2) y + 4 − g (0) (cid:3) ,f (0) = y y + 4 − g (0) ,g (cid:48) (0) + 2 g (0) (cid:2) y + 8 − g (0) (cid:3) y + 4 − g (0) = 0 . One can show that Φ , f and g admit a regular Taylor series around y = 0 , with all higherorder terms in the series being uniquely fixed by y and g (0) .We now turn out attention to the boundary conditions imposed at the conformal bound-ary located at y = 1 . There, we demand f (1) = g (1) = 1 since we are primarily interestedin solutions with a round S at the conformal boundary. Φ(1) ≡ Φ , on the other hand,determines the source. Expansion the fields with a regular Taylor expansion around y = 1 determines f , g and Φ as a function of four unknowns which we take to be y , Φ(1) , Φ (cid:48) (1) and g (cid:48)(cid:48)(cid:48) (1) .The procedure is now clear: we take a given value of y and g (0) and integrate outwardsto the conformal boundary and, in general, we find that g (1) (cid:54) = 1 . This means for a givenvalue of y , we need to adjust g (0) so that g (1) = 1 . Once this is the case, we read offthe corresponding values of Φ , Φ (cid:48) (1) and g (cid:48)(cid:48)(cid:48) (1) . We thus have a one-parameter familyof solutions whose boundary metric is a round three-sphere. This one-parameter family ofsolutions is, in turn, labelled by y .Once we have the desired solution, we can determine its on-shell action numericallyjust as we did when we studied the Einstein- U (1) theory. We call the Euclidean action ofthe wormhole solution S WU (1) . branes on wormhole backgrounds Before discussing the phase diagram, we will pause for a moment and study probe braneson wormhole backgrounds. The point of this study is that in the limit of weak coupling theaction of a probe brane describes the change in the action of the wormhole when a braneis inserted into the solution . As a result, negative probe-brane actions mean that thanour wormhole is not in fact the lowest-action solution. In this case one would in principlethen like to study solutions with such branes to find the true minimum, and to determinewhether it remains a connected wormhole or whether it becomes disconnected. However,this is beyond the scope of the present work. At weak coupling such a true minimum canbe achieved only by including a large number of branes, which one might describe as acondensate. If on the other hand all probe branes have positive action in our wormhole, This property, together with the idea that sources at the AdS boundaries should remain fixed, deter-mines the detailed form of the probe brane action and forbids the addition of arbitrary constants. Withthis understanding, the sign of the probe-brane action becomes physically meaningful – 47 –his would support the idea that our wormhole does indeed dominate the computation ofthe desired partition functions.The appropriate M brane probe action is S M2 = T M2 (cid:90) M M2 d σ (cid:104)(cid:112) det ˜ G − i ε M2 C M2 (cid:105) , (6.14)with the metric on the world-volume of the M branes being given by ˜ G ˙ µ ˙ ν = (11) g AB d x A d σ ˙ µ d x B d σ ˙ ν , (6.15)and ε M2 = ± for brane anti-brane configurations, respectively. Furthermore, the potentialterm is given by C M2 = 13! ε ˙ µ ˙ ν ˙ ρ A (3) ABC d x A d σ ˙ µ d x B d σ ˙ ν d x C d σ ˙ ρ , (6.16)with ε ˙ µ ˙ ν ˙ ρ being the totally anti-symmetric alternating symbol with ε ˙1˙2˙3 = 1 .We are interested in branes that wrap the S so that we take σ ˙ µ = { ψ, θ, ˆ ϕ } with thestandard Euler angles given in (4.6). Applying this procedure to the our wormhole Ansatz gives S M2 ( y ; ε M2 ) = 2 π L y T M2 (cid:34) (cid:112) g ( y )(1 − y ) − ε M2 (cid:90) y (cid:112) f (˜ y ) (cid:112) g (˜ y )(1 − ˜ y ) √ ˜ y √ − ˜ y d ˜ y (cid:35) . (6.17)In deriving the above expression we made a choice in determining A (3) from F (4) . Thischoice was such that upon the change of variable r = L y √ y √ − y − y (6.18)the action S M2 ( r ; ε M2 ) satisfies S M2 ( r ; −
1) = S M2 ( − r ; 1) , (6.19)so that studying S M2 ( r ; 1) covers both the case of brane and anti-brane probes. The questionis then whether S M2 ( r ; 1) is positive definite for all values of r . If S M2 ( r ; 1) < for anyrange of r , we would expect brane nucleation to take place and render our wormhole solutionunstable. Studying negative modes of this novel class of geometries turns out to be more complicatedthan in the Einstein − U (1) case as even the background metric fails to enjoy sphericalsymmetry. However, the current isometry group is now SU (2) × U (1) remains large enoughto reduce the study of the perturbations to ordinary differential equations. We have usedthis observation to analyze such perturbations by generalizing the techniques used for SO (4) symmetry in the above sections. However, due to the extremely cumbersome nature of thisprcedure, we present only a sketch of the analysis below. This sketch should suffice to allowdedicated readers who wish to check and reproduce our results to do so.– 48 –e first introduce charged scalar harmonics on CP ≡ S . Following [62] we definethese to be solutions of the following eigenvalue problem D i D i Y m κ + λ m κ Y m κ = 0 , (6.20)with D = ∇∇ − i m A CP , m ∈ Z , ∇∇ being the standard connection on CP and A CP is theKähler one-form that relates to the standard Kähler 2-form on CP , J CP , as J = d A CP / .Regularity then demands λ m κ = (cid:96) ( (cid:96) + 2) − m with (cid:96) = 2 κ + | m | where κ = 0 , , , . . . (6.21)The construction of the metric and gauge field perturbations then conforms with thosestudied in [62]. Using numerics, we find that there are no negative modes for any m (cid:54) = 0 and κ > . For m = κ = 0 , however, for small wormholes we do find a negative mode onwhich we report further below. Our numerical results are as follows. For each value of the boundary source Φ > Φ min0 ≈ . we find two wormhole solutions which we may again call small and large; seeleft hand side panel of Fig. 12. We find no field-theoretic negative modes anywhere onthe large wormhole branch, though the small wormhole branch has at least one negativemode in the κ = m = 0 sector. Furthermore, the Euclidean action of the large wormholebranch of solutions eventually becomes smaller than twice the corresponding action of thedisconnected solution; see right hand side panel of Fig. 12 where ∆ S U (1) = 2 S DU (1) − S WU (1) - for Φ > Φ HP0 ≈ . .Finally, we also studied the positivity of the probe M brane action (6.17). On thelarge wormhole branch, and for large enough boundary sources Φ > Φ M20 ≈ . , wefind a finite range of r (or, equivalently, values of y ) for which the action from M braneswrapped on the S becomes negative. As a result, all large wormholes which dominateover our disconnected solution turn out to be unstable to brane nucleation. We mark thenucleation threshold at Φ M with a red dot in the right panel of Fig. 12. We note, however,that there exists a range Φ ∈ (Φ min0 , Φ M20 ) in which the large wormholes seem to have nopathology, though in this range they are subdominant with respect to our disconnectedsolution (these non pathological wormholes are represented by the blue disks in Fig. 12).Finally, we comment on a small extension of our result. We have also considered caseswhere the metric at the boundary is not round, i.e. g (1) ≡ g (cid:54) = 1 . These constitute atwo-parameter family of wormhole solutions which we can parametrise with (Φ , g ) . Itturns out that near the conformal boundary, located at y = 1 , one has S M2 ( y ; 1) = π (4 − g ) √ g L y − y + O (1) . (6.22)It is thus clear that we need < g < in order for S M2 ( y ; ε M2 ) to be positive definite nearthe conformal boundary. So we restricted our attention to < g < . In this range we werenot able to find any value of g for which wormholes dominate over the disconnected solution,– 49 – �� ��� ��� ��� ��� ��� ��� ��� ��������������������� ��� ��� ��� ��� ��� - �� - �� - ����� Figure 12 . Left panel: radius of the wormhole solutions as a function of the source Φ . Worm-holes only exists for Φ > Φ min0 ≈ . . Right panel: difference in the Euclidean action ∆ S U (1) = 2 S DU (1) − S WU (1) as a function of Φ . The wormhole solutions have a lower action for Φ > Φ HP0 ≈ . . The orange squares represent small wormholes, the blue disks correspondto large wormholes where the Euclidean action for M is positive and the green diamonds indicatelarge wormholes where the Euclidean action for M is not positive definite. The red disk indicatesthe value of Φ in the large wormhole branch above which the Euclidean action (6.17) for M branesis not positive definite. have positive definite S M2 ( y ; ε M2 ) , and have no negative modes. In fact, we find that forsmall enough values of g even the small wormhole branch has non positive S M2 ( y ; ε M2 ) .The value of g corresponding to the largest ratio Φ M20 / Φ HP0 is g = 4 / , corresponding tothe maximisation of the numerator in the divergent term of S M2 ( y ; 1) near the conformalboundary. Perhaps the simplest asymptotically-AdS Euclidean wormholes are the quotients of Eu-clidean AdS d , which is of course also the hyperbolic plane H d . This space admits a foliationby hyperbolic planes H d − of dimension ( d − , so that the metric can be written d s = d r r L + 1 + ( r + L )d s H d − , (7.1)with d s H d − the metric on the unit H d − . Here r takes values in ( −∞ , ∞ ) . Now, anycompact hyperbolic space of dimension d − can be written as the quotient of H d − withrespect to an appropriate discrete group Γ of H d − isometries. From (7.1), we see thattaking the corresponding quotient of AdS d yields a wormhole with compact hyperbolicslices at each value of r ∈ ( −∞ , ∞ ) and with two separate boundaries at r = ±∞ . In theobvious conformal frame both boundaries are again compact hyperbolic manifolds.– 50 –his construction embeds easily in many UV-complete models. However, as describedin [10], in simple models it is associated witb a dramatic brane-nucleation instability thatoccurs even near the AdS boundary. This makes the entire theory unstable with suchboundaries. The dual field theory interpretation is that conformal field theories requireconformal couplings to curvature, and that such couplings naturally generate negative massterms when the theory is placed on a compact hyperbolic space.As noted in [10], this also suggests that such instabilities can be cured by breakingconformal invariance and adding explicit new couplings to the would-be-CFT that givemasses to various scalars. The question is then what happens to the above wormholesolutions under such deformations. We investigate this issue below using a particular mass-deformation of an AdS compactification of 11-dimensional supergravity; i.e., from the dualgauge theory point of view we study a deformation of the ABJM model [39]. Interestingly,at least in this case, we find wormhole solutions only for small mass deformations µ , andin particular only at deformations µ < µ max where µ max is still too small to stabilize thetheory. At such small deformations we find two branches of wormhole solutions, but theycoalesce µ = µ max . It thus appears that wormholes do not exist in the stable members ofthis family of theories. The deformation of interest can be described using a sub-truncation of the truncation of11-dimensional supergravity detailed in [61]. Our − dimensional metric G takes the form d s d ≡ (11) g AB d x A d x B = Ξ / g ab d x a d x b + Ξ / g (cid:40) d ξ + cos ξ Z (cid:2) d θ + sin θ d φ + (d ψ + cos θ d φ ) (cid:3) + sin ξ Z (cid:2) d θ + sin θ d φ + (d ψ + cos θ d φ ) (cid:3) (cid:41) , (7.2)where Z = e Φ cos ξ + sin ξ , (7.3a) Z = ( e − Φ + χ e Φ ) sin ξ + cos ξ , (7.3b) Ξ = Z Z . (7.3c)For F (4) we have F (4) = − i g U Vol + i sin ξ cos ξg ( (cid:63) dΦ − χ e (cid:63) d χ ) ∧ d ξ − d ˜ A (3) , (7.4)where Vol is the volume form on g and (cid:63) the Hodge operation with respect to g . Further-more, ˜ A (3) = 18 g χe Φ (cid:104) cos ξZ sin θ (d ψ + cos θ d φ ) ∧ d θ ∧ d φ − sin ξZ sin θ (d ψ + cos θ d φ ) ∧ d θ ∧ d φ (cid:105) (7.5)– 51 –nd U = e Φ cos ξ + ( e − Φ + χ e Φ ) sin ξ + 2 . (7.6)Inserting the above expressions for G and F (4) into the eleven-dimensional equations ofmotion (6.2) yields four-dimensional equations of motion for g , χ and Φ that can be derivedfrom the following four-dimensional action S d = − (cid:90) M d x √ g (cid:18) R + 12 ∇ a Φ ∇ a Φ − e ∇ a χ ∇ a χ − V (cid:19) − (cid:90) ∂ M d x √ h K + S B , (7.7)with V = − L (cid:0) e Φ + e − Φ + χ e Φ + 4 (cid:1) , (7.8)where g = L . In the above we will not need to specify S B . From the four-dimensionalperspective, Φ and χ have masses µ L = − . To proceed we need to understand whatboundary conditions we should choose for these fields. By comparing our Ansatz for F (4) with the one presented in [63], we conclude that the sources associated with boundary valuesof the supergravity fields Φ and χ parametrise (possibly supersymmetric) mass deformationsof ABJM. The supersymmetric mass deformation corresponds to deformations for which χ ∼ µ z/L , Φ = O ( z ) , with µ being proportional to the mass deformation and z a Fefferman-Graham coordinate. These are the boundary conditions that we will employ.Rather remarkably, the action above admits yet another sub-truncation in which χ = (cid:112) − e − . (7.9)Perhaps more interestingly, when we performed our numerical studies, we did not imposethe relation above, and yet all our numerically determined solutions were consistent withthe above relation. Our starting point is family of solutions (7.1) with d = 4 . Below, we implicitly assume aquotient by some Γ that makes the r = constant slices compact hyperbolic spaces. Asnoted above, such solutions are unstable to the nucleation of branes, which the in presentcontext are M branes. To understand the effect of the mass deformation on this instabilitywe will study M branes on our deformed wormholes. Before doing this, we provide somedetail regarding our construction.As a metric Ansatz we take d s = L (1 − y ) (cid:20) f d y y (2 − y ) + y d s H (cid:21) , (7.10)with f a function only of y ∈ (0 , and with y to be interpreted as the minimal size of thewormhole neck. The neck is located at y = 0 and the conformal boundary is at y = 1 . Forthe scalars we take χ = (1 − y ) q and Φ = (1 − y ) q . (7.11)The procedure is now very similar to what we have seen before: finiteness of f at y = 0 locks y into a relationship with Φ(0) and χ (0) . We then take these values at the origin– 52 – = 0 and integrate outwards. In general Φ = O [(1 − y )] so we adjust Φ(0) so that nearthe conformal boundary we find
Φ = O [(1 − y ) ] . What remains is a one-parameter familyof solutions parameterized by either y or µ . probe branes We now need to understand what value of µ is required to remove the UV brane-nucleationinstability by making the action for probe M branes positive near the AdS boundary. Weuse the action (6.17) together with the − dimensional Ansätze (7.2) and (7.4). We beginby wrapping our M branes on (the relevant quotient of) H , and obtain an action as afunction of y . Near the conformal boundary we can use the near boundary behaviour ofour fields to determine S M2 near y = 1 . This turns out to be S M2 = y L Vol H − y ) (cid:2) µ −
12 + 2 cos(2 ξ ) (cid:0) µ − y Φ (cid:48)(cid:48) (1) (cid:1)(cid:3) + O (1) , (7.12)where Vol H is the volume of the relevant quotient of H . The condition (7.9) automaticallyensures that the term proportional to cos(2 ξ ) vanishes (we once more note that we find thiscondition to be true numerically). We thus see that we must have µ > to stabilize thetheory in the UV, and in particular to have a hope of S M2 being positive definite. The main result of this section is presented in Fig. 13, where we plot the radius of thewormhole y as a function of µ . Rather interestingly, we find no wormhole solution for µ > µ max ≈ . , which is smaller than √ ≈ . , so that this new class of massdeformed wormholes is still unstable to nucleating M branes. The results of this sectionare thus similar to those found in [27] for mass deformations of N = 4 SYM. ��� ��� ��� ���������������������
Figure 13 . No wormhole solutions seem to exist for µ > µ max ≈ . < √ , which in particularimplies that no stable wormhole solution exist. – 53 – Wormholes from type IIB theory
We can also find a truncation of a UV-complete scenario that is closely related to ourEinstein-scalar model of section 5, though this truncation will live in AdS instead of AdS .As described below, this model is a compactification of type IIB supergravity. After dis-cussing the Ansätze for the fields in section 8.1, we analyze the model as usual in theremaining subsections.
We consider type IIB supergravity with only the ten dimensional metric (10) g , Ramond-Ramond three-form F (3) ≡ d A (2) and dilaton φ . The corresponding equations of motionare (10) R AB = 12 (10) ∇ A φ (10) ∇ B φ + e φ (cid:34) F (3) ACD F CD (3) B − (10) g AB F (3) CDE F CDE (3) (cid:35) , (8.1a) d (cid:16) e φ (cid:63) F (3) (cid:17) = 0 , (8.1b) (cid:3) φ − e φ F (3) ABC F ABC (3) = 0 , (8.1c)where (cid:63) is the Hodge operation associated with the ten-dimensional metric (10) g , (10) ∇ itsassociated metric-compatible connection and upper case Latin indices are ten-dimensional.Consider the following ten-dimensional field configuration d s = ( g ab d x a d x b + L dΩ ) e φ + e − φ (cid:104) ( e √ φ d z + e −√ φ d z ) e φ + ( e √ φ d z + e −√ φ d z ) e − φ (cid:105) (8.2a) F (3) = 2 iL Vol + 2 L d Ω (8.2b)and φ = φ , (8.2c)where d Ω is the volume form on a round − sphere, Vol is the volume form of thethree-dimensional metric g , φ i , with i = 1 , , , , depend only on the three-dimensionalcoordinates x a and the coordinates { z , z , z , z } parametrise a 4-torus. Here, lower caseLatin indices are three-dimensional. Inserting the Ansätze (8.2) into the -dimensionalequations equations of motion (8.1) yields a set of three-dimensional equations for g , and φ i which can be derived from the following three-dimensional action S d = − (cid:90) M d x √ g (cid:34) R + 1 L − (cid:88) i =1 ∇ a φ i ∇ a φ i (cid:35) . (8.3)This action will have to be supplemented by appropriate boundary terms. As expected,the three dimensional metric g is asymptotically AdS . The scalars φ i appear as masslessAdS scalars that are minimally coupled to gravity.– 54 –e wish to turn on a source for all the φ i , and we should thus consider adding boundaryterms to (8.3) appropriate to such a choice and which render the on-shell action finite onsolutions to corresponding equations of motion. We now introduce Fefferman-Grahamcoordinates for g , where the conformal boundary is located at z = 0 . Since φ i is massless, itwill have in general a log z divergence close to the conformal boundary. In fact, the genericbehaviour of φ i close to the conformal boundary takes the form φ i = V i ( x µ ) + z Z i ( x µ ) + z log z ˜ Z i ( x µ ) + . . . , (8.4)where µ runs over the boundary directions, and ˜ Z i is a function of V i ( x µ ) only. For thisreason, the counterterms to be added to (8.3) will explicitly depend on a UV cut off z = ε .The boundary terms that lead to a well defined variational problem for the metric andscalar field and render the on-shell action finite read S c ( ε ) = − (cid:90) ∂ M ε d x √ h K + 2 L (cid:90) ∂ M ε d x √ h − L log εL (cid:88) i =1 (cid:90) ∂ M ε d x √ h ˜ ∇ µ φ i ˜ ∇ µ φ i − L log εL (cid:90) ∂ M ε d x √ h ˜ R , (8.5)where h the induced metric on a surface of constant z = ε . We denote this surface by ∂ M ε ,and ˜ ∇ is the natural metric-compatible connection on ( ∂ M ε , h ) . Furthermore, ˜ R the Ricciscalar on ∂ M ε and K the trace of the extrinsic curvature associated to an outward-pointingnormal to ∂ M ε . The first term in (8.5) is the usual Gibbons-Hawking-York term, and theremaining are boundary counterterms. The total on-shell action is then S on − shell = lim ε → + ˜ S ( ε ) + S c ( ε ) . (8.6)where ˜ S ( ε ) is obtained from (3) S in (8.3) by replacing M with M ε , which is obtained from M by chopping off from spacetime the region z > ε .In the remainder of this section we will set φ = 0 , which in particular restricts usto trivial dilaton fields. We stress however, that appendix C describes wormhole solutionswith a toroidal boundary metric T for which φ (cid:54) = 0 . The solutions we seek to construct enjoy spherically symmetric metrics, but the scalars φ i will break said symmetry in a special way. In particular, we introduce standard polar andazimuthal coordinates ( θ, ϕ ) on the S , so that the standard metric on the S reads dΩ = d θ + sin θ d ϕ . (8.7)We then take the scalars to satisfy φ = ψ ( r ) sin θ cos ϕ , φ = ψ ( r ) sin θ sin ϕ and φ = ψ ( r ) cos θ . (8.8)These forms are chosen so that (cid:88) i =1 φ i = ψ ( r ) . (8.9)– 55 –or the metric we take d s = d r f ( r ) + g ( r )dΩ . (8.10)Wormhole solutions will have f ( r ) , g ( r ) > throughout spacetime, whereas the discon-nected solutions will have f ( r ) = 1 + O ( r ) , with g ( r ) = O ( r ) near r = 0 . We again wish to address potential brane nucleation instabilities. Since we are in type IIBand the only non-zero Ramond-Ramond field is the three-form F (3) , the relevant branesare D ’s and D ’s. We shall present here results for the D1’s, but the invariance of thebackground under Hodge-duality of the Ramond-Ramond 3-form field strength and thetrivial dilaton imply that equivalent results can be obtained for D ’s wrapped on the 4-torus (or for related D1D5 bound states).For a generic spacetime ( M , g, A (2) ) , the Euclidean action of a probe D1 with world-volume coordinates σ ˙ µ (for ˙ µ = 1 , ) takes the simple form S D1 ± E = (cid:90) M d σ (cid:34)(cid:115) det (cid:18) g AB ∂x A ∂σ ˙ µ ∂x B ∂σ ˙ ν (cid:19) ± i ε ˙ µ ˙ ν ∂x A ∂σ ˙ µ ∂x B ∂σ ˙ ν A (2) AB (cid:35) , (8.11)where the lower and upper signs stand for D1 and D1 , respectively. We will choose ourbrane world-volume coordinates to wrap the S , so that σ ˙1 = θ , σ ˙2 = ϕ . We also introducecoordinates on the S for which the metric is dΩ = d˜ θ + sin ˜ θ (cid:16) dˆ θ + sin ˆ θ d ˜ ϕ (cid:17) , (8.12)with ˜ θ, ˆ θ ∈ (0 , π ) and ˜ ϕ ∈ (0 , π ) .Within our symmetry class, we can write A (2) locally as A (2) = 2 iL λ ( r ) sin θ d θ ∧ d ϕ + L ˜ λ (˜ θ ) sin ˆ θ dˆ θ ∧ d ˜ ϕ , (8.13)with d λ d r = g ( r ) (cid:112) f ( r ) , and d˜ λ d˜ θ = sin ˜ θ . (8.14)One can thus write S D1 E as S D1 ± E = 4 π (cid:20) g ( r ) ∓ L λ ( r ) (cid:21) . (8.15)To simplify our analysis, we take as a boundary condition λ (0) = 0 for both the wormholesand disconnected solutions. Using these boundary conditions, it is easy to check that λ isan odd function of r . Thus, if we check that S D1 E > for the upper sign and for all valuesof r , we automatically guarantee positivity for the lower sign as well.Even without actually solving the equations of motion for all values of r , we can ob-tain useful information by studying the asymptotic behavior of solutions. We first mapeverything to Fefferman-Graham coordinates, defined via d r d z = − (cid:112) f ( r ( z )) Lz (8.16)– 56 –ith r z = L as z → . It is also useful to define g = L z G , (8.17)in terms of which the equation for λ becomes: d λ d z = − L z G . (8.18)Solving the resulting equations of motion asymptotically yields G = 1 + 12 (cid:0) V − (cid:1) z + O (cid:0) z (cid:1) , (8.19a) ψ = V + ( λ (2) + V log z ) z + O (cid:0) z (cid:1) , (8.19b)where λ (2) is a constant. Integrating (8.18) then gives λ = L z − L V −
1) log z + O (1) , (8.20)which in turn yields S D1+ E = L ( V −
1) log z + O (1) . (8.21)This result suggests that any solution with V > will be unstable to spontaneouslynucleating a D1 brane. The question is then whether we can find any wormhole solutionwith V < . If so, we must then further check to see whether the probe brane action onthe resulting background is positive definite for all z (and not just near z = 0 ). We introduce radial coordinates, for which g in (8.10) is given by g ( r ) = r . (8.22)To work with compact coordinates, we also introduce a new coordinate y ∈ (0 , so that r = L y (cid:112) − y − y , (8.23)with y = 1 being the location of the conformal boundary and y = 0 the centre where the S shrinks to zero size. Regularity at r = y = 0 demands that ψ ( r ) = O ( r ) = O ( y ) . (8.24)To sum up, we take the Ansatz d s = L (1 − y ) (cid:20) y ˜ f ( y ) (2 − y ) + y (2 − y )dΩ (cid:21) and ψ = y (cid:112) − y q ( y ) , (8.25)– 57 –here have translated the f ( r ) in (8.10) into a function ˜ f ( y ) = f ( r ( y )) . The Einstein andKlein-Gordon equations yield ˜ f ( y ) = 8 − y (cid:0) − y (cid:1) (cid:0) − y (cid:1) (cid:2) (cid:0) − y (cid:1) q ( y ) + y (cid:0) − y (cid:1) q (cid:48) ( y ) (cid:3) (cid:104) − y (2 − y ) (1 − y ) q ( y ) (cid:105) , (8.26a) − y y (cid:113) ˜ f ( y ) y (cid:0) − y (cid:1) / (cid:113) ˜ f ( y ) (1 − y ) (cid:16) y (cid:112) − y q ( y ) (cid:17) (cid:48) (cid:48) − q ( y ) = 0 . (8.26b)Note that Eq. (8.26b) is a second order differential equation for q , since ˜ f is given in termsof q and q (cid:48) in Eq. (8.26a). As boundary conditions, we take q (cid:48) (0) = 0 (which follows fromregularity at y = 0 ) and q (1) = V . For the wormhole phase, we take g in (8.10) to have the form g ( r ) = r + r (8.27)with r denoting the wormhole radius. As we shall see, it will correspond to the minimumsize of the S . We change coordinates to r = r √ y √ − y − y , (8.28)where the Z symmetry plane of the wormhole solution is identified with y = 0 , and theconformal boundary is located at y = 1 . It is also convenient to define r ≡ y L . Therelevant Ansatz now reduces to d s = L (1 − y ) (cid:34) ˆ f ( y ) d y (2 − y ) y + y dΩ (cid:35) , and ψ = q ( y ) , (8.29)where, once again, we rewrite f in (8.10) in terms of ˆ f ( y ) .The Einstein equation and Klein gordon equations now yield ˆ f ( y ) = y (2 − y ) y (cid:2) − (1 − y ) q (cid:48) ( y ) (cid:3) (cid:2) y − (1 − y ) ( q ( y ) − (cid:3) , (8.30a) √ y √ − y (cid:113) ˆ f ( y ) √ y √ − y (1 − y ) (cid:113) ˆ f ( y ) q (cid:48) ( y ) (cid:48) − q ( y ) y (1 − y ) = 0 . (8.30b)Reflection symmetry around y = 0 , and smoothness of the corresponding solution imply q (0) = (cid:113) y , and q (cid:48) (0) = 2 (cid:112) y y . (8.31)At the conformal boundary, we wish to set q (1) = V . The strategy is now simpleenough: we take Eq. (8.30b) and integrate it all the way to y = 1 , where we read off V for any value of y we choose. We found it convenient to preform this integration using animplicit fourth order Runge-Kutta method.– 58 – .6 Results There are a couple of surprising results in this setup. First, we only find one branch ofwormhole solutions, which we coin as large since it extends to arbitrarily large values of thesource V . Nevertheless wormholes only seem to exist for V ≥ V min ≈ . . These resultscan be seen on the left hand side panel of Fig. 14 where we plot y as a function of V . Thefact that V > in order for the wormhole solution to exists immediately reveals that thewormholes we found suffer from nucleation instabilities associated with D and ¯D1 branes.Finally, on the right hand side panel of Fig. 14 we plot the different in on-shell action ∆ S d = 2 S D d − S W d (8.32)where S D d and S W d are the three-dimensional on-shell actions for the disconnected andwormhole solutions (respectively). We can see that the wormhole solution dominates overthe disconnected solution for V ≥ V HP ≈ . . � � � � � ��������� ��� ��� ��� ��� - ������������������� Figure 14 . Left panel: radius of the wormhole solutions as a function of the source V . Wormholesonly exists for V > V min ≈ . . Right panel: difference in the Euclidean action ∆ S d =2 S D d − S W d as a function of V . The wormhole solutions have a lower action for V > V HP ≈ . .Unlike the higher-dimensional examples, we find no evidence for a small wormhole branch. We also studied the field theoretical stability of the wormhole solutions. The analysis iscompletely analogous to section 5.4, except it is easier because there are no tensor harmonicson S and vectors harmonics can be obtained from scalar harmonics via the Hodge operationon gradients of the scalar harmonics. We found no negative modes, irrespectively of thevalue of V . Our work above explored the construction and stability of asymptotically anti-de Sitter Eu-clidean wormholes in a variety of models. Indeed, we have studied many more models than– 59 –ere described above, but for brevity we limited our presentation to a few representativecases with spherical (or squashed sphere) boundaries. A few low energy models with torusboundaries are discussed in appendix C, and a table showing a longer list of 22 string/M-theory compactifications and 14 ad hoc low energy models and associated results obtainedin a variety of dimensions is presented in appendix D. While not all issues were analyzedfor all models in the table, we hope it will nevertheless be of use in guiding future studies.Perhaps notably, our list does not include the model described in section 5 of [10] where thedisconnected solution remains to be found in order to determine if the wormhole describedthere will dominate.In simple ad hoc low-energy models, it was straightforward to find two-boundary Eu-clidean wormholes that dominate over disconnected solutions and which are stable (lackingnegative modes) in the usual sense of Euclidean quantum gravity. Similar results werefound previously in the context of JT gravity coupled to matter [17] and in studies of con-strained wormholes in pure gravity [37]. In particular, resulting phase diagram was a directanalogue of the Hawking-Page phase transition for AdS-Schwarzschild black holes in which,for boundary sources above some threshold, we find both a ‘large’ and a ‘small’ branchof wormhole solutions with the latter being stable and dominating over the disconnectedsolution for large enough sources.We also studied two-boundary Euclidean wormholes in a variety of string and M-theorycompactifications. At first glance the solutions are generally similar to those in the ad hocmodels, and we find several contexts where large wormholes dominate over the disconnectedsolutions we find and where the large wormholes are stable with respect to field-theoreticperturbations. However, wormholes in these UV-complete settings that are large enough todominate over our disconnected solution always suffer from brane-nucleation instabilities(even when sources that one might hope would stabilize such instabilities are tuned to largevalues). This implies the existence of additional solutions with lower action. It is naturalto expect that the lowest-action such solutions are again disconnected, but this remains tobe studied in detail. Including finite back-reaction from such branes would be a naturalnext step in understanding wormhole solutions in string theory.The overall picture of UV-complete models is thus rather similar to that obtainedby Maldacena and Maoz [10] with the following exceptions. First, we have performed athorough analysis of potential field-theoretic negative modes and shown our large wormholesto be free of such pathologies. Second, we have identified subdominant wormholes that arefree of both field-theoretic and brane-nucleation instabilities. In particular, this was thecase for the U (1) large wormholes with round boundaries and . ≈ Φ M > Φ > Φ min0 ≈ . described section 6.6, though in other models subdominant wormholescan suffer from brane nucleation instabilities as well (see e.g. section 8 and the discussionof general squashed boundaries in section 6.6). Third, section 7 investigated finite values ofsources that provide mass-deformations of the form that were suggested in [10] to stabilizetheories against brane-nucleation. As predicted by [10], this does in fact stabilize the theoryin the sense that it removes the UV brane-nucleation instability near the asymptotically-AdS boundaries. One thus expects that the disconnected solution is fully stabilized at suchvalues of the deformation parameter. However, we were able to find wormhole solutions only– 60 –hen the deformation is small enough that the UV remains unstable. This is perhaps thestrongest evidence yet that, at least without taking parameters to exponentially large values(see below), Euclidean wormholes will not dominate partition functions in UV-completetheories.The interesting question is of course what such results imply for the AdS/CFT fac-torization problem described in the introduction. To begin this discussion, we note thata brane-nucleation instability is really the statement that adding a brane to the given so-lution will lower the action. Since branes are discrete, if these are the only “instabilities”this means that sufficiently small fluctuations around the solution must in fact increase theaction. So in a technical sense the wormhole saddles we found in the string-compactifiedmodels are in fact stable. The point is simply that the wormholes constructed thus far willbe sub-dominant saddles and will not control leading-order effects.This result is natural even if one believes that bulk AdS gravity in UV-complete theoriesshould be dual to an ensemble of quantum theories. Had we found a case where a sim-ple semi-classical wormhole dominates the computation of a partition function, this wouldhave indicated that ensemble-fluctuations of that partition function are large, or at leastthat they are not particularly small when compared with its ensemble expectation-values .In other words, it would have implied that an ensemble dual to bulk string theory is notsharply peaked in the semi-classical limit. On the other hand, evidence to date suggeststhat ensembles associated with quantum gravity are generally peaked very sharply indeed.This is the case whether one looks at the ensembles associated with low-dimensional grav-itational wormholes [12–15, 26, 64] or at fluctuations in ensembles [65] of states associatedwith black hole interiors . Indeed, the fluctuations associated with the double cone so-lution [13] are visible only when one probes the fine structure of the associated ensembleby studying exponentially large times. One might similarly expect that more standardpartition functions become dominated by wormholes only at exponentially large sources,which is a regime that we have certainly not probed (and which may involve additionalUV physics due large field values and strong gradients)! It is interesting that UV-completemodels appear to generally achieve this expectation while ad hoc low-energy models oftenadmit exceptions.What then are the implications of our (sub-dominant) wormhole saddles in the string-compactified models? We emphasize that they are in fact stable with respect to sufficientlysmall perturbations (within the truncations studied), so that they cannot be immediatelydismissed. This is in particular a technical advance beyond the analysis performed in [10].While in general the semi-classical approximation leads to a sum over all (stable) Eu-clidean saddles, in many contexts it would be dangerous to draw conclusions about physicsbased on sub-dominant saddles. This is simply because the effects of sub-dominant saddlesare small, and so in many cases can be easily dwarfed by even small corrections to thephysics of dominant saddles.However, the present context appears to be rather different. In studying quantities like We thank Henry Maxfield for discussions on this point. And indeed, these may be closely related [26]. – 61 – Z := (cid:104) Z (cid:105) − (cid:104) Z (cid:105) the contribution of any disconnected saddle will vanish identically, andwithout error. So only contributions from connected saddles remain. Unless such contribu-tions fully cancel amongst themselves, δZ will be non-zero. Though we are certainly notable to analyze the possibility of a conspiracy that might enforce such cancellations, even inthe string compactifications we studied the most naive interpretation of our sub-dominantwormhole saddles remains that they will make δZ non-zero and require an ensemble ofdual field theories. This is of course also the picture implied by the dominance that of thedouble-cone wormholes of [13] that appears to hold in late time computations of the spec-tral form factor in generic models . The factorization problem of AdS/CFT thus remainsfar from resolved and will surely be the object of much future investigation. Acknowledgments
We thank Xi Dong, Jerome Gauntlett, Juan Maldacena, Henry Maxfield, Douglas Stanfordand Edward Witten for useful conversations. We also thank the Kavli Institute for Theo-retical Physics for its hospitality during a portion of this work. As a result, this researchwas supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. D. M. was primarily supported by NSF grant PHY-1801805 and funds from theUniversity of California. J. E. S. is supported in part by grant ST/T000694/1. J. E. S.was also supported by a J. Robert Oppenheimer Visiting Professorship at the Institute forAdvanced Studies in Princeton.
A Symmetric matrices for the Einstein- U (1) wormholes A.1 The symmetric Matrix V for the scalars The symmetric matrix V is given by V IJ = ( K − ) IK ( K − ) JM V KM . (A.1)We then have V = ( (cid:96) S + 1)64 π g ( λ S − λ S (cid:2) g ( λ S − λ S + 24Φ (cid:0) gL λ S + 4 r L + 4 r (cid:1)(cid:3) . (A.2)The remaining two components take a more complicated form.For V we find V = g ( (cid:96) S + 1)32 π L r f ( λ S −
3) ( gλ S + 24 L Φ ) (cid:34) (cid:88) i =0 Φ i p ( i )11 ( r ) + Φ Φ (cid:48) (cid:88) i =0 Φ i m ( i )11 ( r ) (cid:35) , (A.3a)where p (3)11 ( r ) = − L r ( λ S −
4) ( λ S − g , (A.3b) Though it would be interesting to further investigate higher-dimensional double cones in detail – 62 – (2)11 ( r ) = − L rg (cid:104) g ( λ S − λ S −
3) + gL ( λ S − λ S + 72) − gr ( λ S − λ S + 3 r ( r − L )( λ S − λ S (cid:105) , (A.3c) p (1)11 ( r ) = rλ S (cid:110) L (cid:2) gr λ S (2 λ S − − g (cid:0) λ S − λ S + 42 (cid:1) + r (cid:0) − λ S + λ S + 12 (cid:1)(cid:3) + L (cid:2) g (cid:0) λ S − λ S − (cid:1) + r (cid:0) λ S − λ S − (cid:1)(cid:3) + 24 gr ( λ S − (cid:111) , (A.3d) p (0)11 ( r ) = − rλ S L (cid:110) L (cid:0) λ S + 3 λ S − (cid:1) − g (cid:2) L r ( λ S + 1) + 12 r ( λ S + 1) (cid:3) − g L (cid:0) λ S + 3 λ S − (cid:1) + gL r (cid:0) L + r (cid:1) (cid:0) λ S − λ S − (cid:1) + 4 r (cid:0) L + r (cid:1) ( λ S + 1) (cid:111) , (A.3e) m (2)11 ( r ) = − L (cid:0) g + L (cid:1) ( λ S − , (A.3f) m (1)11 ( r ) = − L λ S (cid:110) g L ( λ S −
3) + g (cid:2) L r − ( λ S − L + 12 r (cid:3) + 5 L r (cid:0) L + r (cid:1) ( λ S − (cid:111) , (A.3g)and m (0)11 ( r ) = λ S (cid:104) g L ( λ S + 2) + 3 g L λ S + 3 g − gL r (cid:0) L + r (cid:1) (3 λ S − r (cid:0) L + r (cid:1) (cid:105) . (A.3h)While for V we have V = ( (cid:96) S + 1)32 π L r f ( λ S −
3) ( gλ S + 24 L Φ ) (cid:34) (cid:88) i =0 Φ i p ( i )12 ( r ) + Φ Φ (cid:48) (cid:88) i =0 Φ i m ( i )12 ( r ) (cid:35) , (A.4a)where p (3)12 ( r ) = 2304 L (cid:0) g + L (cid:1) , (A.4b) p (2)12 ( r ) = − L (cid:110) g L ( λ S + 6) + g (cid:2) L r + L ( λ S + 6) + 12 r (cid:3) − L r ( L + r ) λ S (cid:111) , (A.4c) p (1)12 ( r ) = − λ S (cid:110) g L − g (cid:2) r ( L + r ) + L ( λ S − (cid:3) + gL r ( L + r )(2 λ S −
21) + 6 r ( L + r ) (cid:111) , (A.4d) p (0)12 ( r ) = gλ S L (cid:110) g (cid:2) L r + L ( λ S −
3) + 6 r (cid:3) + g L ( λ S − − gL r ( L + r )( λ S − − r ( L + r ) (cid:111) , (A.4e)– 63 – (2)12 ( r ) = − f L r , (A.4f) m (1)12 ( r ) = 144 f L r λ S , (A.4g)and m (0)12 ( r ) = 2 grλ S (cid:2) g (7 L + 3 r ) + 3 g + g (4 L − r )( L + r ) − r ( L + r ) (cid:3) . (A.4h) A.2 The symmetric Matrix V for the tensors The symmetric matrix V is given by V IJ = ( K − ) IK ( K − ) JM V KM . (A.5)We then have V = 64 π g L (cid:110) gL ( m − m + [6 m (3 m + 4) − r (cid:0) L + r (cid:1) − L ( m − m + 7)Φ (cid:111) , (A.6a) V = − √ π ( m − g L (6 m − (cid:110) gL (cid:2) − m )Φ + 17 m ( m − (cid:3) +17( m − (cid:2) L (cid:0) m − m + 19 (cid:1) Φ + (6 m − m − r (cid:0) L + r (cid:1)(cid:3) (cid:111) (A.6b) V = 32 π ( m − L ( g − gm ) (cid:110) f gL (1 − m ) r Φ (cid:48) + 1250 g L (1 − m ) + 25 g (cid:104) L (1 − m ) r − − m ) Φ L − m − m + 1)Φ L + 289 m ( m − L +36(1 − m ) r (cid:105) +289 L ( m − (cid:2) (18 m − m − r (cid:0) L + r (cid:1) − L (3 m − m + 2)Φ (cid:3) (cid:111) (A.6c) B Symmetric matrices for the scalars
The symmetric matrix V is given by V IJ = ( K − ) IK ( K − ) JM V KM . (B.1)We then have V IJ = 1 f Π y n y m y ( m y Π (cid:48) + λ S − d IJ (cid:88) i =0 Π (cid:48) i V ( i ) IJ . (B.2)with V (9)11 = V (9)22 = 0 , n y = y (2 − y ) , m y = 1 − y and d = (cid:0) m y Π (cid:48) + λ S − (cid:1) , (B.3) d = (cid:0) m y Π (cid:48) + λ S − (cid:1) (cid:2) Π (cid:0) m y Π (cid:48) − (cid:1) − m y Π (cid:48) − λ S + 3 (cid:3) , (B.4) d = ( λ S − (cid:2) Π (cid:0) m y Π (cid:48) − (cid:1) − m y Π (cid:48) − λ S + 3 (cid:3) . (B.5)– 64 –urthermore, V (8)11 = y (cid:2) ∆Π − (∆ − + 1 (cid:1) ] m y n y , (B.6) V (7)11 = 2Π y m y n y (∆ + λ S ) , (B.7) V (6)11 = Π (cid:0) f y m y n y λ S − ∆ f y m y n y − f y m y n y (cid:1) + Π (cid:16) f y m y n y − f y m y n y λ S + 8 f y m y n y + 4∆ y m y n y λ S − y m y n y λ S + 3 y m y n y λ S + ∆ y m y n y − y m y n y (cid:17) + Π (cid:16) f y m y n y λ S − ∆ f y m y n y − f y m y n y − f y m y n y − y m y n y λ S + 4 y m y n y λ S − ∆ y m y n y + 12∆ y m y n y − y m y n y (cid:17) + 3 f y m y n y + 3 f y m y n y + 3 y m y n y λ S − y m y n y , (B.8) V (5)11 = Π (cid:0) f y m y n y λ S + 6∆ y m y n y λ S − y m y n y λ S − y m y n y (cid:1) − f Π y m y n y λ S , (B.9) V (4)11 = Π (7∆ f y m y n y λ S − f m y λ S − f y m y n y λ S + 47 f y m y n y λ S − f y m y n y − f y m y n y +2∆ y m y n y λ S +3∆ y m y n y λ S − y m y n y λ S +6 y m y n y λ S − y m y n y λ S − y m y n y +54∆ y m y n y )+Π (4 f m y λ S − f y m y n y λ S +4 f y m y n y λ S − f y m y n y λ S + 9∆ f y m y n y + 36 f y m y n y ) − f Π m y λ S + Π (3 f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 9∆ f y m y n y + 27 f y m y n y + 63 f y m y n y − y m y n y λ S − y m y n y λ S + 27∆ y m y n y λ S + 2 y m y n y λ S − y m y n y λ S + 9∆ y m y n y − y m y n y ) + 9 f y m y n y λ S + 9 f y m y n y λ S − f y m y n y − f y m y n y + 3 y m y n y λ S − y m y n y λ S + 54 y m y n y , (B.10) V (3)11 = Π (6∆ f y y m y n y λ S − f y m y λ S − f m y λ S + 6 f y y m y n y λ S + 6∆ f y m y n y λ S + 6∆ f y m y n y λ S − f yy m y n y λ S − f y m y n y λ S + 18 f y m y n y λ S + 6 f y m y n y λ S − f yy m y n y λ S + 4∆ y m y n y λ S − y m y n y λ S + 6 y m y n y λ S − y m y n y λ S + 54∆ y m y n y )+ Π (6 f m y λ S + 6 f y m y λ S ) , (B.11)– 65 – (2)11 = (9 f m y λ S − f m y λ S )Π + (6 f λ S m y − f n y y m y − f ∆ n y y m y − f n y y λ S m y − f ∆ n y y λ S m y − f λ S m y + 84 f n y y λ S m y + 18 f ∆ n y y λ S m y )Π + (27∆ m y n y y − m y n y y + ∆ m y n y λ S y − m y n y λ S y − m y n y λ S y − m y n y λ S y + 72∆ m y n y λ S y + 27 m y n y λ S y − f m y n y λ S y + 40 f m y n y λ S y + 6 f ∆ m y n y λ S y + 216 f m y n y y + 54 f ∆ m y n y y − f m y n y λ S y − f ∆ m y n y λ S y − f m y λ S + 9 f m y λ S )Π + (27∆ m y n y y − ∆ m y n y λ S y + 108∆ m y n y y + 54 m y n y y − m y n y λ S y + 18∆ m y n y λ S y − f m y n y λ S y − f m y n y y + 18∆ m y n y λ S y − m y n y λ S y − m y n y λ S y + 36 f m y n y λ S y + 3 f m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y − f m y n y y − f ∆ m y n y y + 135 f m y n y λ S y + 18 f ∆ m y n y λ S y )Π − m y n y y + 81 f m y n y y + m y n y y λ S + 81 f m y n y y − m y n y y λ S + 9 f m y n y y λ S + 9 f m y n y y λ S + 81 m y n y y λ S − f m y n y y λ S − f m y n y y λ S , (B.12) V (1)11 = Π (12 f y m y λ S − f y m y λ S + 12 f m y λ S − f m y λ S − f y m y n y λ S + 18∆ f y m y n y λ S − f y m y n y λ S + 18∆ f y m y n y λ S − f y m y n y λ S + 54 f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 54 f y m y n y λ S + 12∆ y m y n y λ S − y m y n y λ S + 18 y m y n y λ S − y m y n y λ S + 54∆ y m y n y ) + Π (36 f m y λ S − f m y λ S − f y m y λ S + 36 f y m y λ S + 6 f y m y n y λ S − f y m y n y λ S ) , (B.13) V (0)11 = (12 f m y n y λ S y + 3 f ∆ m y n y λ S y + 108 f m y n y y + 27 f ∆ m y n y y − f m y n y λ S y − f ∆ m y n y λ S y )Π + ( − n y y + 81∆ n y y − n y λ S y + 9∆ n y λ S y + 18∆ n y λ S y − n y λ S y + 7 f m y n y λ S y + f ∆ m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y − f m y n y y − f ∆ m y n y y + 207 f m y n y λ S y + 45 f ∆ m y n y λ S y )Π + (3∆ n y λ S y − ∆ n y λ S y + 27∆ n y y − n y y − n y y − f λ S y + 9∆ n y λ S y − n y λ S y − n y λ S y + 27 f n y λ S y + 81 f n y y + 27 f λ S y − n y λ S y + 81∆ n y λ S y + 81 n y λ S y − f n y λ S y + f m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y − f m y λ S y + 90 f m y n y λ S y + 9 f ∆ m y n y λ S y + 189 f m y n y y + 27 f ∆ m y n y y + 54 f m y λ S y − f m y n y λ S y − f ∆ m y n y λ S y − f m y λ S + 27 f m y λ S )Π + 81 n y y − f n y y − n y y λ S + 3 f n y y λ S + 3 f m y n y y λ S − f m y n y y + 27 n y y λ S − f n y y λ S − f m y n y y λ S − n y y λ S + 81 f n y y λ S + 81 f m y n y y λ S , (B.14)– 66 – (9)12 = 2Π y m y n y λ S − y m y n y λ S (B.15) V (8)12 = 3Π y m y n y λ S − Π y m y n y λ S , (B.16) V (7)12 = Π (cid:0) f y m y n y λ S − y m y n y λ S (cid:1) − f Π y m y n y λ S , + Π (cid:0) y m y n y λ S − f y m y n y λ S (cid:1) , (B.17) V (6)12 = − f Π y m y n y λ S +Π (12 f y m y n y λ S +18 f y m y n y λ S − y m y n y λ S − y m y n y λ S )+ Π (6 y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 9 y m y n y λ S ) , (B.18) V (5)12 = Π (40 f y m y n y λ S − f m y λ S − f y m y n y λ S ) + Π (12 f m y λ S + 6 f y m y n y λ S − f y m y n y λ S + 54 y m y n y λ S ) + Π (18 f y m y n y λ S − f m y λ S + 4 f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 58 f y m y n y λ S + 2 y m y n y λ S − y m y n y λ S )+ 4 f Π m y λ S + Π(4 y m y n y λ S − f y m y n y λ S − f y m y n y λ S ) , (B.19) V (4)12 = Π ( f y m y n y λ S − f m y λ S − f y m y λ S + 18 f y m y n y λ S )+ Π (48 f y m y λ S + 48 f m y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 6 y m y n y λ S + 81 y m y n y λ S ) + Π (72 f y m y n y λ S − f y m y λ S − f m y λ S − f y m y n y λ S + 4 f y m y n y λ S − f y m y n y λ S + 84 f y m y n y λ S + 7 y m y n y λ S − y m y n y λ S − y m y n y λ S ) , (B.20) V (3)12 = Π (36 f m y λ S − f m y λ S + 10 f y m y n y λ S − f y m y n y λ S )+ Π (10 f m y λ S − f m y λ S − f y m y n y λ S − f y m y n y λ S + 156 f y m y n y λ S − y m y n y λ S ) + Π (36 f y m y λ S + 72 f y m y λ S − f m y λ S + 36 f m y λ S + 12 f m y λ S + 6 f y m y n y λ S − f y m y n y λ S + 2 f y m y n y λ S + 26 f y m y n y λ S − f y m y n y λ S − y m y n y λ S + 108 y m y n y λ S ) + Π( − f y m y λ S − f y m y λ S − f m y λ S +72 f y m y n y λ S +72 f y m y n y λ S +4 y m y n y λ S − y m y n y λ S )+Π (2 f m y λ S − f m y λ S ) , (B.21) V (2)12 = Π (72 f m y λ S − f m y λ S − f y m y λ S + 72 f y m y λ S − f y m y n y λ S − f y m y n y λ S ) + Π (36 f y m y λ S − f y m y λ S + 36 f m y λ S − f m y λ S + 108 f y m y n y λ S − f y m y n y λ S + 33 f y m y n y λ S + 126 f y m y n y λ S − y m y n y λ S − y m y n y λ S ) + Π (72 f y m y λ S − f y m y λ S − f m y λ S + 72 f m y λ S + 12 f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 21 f y m y n y λ S + 30 f y m y n y λ S − f y m y n y λ S +99 f y m y n y λ S − f y m y n y λ S − y m y n y λ S +54 y m y n y λ S +27 y m y n y λ S ) , (B.22)– 67 – (1)12 = Π (108 f y m y λ S − f y m y λ S − f y m y λ S + 216 f y m y λ S − f m y λ S + 108 f m y λ S + 18 f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 42 f y m y n y λ S − f y m y n y λ S + 54 y m y n y λ S ) + Π(36 f y m y λ S − f y m y λ S + 72 f y m y λ S − f y m y λ S + 36 f m y λ S − f m y λ S − f y m y n y λ S + 108 f y m y n y λ S − f y m y n y λ S + 108 f y m y n y λ S + 12 y m y n y λ S − y m y n y λ S )+Π (12 f y m y n y λ S − f y m y n y λ S )+Π (4 f y m y n y λ S − f y m y n y λ S +72 f y m y n y λ S ) , (B.23) V (0)12 = 2 f Π y m y n y ( λ S − λ S (cid:0) + λ S − (cid:1) , (B.24) V (8)22 = (cid:0) − Π (cid:1) y m y n y λ S (cid:2) Π (3∆ − λ S ) + 1 (cid:3) , (B.25) V (7)22 = 2Π(1 − Π ) y m y n y λ S (cid:2) f m y + y n y (∆ − λ S ) (cid:3) , (B.26) V (6)22 = Π (5 f y m y n y λ S − f y m y n y λ S − ∆ y m y n y λ S − y m y n y λ S + 30∆ y m y n y λ S − y m y n y λ S − y m y n y λ S ) + Π (3∆ f y m y n y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + ∆ y m y n y λ S + 7∆ y m y n y λ S − y m y n y λ S − y m y n y λ S + 18 y m y n y λ S + 12 y m y n y λ S ) + Π (3∆ f y m y n y λ S − f y m y n y λ S )+ 9 f y m y n y λ S + 9 f y m y n y λ S + y m y n y λ S − y m y n y λ S , (B.27) V (5)22 = Π (12 f y m y n y λ S − f y m y λ S − f m y λ S − f y m y n y λ S − f y m y n y λ S + 14 f y m y n y λ S + 18 f y m y n y λ S − y m y n y λ S + 18∆ y m y n y λ S − y m y n y λ S − y m y n y λ S ) + Π(6 f y m y λ S + 6 f m y λ S + 6∆ f y m y n y λ S + 6∆ f y m y n y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 4∆ y m y n y λ S − y m y n y λ S − y m y n y λ S + 36 y m y n y λ S ) + Π (6 f m y λ S + 6 f y m y λ S − f y m y n y λ S ) , (B.28) V (4)22 = (4 f m y n y y λ S − f m y n y y λ S + 4 f ∆ m y n y y λ S − f ∆ m y n y y λ S )Π + (6 m y n y λ S y − ∆ m y n y λ S y − m y n y λ S y + 30∆ m y n y λ S y + 9∆ m y n y λ S y − m y n y λ S y + 2 f m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y + 42 f ∆ m y n y λ S y )Π + (5∆ m y n y λ S y − m y n y λ S y + 22 m y n y λ S y + 3∆ m y n y λ S y − m y n y λ S y − m y n y λ S y − f m y n y λ S y − m y n y λ S y + 108∆ m y n y λ S y − m y n y λ S y − f m y λ S y + 81 f m y n y λ S y − f m y n y λ S y + 15 f m y n y λ S y + 7 f ∆ m y n y λ S y − f m y λ S y + 81 f m y n y λ S y − f ∆ m y n y λ S y − f m y λ S )Π − m y n y y λ S − m y n y y λ S + 15 f m y n y y λ S + 15 f m y n y y λ S + 54 m y n y y λ S + 18 f m y y λ S − f m y n y y λ S + 18 f m y λ S + 36 f m y y λ S − f m y n y y λ S , (B.29)– 68 – (3)22 = Π (72 f y m y λ S − f y m y λ S − f m y λ S + 72 f m y λ S − f y m y n y λ S + 36∆ f y m y n y λ S − f y m y n y λ S + 36∆ f y m y n y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 12∆ y m y n y λ S − y m y n y λ S + 12 y m y n y λ S ) + Π(12 f y m y λ S − f y m y λ S + 12 f m y λ S − f m y λ S + 12∆ f y m y n y λ S − f y m y n y λ S + 12∆ f y m y n y λ S − f y m y n y λ S − f y m y n y λ S + 72 f y m y n y λ S − f y m y n y λ S + 48 f y m y n y λ S + 54 f y m y n y λ S + 2∆ y m y n y λ S − y m y n y λ S + 54∆ y m y n y λ S − y m y n y λ S + 48 y m y n y λ S − y m y n y λ S ) + Π (6 f m y λ S − f m y λ S + 6 f y m y λ S − f y m y λ S + 2 f y m y n y λ S + 6 f y m y n y λ S ) , (B.30) V (2)22 = (9 f m y λ S − f m y λ S )Π + (6 f m y λ S − f m y n y y λ S + f ∆ m y n y y λ S − f m y λ S + 36 f m y n y y λ S − f ∆ m y n y y λ S + 45 f ∆ m y n y y λ S )Π + (6∆ m y n y λ S y − m y n y λ S y − ∆ m y n y λ S y + 12∆ m y n y λ S y − m y n y λ S y + 27 m y n y λ S y − m y n y λ S y + 162∆ m y n y λ S y + 3 f m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y + 9 f m y n y λ S y + 48 f ∆ m y n y λ S y − f ∆ m y n y λ S y − f m y λ S + 9 f m y λ S )Π + (∆ m y n y λ S y + 6 m y n y λ S y + 3∆ m y n y λ S y − m y n y λ S y − m y n y λ S y − f m y n y λ S y − m y n y λ S y + 117∆ m y n y λ S y − f m y λ S y + 90 f m y n y λ S y + 27∆ m y n y λ S y − m y n y λ S y + 108 m y n y λ S y + 108 f m y λ S y − f m y n y λ S y − f m y n y λ S y + 15 f m y n y λ S y + 5 f ∆ m y n y λ S y − f m y λ S y + 45 f m y n y λ S y − f ∆ m y n y λ S y + 216 f m y λ S y − f m y n y λ S y + 45 f ∆ m y n y λ S y − f m y λ S + 108 f m y λ S )Π − m y n y y λ S + 6 m y n y y λ S + 3 f m y n y y λ S + 3 f m y n y y λ S + 27 m y n y y λ S + 36 f m y y λ S − f m y n y y λ S + 36 f m y λ S + 72 f m y y λ S − f m y n y y λ S − m y n y y λ S − f m y y λ S + 243 f m y n y y λ S − f m y λ S − f m y y λ S + 243 f m y n y y λ S , (B.31)– 69 – (1)22 = (72 f λ S m y − f λ S m y − f λ S m y − f y λ S m y + 6 f n y y λ S m y + 72 f y λ S m y − f n y y λ S m y − f y λ S m y )Π + (18 m y n y λ S y − f m y n y λ S y + 18∆ m y n y λ S y − m y n y λ S y + 54 f m y n y λ S y − f ∆ m y n y λ S y − m y n y λ S y + 54 f ∆ m y n y λ S y − f m y n y λ S y + 24 f m y λ S y − f m y n y λ S y − f m y λ S y + 90 f m y n y λ S y − f ∆ m y n y λ S y + 108 f m y λ S y − f m y n y λ S y + 54 f ∆ m y n y λ S y + 24 f m y λ S − f m y λ S +108 f m y λ S )Π +(12 f m y n y λ S y − m y n y λ S y +6∆ m y n y λ S y +72 m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y − m y n y λ S y − m y n y λ S y + 108 f m y n y λ S y + 36 f ∆ m y n y λ S y + 54∆ m y n y λ S y − f ∆ m y n y λ S y + 12 f y m y n y λ S y + 12 f m y n y λ S y − f ym y n y λ S y − f m y λ S y − f y m y n y λ S y − f m y n y λ S y + 132 f ym y n y λ S y − f y ∆ m y n y λ S y − f ∆ m y n y λ S y + 12 f y ∆ m y n y λ S y + 36 f m y λ S y +72 f y m y n y λ S y +72 f m y n y λ S y − f ym y n y λ S y +36 f y ∆ m y n y λ S y +36 f ∆ m y n y λ S y − f y ∆ m y n y λ S y − f m y λ S y + 54 f y m y n y λ S y + 54 f m y n y λ S y − f ym y n y λ S y − f y ∆ m y n y λ S y − f ∆ m y n y λ S y + 108 f y ∆ m y n y λ S y − f m y λ S + 36 f m y λ S − f m y λ S )Π , (B.32) V (0)22 = (18 f ∆ m y n y y λ S − f ∆ m y n y y λ S − f ∆ m y n y y λ S )Π + (3∆ n y λ S y − n y λ S y − n y λ S y + 54∆ n y λ S y + 27∆ n y λ S y − n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y + 18 f m y n y λ S y + 12 f ∆ m y n y λ S y − f m y n y λ S y − f ∆ m y n y λ S y + 54 f ∆ m y n y λ S y )Π + (∆ n y y λ S − f m y n y y λ S − n y y λ S + 9 f m y n y y λ S + f ∆ m y n y y λ S − f y λ S − n y y λ S + 27∆ n y y λ S − n y y λ S + 45 f n y y λ S − f m y λ S − f m y y λ S + 18 f m y n y y λ S − f ∆ m y n y y λ S + 135 f y λ S + 27∆ n y y λ S − n y y λ S + 81 n y y λ S − f n y y λ S + 135 f m y λ S + 270 f m y y λ S − f m y n y y λ S + 27 f ∆ m y n y y λ S − f y λ S − n y y λ S + 81∆ n y y λ S − n y y λ S + 243 f n y y λ S − f m y λ S − f m y y λ S + 243 f m y n y y λ S − f ∆ m y n y y λ S )Π + 3 n y y λ S − f n y y λ S − f m y n y y λ S + 18 f y λ S − n y y λ S − f n y y λ S + 18 f m y λ S + 36 f m y y λ S − f m y n y y λ S − f y λ S − n y y λ S + 135 f n y y λ S − f m y λ S − f m y y λ S + 135 f m y n y y λ S + 162 f y λ S + 81 n y y λ S − f n y y λ S + 162 f m y λ S + 324 f m y y λ S − f m y n y y λ S . (B.33) C Wormholes with toroidal boundaries in simple low-energy theories
This appendix collects some results regarding wormholes with toroidal boundaries. As inthe spherical case, we consider both the U (1) theory of section 4 (see section C.1) and ascalar theory (see section C.2), though now the latter will contain three complex scalars.These results are less complete than for the spherical-boundary cases of sections 4 and 5,in part because we construct only wormhole solutions and do not construct a disconnectedsolution. Indeed, in the torus case a smooth disconnected solution must feature a preferred– 70 –ycle of the torus that shrinks to zero size while the other cycles remain finite (much as inthe familar AdS soliton [66, 67]). This requires a metric Ansatz that breaks the discretesymmetries we impose below.Despite this lack of completeness, the results below indicate that toroidal solutions arebroadly similar to those with spherical boundaries. In particular, in the scalar case weagain find a Hawking-Page-like structure for the wormhole phases, and in particular thelarge wormhole branch is stable. In contrast, in the U (1) case we identify only a singlebranch of wormhole solutions which we find exists for arbitrarily small values of appropriateboundary sources. C.1 U (1) with a toroidal boundary As a short aside we mention that there is a very simple four-dimensional example of awormhole with three gauge fields when the boundary metric is a torus. We take the sametheory as in (4.1) but with three gauge fields of the form A I = L B ε IJK x J d x K (C.1)with B constant, and a metric of the form d s = d r f + ( r + r ) (cid:0) d x + d x + d x (cid:1) . (C.2)A solution exists provided f ( r ) = r + 2 r L and B = r L . (C.3)In this section we are assuming that x , x and x are periodic coordinates with periods ∆ x , ∆ x , ∆ x . In this context we believe that the solution we have found is the uniqueconnected geometry, though we have not found a way to construct a disconnected solution.It is a simple exercise to compute the regulated on-shell action for this solution which yields S = 8 √ K (cid:0) (cid:1) r L ∆ x ∆ x ∆ x . (C.4)We have not attempted to study the negative modes of this solution, though we believethat this wormhole will again be stable. The reason for this is that the infinite-radius limitof the U (1) wormholes with a spherical boundary we constructed are connected coincideswith the ∆ x i → ∞ limit of the wormholes discussed in this section. Since for the sphericalwormholes we found no negative modes, we expect the same to hold here. C.2 Wormholes sourced by scalar fields with a toroidal boundary
We now consider wormholes sourced by scalar fields which have a toroidal boundary. Ifwe want to keep isotropy and homogeneity, we seem to need at least three complex scalar– 71 –elds, which we label by ψ I and collectively assemble in a vector (cid:126)ψ . We will proceed muchas in section 5, using the simple low-energy action S = − (cid:90) M d x √ g (cid:20) R + 6 L − ∇ a (cid:126)ψ ) · ( ∇ a (cid:126)ψ ) ∗ − µ (cid:126)ψ · (cid:126)ψ ∗ (cid:21) − (cid:90) ∂ M d x √ h K + S µ B . (C.5)Here L is the four-dimensional AdS length scale and ∗ denotes complex conjugation, thesecond term is the usual Gibbons-Hawking term and S µ B the boundary counter-term tomake the action finite and the variational problem well defined from the perspective of (cid:126)ψ .We again consider both the massless example and the effective mass that would descirbeconformal coupling, so the boundary terms S µ B are chosen as in Section 5 (but now withthree complex scalar fields). The Einstein equation and scalar field equation (ignoringboundary terms) derived from this action read R ab − R g ab − L g ab = 2 ∇ ( a (cid:126)ψ · ∇ b ) (cid:126)ψ ∗ − g ab ∇ c (cid:126)ψ · ∇ c (cid:126)ψ ∗ − µ g ab (cid:126)ψ · (cid:126)ψ ∗ , (C.6a) (cid:3) (cid:126)ψ = µ (cid:126)ψ . (C.6b)Before describing our (numerical) solutions with torus boundary, we would like tocomment on a simple analytic wormhole that arises for the massless case when the torusboundary is replaced by R . For this solution one considers the configuration (cid:126)ψ = Φ x Φ x Φ x , (C.7a)with metric d s = d r f ( r ) + ( r + r )(d x + d x + d x ) , (C.7b)taking r = Φ L and f = r + r L . (C.7c)This shows rather explicitly that wormhole solutions can exist for any value of A , thoughthe solution does not satisfy standard boundary conditions at large x i .To fix this, we introduce harmonic dependence on x i in the scalar field Ansatz and alsoconsider µ (cid:54) = 0 to write (cid:126)ψ = (cid:126)X k ψ ( r ) , (C.8)with (cid:126)X k = e i k x e i k x e i k x . (C.9)We also consider the metric d s = d r f ( r ) + ( r + r )(d x + d x + d x ) , (C.10)– 72 –here f , ψ and r are to be determined numerically for a given source Φ associated with theboundary value of ψ . In performing such numerics it is wise to use a compact coordinate,so we introduce y = 1 − r (cid:112) r + r . (C.11)The conformal boundary is now located at y = 1 , and the Z symmetry plane at y = 0 .We are interested in the case where x , x and x span a cubic 3-torus T with period (cid:96) = 2 π/k . By construction, the torus has minimal volume at r = y = 0 . It is a simpleexercise to determine f from the Einstein equation to find f = L (cid:2) k (1 − y ) + r µ (cid:3) ψ − r L (2 − y )(1 − y ) y [(1 − y ) ψ (cid:48) − . (C.12)Since we are interested in solutions for which ψ is smooth at y = 0 and f is finite there, weneed to have L ( k + r µ ) ψ (0) = r . (C.13)At this stage we introduce ψ (0) = A and write r in terms of A in the equation for ψ .This yields m y ψ (cid:48)(cid:48) + p (0 , , y ψ + m y p (2 , , y ψ (cid:48) − m y p (0 , , ψ (cid:48) ψ − m y p (3 , , y ψ (cid:48) = 0 , (C.14a)where p ( a,b,c ) y = a A − ( b m y + c A L µ ) ψ A − ( m y + A L µ ) ψ , (C.14b)and we again recall that m y = 1 − y . The equation for ψ depends only on A , and thedependence in r and k has dropped out. This is to be expected. If the boundary metricis flat, there is a residual gauge freedom when one simultaneously scales all the x i anduses conformal invariance. A priori one might have thought that the wormholes we seek toconstruct formed a two-dimensional family of solutions parametrised by Φ and k with r being fixed by the former. However, due to conformal invariance this is not the case, andinstead only the ratio V /k ∆ − is physically meaningful in the bulk. One might erroneouslythink that this should have reduced the moduli space of solutions in the spherical case to 0dimensions. However, the sphere there introduces a new scale in the problem which cannotbe removed.Since we do not construct a disconnected solution for comparison, computing the on-shell action is not of much interest. Instead, we will focus on trying to understand whetherwormholes in this class of theories exist to arbitrary small values of Φ and whether they arefree of negative modes. The answer to the first question appears to be negative. Even forwormholes with toroidal boundary conditions we find a minimal critical amplitude V /k ∆ − above which they can exist. This is perhaps surprising, as we find no smooth solutions atall within our ansatz for V /k less than this critical value. For the massless case we find thatwormholes only exist for V ≥ V min ≈ . (see left hand side of Fig. 15), whereas forthe conformal case we need V /k > ( V /k ) min ≈ . (see right hand side of Fig. 15).– 73 –ust as for the spherical case, for each value of V ≥ V min two wormhole solutions exist. Weagain call the phase with smallest y /k (with y ≡ r /L ) the small wormhole phase andwe call the phase with largest y /k the large wormhole phase. Precisely at V /k ∆ − min we have y /k = ( y /k ) min with ( y /k ) min ≈ . and ( y /k ) min ≈ . for the masslessand conformal cases, respectively. The small wormhole phase is shown in Fig. 15 as orangesquares, while the large wormhole phase is represented by the blue disks. � � � � � � �������� � � �� �� �� �� �� ���������������������� Figure 15 . Wormholes with toroidal boundary conditions.
Left panel : wormholes sourced bymassless scalars, with the dashed red line being analytically generated by Eq. (C.15) with terms upto i = 8 . Right panel : wormholes generated by conformally coupled scalars. In both panels, thesmall wormhole phase is represented by orange squares, while large wormholes are given by bluedisks.
For the massless case we can actually do better and find a uniform expansion for ψ inpowers of /A . This tuns out to be a very useful expansion because this allows us to checkour numerics. The expansion takes the rather simple form ψ = A + + ∞ (cid:88) i =0 A i +10 ψ ( i ) (cid:18) rr (cid:19) . (C.15)We have carried out this expansion to i = 8 , but for the sake of brevity we only presenthere results for i = 0 , ψ (0) ( z ) = 12 z z , (C.16) ψ (1) ( z ) = 148 (cid:34) z (cid:0) z + 14 z (cid:1) (1 + z ) − z z arctan z − z (cid:35) . (C.17)On the left panel of Fig. 15 we compare our analytic expansion up to i = 8 , with thenumerical data and find excellent agreement for a large range of A .– 74 – .2.1 Negative modes with toroidal boundaries Studying perturbations of wormholes with planar boundaries turns out to be schematicallsimilar to studying those with spherical boundaries, though it is easier in practice. Again,we expand all our perturbations in terms of scalar, vector and tensor harmonics on T obeying to (cid:3) T S k S + k S S k S = 0 , (C.18a) (cid:3) T S k V i + k V S k V i = 0 with ∇∇ i S k V i = 0 , (C.18b)and (cid:3) T S k T ij + k T S k T i = 0 , with ∇∇ i S k T ij = 0 and g ij S k T ij = 0 , (C.18c)respectively. Modes with k S = k V = 0 have to be studied separately, but k T = 0 can beobtained from the result with k T = 0 . C.2.1.1 Scalar-derived perturbations with k S = 0 This section is similar in many respects to the section where we studied the negative mode ofa spherically symmetric wormhole with respect to scalar-derived perturbations with (cid:96) = 0 .We shall see that the large wormhole phase has no negative modes, whereas the smallwormhole phase does seem to possess such a mode.Our perturbations read δ d s = L (1 − y ) (cid:20) δf ( y ) d y (2 − y ) y + δp ( y )(d x + d x + d x ) (cid:21) , (C.19a)and for the scalars δ (cid:126)ψ = (cid:126)X k δψ . (C.19b)Under an infinitesimal diffeomorphism ξ = ξ y d y these perturbations transform as δf = 2(1 − y ) L (cid:20) − y + 2 y − y (2 − y )(1 − y ) f (cid:48) f (cid:21) ξ y + 2 y (2 − y )(1 − y ) L ξ (cid:48) y , (C.20a) δp = 2(2 − y )(1 − y ) yy L f ξ y , (C.20b) δψ = (2 − y )(1 − y ) yψ (cid:48) L f ξ y . (C.20c)By now the procedure is familiar. We first expand the action (C.5) to second orderin the perturbations. The resulting action, S (2) is a function of δψ , δf and their firstderivatives with respect to y . Additionally, S (2) is also a function of δp and its first andsecond derivatives. We first integrate by parts the term proportional to δp (cid:48)(cid:48) with theresulting boundary term cancelling the Gibbons-Hawking-York perturbed boundary action. S (2) is now a function of δψ , δf , δp and their first derivatives. One can also integrate onemore my parts terms proportional to δf (cid:48) (whose boundary terms cancel with the perturbedboundary counter terms appearing in (C.5) ). The second order action S (2) is then a function– 75 – ψ , δp and their first derivatives and of δf . Crucially, δf enters the action algebraically.This means we can perform the Gaussian integral over δf (again using the Wick rotationdescribed in section 3) and find an effective action ˇ S (2) that is a function of δψ , δp andtheir first derivatives only. At this stage we introduce the gauge invariant quantity Q = √ (cid:20) δψ − (1 − y ) ψ (cid:48) y δp (cid:21) , (C.21)where the factor of √ was chosen for later convenience of presentation. Clearly Q isinvariant under the infinitesimal gauge transformations (C.20). Solving the above relationwith respect to δψ − gives an action for Q , where the dependence in δp completely dropsout because of gauge invariance. The final action for Q reads ˇ S (2) = 2 L (cid:96) (cid:90) + ∞ d y y √ f √ − y (1 − y ) √ y (cid:20) (2 − y )(1 − y ) y − (1 − y ) ψ (cid:48) Q (cid:48) f + V Q (cid:21) , (C.22)where V = 1 ψ (cid:40) (2 − y ) yf + 2 f (cid:2) y − k (1 − y ) ψ ψ (cid:48) (cid:3) (2 − y ) yy [1 − (1 − y ) ψ (cid:48) ] − − (1 − y ) ψ (cid:48) (cid:41) . (C.23)As expected, V is not positive definite for all wormholes. The combination − (1 − y ) ψ (cid:48) which appears multiplying the kinetic term for Q is positive definite so long as y /k = 1 and y /k (cid:38) . , for the massless and conformally coupled cases, respectively. Inparticular, for the large wormhole branch − (1 − y ) ψ (cid:48) is positive definite.To proceed, we use numerical methods. We first note that in the original r coordinatesof (C.7b), V would have been even around r = 0 . This means perturbations that are evenand odd with respect to r = 0 will be orthogonal, so we can study them separately. In termsof the y coordinates, these correspond to very distinct behaviours near y = 0 . Namely, inthe odd sector we have Q ∼ √ y near the origin, while in the even sector Q admits a regularTaylor expansion around y = 0 . We have not found any negative mode on the odd sectorof perturbations.To search for negative modes λ , we integrate (C.7b) by parts and set − √ − y (1 − y ) √ y √ f (cid:20) − y ) √ − y √ y − (1 − y ) ψ (cid:48) Q (cid:48) √ f (cid:21) (cid:48) + V Q = λQ . (C.24)Near the boundary, we find that admits two possible boundary behaviours Q = C + (1 − y ) + (cid:113) ( ∆ − ) − λ [1 + . . . ] + C − (1 − y ) − (cid:113) ( ∆ − ) − λ [1 + . . . ] , (C.25)with normalisability demanding we set C − = 0 . We thus have a well defined Sturm-Liouvilleproblem, which we can readily solve numerically.The results of this analysis can be seen in Fig. 16 where we plot λ as a function of y /y min0 for the massless (left panel) and conformal (right panel) cases. In both cases,a negative mode exists for the small wormhole phase, but becomes positive on the largewormhole phase. This establishes that large wormholes are stable with respect to scalar-derived perturbations with k S = 0 . This can be analytically proved by manipulating the scalar equation (C.14a) and numerically checkedto be the case for the first digits. – 76 – ��� ���� ���� ���� ���� - �� - �� - �� - �� - �� - ��� ���� ���� ���� ���� ���� - �� - �� - �� - �� Figure 16 . Negative mode for the k = 0 perturbations as a function of y /y min0 . Left panel : neg-ative modes of wormholes sourced by massless scalars.
Right panel : negative modes of wormholesgenerated by conformally coupled scalars.
C.2.1.2 Scalar-derived perturbations with k S (cid:54) = 0 We have chosen our boundary metric to be a torus so our fundamental scalar harmonictakes a very simple form S k S = cos( k S · x + γ S ) , (C.26)with k S = | k S | and x = { x , x , x } . Note also that k S = k { n , n , n } , where n i areintegers because our 3-torus is cubic. Finally, γ S is an arbitrary phase which will play norole. While we were able to keep all n i distinct, it is clear the most dangerous sector occurswhen we take all n i = n . This is the sector we present here. We also define k S = √ κ , with κ = n k .The metric perturbations take the already familiar form δ d s k S = h k S yy ( y ) S k S d y + 2 h k S y ( y ) ∇∇ i S k S d y d x i + H k S T ( y ) S k S ij d x i d x j + H k S L ( y ) S k S g ij d x i d x j (C.27a)where S k S ij = ∇∇ i ∇∇ j S k S − g ij ∇∇ k ∇∇ k S k S , (C.27b)while for the scalar perturbation we choose δ (cid:126)ψ k S = (cid:126)X k B k S ( y ) S k S + ( ∇∇ i S k S ∇∇ i (cid:126)X k ) A k S ( y ) . (C.27c)In the above g is the metric on T and ∇∇ its associated metric preserving connection.Under an infinitesimal diffeomorphism of the form ξ k S = ξ k S y ( y ) d y + L k S y ( y ) ∇∇ i S k S d x i (C.28)– 77 –he metric and scalar perturbations transform as δA k S = (1 − y ) ψL y L k S y , (C.29a) δB k S = (2 − y ) y (1 − y ) ψ (cid:48) L f ξ k S y , (C.29b) δh k S yy = 2 (cid:0) − y + 2 y (cid:1) y (2 − y )(1 − y ) ξ k S y − f (cid:48) f ξ k S y + 2 ξ k S y (cid:48) , (C.29c) δh k S y = ξ k S y + L k S y (cid:48) − − y L k S y , (C.29d) δH k S T = 2 L k S y , (C.29e) δH k S L = 2(2 − y ) yy (1 − y ) f ξ k S y − κ L k S y . (C.29f)Had we taken all n i distinct, we would have to consider three different perturbations similarto A k S and B k S , parametrising each of the complex scalars in (cid:126)ψ .The remaining procedure is very similar to what we have seen when studying negativemodes of the womholes sourced by the scalars with a spherical boundary. First, we writethe second order action S (2) in first order form by integrating by parts and find it it canbe written in term of A k S , B k S , H k S L , H k S T , h k S y and their first derivatives. However, h k S yy appears algebraically and we again apply the Wick-rotation procedure of section 3. Wecan thus perform the Gaussian integral and find a new second order action ˜ S (2) that is afunction of A k S , B k S , H k S L , H k S T and their first derivatives, but now h k S y enters algebraicallyand again we can perform the corresponding Gaussian integral finding an action ˇ S (2) thatis a function of A k S , B k S , H k S L , H k S T and their first derivatives.At this point we introduce gauge invariant variables Q k S and Q k S defined by Q k S = A k S − (1 − y ) ψ L y H k S T , (C.30a) Q k S = B k S − (1 − y ) κ ψ (cid:48) L y H k S T − (1 − y ) ψ (cid:48) L y H k S L . (C.30b)which are invariant under the infinitesimal transformations (C.29). Solving the above rela-tions with respect to A k S and B k S and inputting those in ˇ S (2) gives an action for Q k S , Q k S and their first derivatives only. The dependence in H k S T and H k S L , after using the equationsof motion for ψ , completely drops out by virtue of gauge invariance. To ease presentationwe define further Q k S = 12 √ π / √ k m ∆ y ψ (cid:20) q k S − (1 − y ) n ψ (cid:48) q k S (cid:21) , (C.31) Q k S = 12 √ π / √ k m ∆ y q k S . (C.32)The second order action ˇ S (2) can then be written as ˇ S (2) = 2 L (cid:90) + ∞ d y y m − y √ f √ n y (cid:34) m y n y f q k S i (cid:48) K ij q k S j (cid:48) + q k S i V ij q k S j (cid:35) , (C.33)– 78 –here K − = (cid:34) n (cid:16) n ψ + 3 m y ψ (cid:48) (cid:17) m y ψ (cid:48) n m y ψ (cid:48) n (cid:35) . (C.34)It is a simple exercise to show that K is positive definite so long as − (1 − y ) ψ (cid:48) ispositive. However, we have argued that this is the case for all large wormhole. It thenall boils down to the positivity properties of V , whose explicit form we present in sectionC.2.1.2.1. This easy to study numerically, and we find that, for | n | = 1 , V is positivedefinite for y /k (cid:38) . and y /k ≥ . for the conformal and massless cases,respectively. Both these values are well within the small wormhole branch, so these resultsestablish stability in the large wormhole branch. C.2.1.2.1 Symmetric matrices for the scalars with toroidal boundary con-ditions
The symmetric matrix V is given by V IJ = ( K − ) IK ( K − ) JM V KM . (C.35)We then have V IJ = 1 A f ψ d IJ (cid:88) i =0 ψ (cid:48) i V ( i ) IJ . (C.36)with V (6)12 = V (6)22 = V (5)12 = 0 , n y = y (2 − y ) , m y = 1 − y and d = A n ψ n y , (C.37) d = A n n y , (C.38) d = 1 . (C.39)Furthermore V (6)11 = 72 A ψ m y n y , (C.40) V (5)11 = 48 A ψ m y n y , (C.41) V (4)11 = A (cid:0) f ψ m y n y − f ψ m y n y − ψ m y n y − ψ m y n y + 4 ψm y n y (cid:1) − A f ψ m y n y , (C.42) V (3)11 = A (cid:0) f ψ m y n y − f ψ m y n y + 48 f ψ m y n y − ψ m y n y − ψ m y n y (cid:1) − A f ψ m y n y , (C.43) V (2)11 = A (cid:0) f ψ m y − f ψ m y + 9 f n ψ m y n y + 9∆ f ψ m y n y + 14 f ψ m y n y (cid:1) + A (6∆ f ψ m y − f ψ m y + 54∆ f ψ m y − f n ψ m y n y + 27∆ f n ψ m y n y − f ψ m y n y + 13∆ f ψ m y n y + 42∆ f ψ m y n y + 4 f ψm y n y − n ψm y n y − ψ m y n y + 24∆ ψ m y n y − ψm y n y ) + 6 f ψ m y , (C.44) Other values of | n | > are even more stable. – 79 – (1)11 = A (6∆ f ψ m y − f ψ m y + 4∆ f n ψ m y n y − f n ψ m y n y − f ψ m y n y + 2∆ n m y n y + 6∆ ψ m y n y ) − A (cid:0) f ψ m y + 4 f n ψ m y n y (cid:1) , (C.45) V (0)11 = A (21∆ f n ψ m y n y − ∆ f n ψ m y n y − f n ψ m y n y − f n ψm y n y + 9∆ f n ψm y n y + ∆ ( − f ) ψ m y n y − ∆ f ψ m y n y + 12∆ f ψ m y n y − ∆ n ψn y + 3∆ n ψn y − ∆ ψ n y + 3∆ ψ n y ) + A (∆ f n ψ m y n y + 7 f n ψ m y n y + 3 f n ψm y n y + ∆ f ψ m y n y + 4 f ψ m y n y ) , (C.46) V (5)12 = 36 A ψ m y n y , (C.47) V (4)12 = 24 A ψm y n y , (C.48) V (3)12 = A (cid:0) f ψ m y n y − f ψ m y n y − ψ m y n y − ψ m y n y + 2 m y n y (cid:1) − A f ψ m y n y , (C.49) V (2)12 = A (cid:0) f ψ m y n y − f ψ m y n y + 24 f ψm y n y − ψm y n y − ψm y n y (cid:1) − A f ψ m y n y , (C.50) V (1)12 = A (12∆ f ψ m y − f ψ m y + 6 f n ψ m y n y + 4∆ f ψ m y n y + 6 f ψ m y n y )+ A (2∆ f ψ m y − f ψ m y + 18∆ f ψ m y − f n ψ m y n y + 18∆ f n ψ m y n y − f ψ m y n y + 6∆ f ψ m y n y + 18∆ f ψ m y n y + 2 f m y n y − ψ m y n y + 12∆ ψ m y n y − m y n y ) + 2 f ψ m y , (C.51) V (0)12 = A (2∆ f ψ m y − f ψ m y + 2∆ f n ψm y n y − f n ψm y n y − f ψn y + 2∆ ψn y ) − A (cid:0) f ψ m y + 2 f n ψm y n y (cid:1) (C.52) V (4)22 = 18 A ψ m y n y , (C.53) V (3)22 = 12 A ψm y n y , (C.54) V (2)22 = A (6∆ f ψ m y − f ψ m y − ψ m y n y − ψ m y n y + m y n y ) − f ψ m y , (C.55) V (1)22 = A (4∆ f ψ m y − f ψ m y + 12 f ψm y − ψm y n y ) − f ψ m y (C.56) V (0)22 = A (9∆ f n ψ m y − f n ψ m y + ∆ ( − f ) ψ m y + 3∆ f ψ m y + f − ∆ ψ n y + 3∆ ψ n y − n y ) + 3 f n ψ m y + ∆ f ψ m y . (C.57)– 80 – .2.1.3 Vector-derived perturbations with k V = 0 This mode is special, and must be studied separately. It is the analogue of the mode with (cid:96) V = 1 in the spherical case. Again we find that the associated S (2) vanishes identically afterintegrating out h y (whose quadratic coefficient has the correct sign to make the integralover h y converge). Thus the other parts of this mode are pure-gauge. C.2.1.4 Vector-derived perturbations with k V (cid:54) = 0 Vector derived perturbations with k V (cid:54) = 0 follow a similar pattern to the spherical case.Recall that vector harmonics on T must be transverse, i.e. ∇∇ i S i = 0 and obey to (cid:3) T S k V i + k V S k V i = 0 . (C.58)One such example is for instance S k V i d x i = cos( k V x + γ )d x , (C.59)where γ is an unimportant phase and k V = nk . From S k V i we can construct the followingsymmetric tensor S k V ij = ∇∇ i S k V j + ∇∇ j S k V i , (C.60)which we use to build the most general vector-derived metric perturbation. Vector-derivedperturbation with k V read δ d s k V = 2 h k V y ( y ) S k V i d y d x i + H k V T ( y ) S k V ij d x i d x j (C.61a)while for the scalar perturbation we choose δ(cid:126) Π k V = ( S k V i ∇∇ i (cid:126)X k ) A k V ( y ) . (C.61b)The most general vector-derived infinitesimal diffeomorphism can be constructed via ξ (cid:96) V = L k V y S k V i d x i . (C.62)The above infinitesimal diffeomorphism induces the following gauge transformations δh k V y = − L k V y − y + L k V y (cid:48) , (C.63a) δH k V T = L (cid:96) V y , (C.63b) δA k V = (1 − y ) ψL y L k V y . (C.63c)By now the procedure should be very familiar. We expand the action to second orderin perturbations, and write it in first order form. To do this, we have to integrate by parts,and the boundary terms cancel with the perturbed Gibbons-Hawking-York term. It turnsout that h k S y enters the second order action S (2) algebraically, which means we can do the– 81 –aussian integral and find a new action ˇ S (2) which is a function of H k S T and A k S only. Atthis stage we introduce a gauge invariant variable Q k V defined through the relation A k V = √ km ∆ y √ π / (cid:114) ψ n Q k V + (1 − y ) ψL y H k V T (C.64)which leads to the following second order action ˇ S (2) = 2 L (cid:90) ∞ d y √ f y m − y √ n y (cid:34) m y n y f Q (cid:48) k V + V Q k V (cid:35) (C.65)with V = m y (cid:18) A − µ L (cid:19) (cid:0) n + 4 ψ + ∆ ψ (cid:1) − n y µ L f + n ( n + 4 ψ ) ψ − n y f (cid:40) m y ψ (cid:48) + 1( n + 4 ψ ) ψ (cid:34) n − m y n (cid:0) n − ψ (cid:1) ψ (cid:48) n + 4 ψ (cid:35)(cid:41) , (C.66)where we recall that n y ≡ y (2 − y ) , m y ≡ − y and k V = n k . Though it is not apparent fromthe above expression, it turns out that V seems positive definite for all wormholes we haveconstructed. In particular, it appears positive for the large wormholes, thus establishingstability with respect to vector-derived perturbations in this sector as well. C.2.1.5 Tensor-derived perturbations
We now come to the easiest sector of perturbations. The building blocks for perturbationsin this sector are given by tensor harmonics on T , which obey to (cid:3) T S k T ij + k T S k T ij = 0 (C.67)with ∇∇ i S k T ij = 0 and g ij S k T ij = 0 . An example of such an harmonic is S k T ij d x i d x j = cos( k T x + γ )d x d x (C.68)where γ is again an arbitrary phase and k T = nk .The metric perturbation is simply given by δ d s k T = √ k / L y π / m y H k T ( y ) S k T ij d x i d x j , (C.69)while for ψ we demand δψ = 0 . In this case H k T is automatically gauge invariant, andbringing the quadratic action to first order form yields ˇ S (2) = 2 L (cid:90) + ∞ d y √ f y m y √ n y (cid:34) m y n y f H (cid:48) k T + V H k T (cid:35) (C.70)with V = k m y y (cid:0) n + 4 ψ (cid:1) , (C.71)which is manifestly positive. This thus establishes that no negative modes exist in thetensor sector of perturbations. – 82 – Table with longer list of models studied
This appendix provides a table (below) listing the 22 string/M-theory compactifications andthe 14 ad hoc low energy models that we have studied most fully, along with the resultsobtained. The methods applied to obtain these results are much like those explained indetail in the main text and in appendix C for the cases reported there. Models 1-4, 20, 23,27, and the d = 4 version of 26 were analyzed in the main text, and models 24, 29, and the d = 4 version of 28 were discussed in appendix C.In most cases, models are described by giving the bibliography reference that describesthem. The consistent truncations leading to the models labelled by IIB were discussed insection 8. Models labelled by E are pure gravity models, labelled by ES are Einstein-scalar models and labelled by EM are Einstein-Maxwell models. The model labelled by M † consists of a reduction of 11-dimensional supergravity on AdS × S × T of the form d s = g ab d x a d x b + L dΩ + e √ φ d x + e −√ φ d x + e √ φ d x + e −√ φ d x + e √ φ d x + e −√ φ d x (D.1a) F (4) = L Ω ∧ (d z ∧ d z + d z ∧ d z + d z ∧ d z ) . (D.1b)where d Ω is the two-dimensional volume form on the round S . Perhaps notably, our listdoes not include the model described in section 5 of [10] where the disconnected solutionremains to be found in order to determine if the wormhole described there will dominate.– 83 – Model Fields ∂ M E WD NM BN φ = χ = 0 , A i (1) = Φ δ i , (cid:101) A i (1) = Φ δ i , i = 1 , , S Y Y ND YD2 [61] φ = χ = 0 , A i (1) = Φ δ i , (cid:101) A i (1) = Φ δ i , i = 1 , , (cid:101) S Y Y ND YD3 [61] φ (cid:54) = 0 , χ (cid:54) = 0 , A i (1) = (cid:101) A i (1) = 0 , i = 1 , , H Y ? ? Y4 [61] φ (cid:54) = 0 , χ (cid:54) = 0 , A i (1) = (cid:101) A i (1) = 0 , i = 1 , , (cid:101) H Y ? ? Y5 [68] A i = Φ σ i , ϕ = ϕ = − ϕ , σ i = A = 0 , i = 1 , , S Y Y ND YD6 [68] A i = Φ σ i , ϕ = ϕ = − ϕ , σ i = A = 0 , i = 1 , , (cid:101) S Y Y ND YD7 [68] A i = Φ , ϕ = ϕ = − ϕ , σ i = A = 0 , i = 1 , , T Y ? ND Y8 [68] A i = A = 0 , ϕ i = ϕ, σ i = k x i , i = 1 , , T N9 [69, 70] φ, χ H Y ? ? Y10 [71, 72] z = − z = − z , β = β = 0 T N11 [71, 72] z = z = − z = − z , β (cid:54) = 0 , β = 0 T N12 [71, 72] z = z , z = z = β = 0 , β (cid:54) = 0 T N13 [73] A i = B A , ϕ = ϕ = 0 , i = 1 , , CP × H Y ? ? Y14 [73] A i = B x d x , ϕ = ϕ = 0 , i = 1 , , T N15 [73] A i = B A , ϕ = ϕ = 0 , i = 1 , , CP × T N16 [73] A i = Φ ˆ σ i , ϕ = ϕ = 0 , i = 1 , , S × S N17 [73] A i = Φ ˆ σ i , ϕ = ϕ = 0 , i = 1 , , S × (cid:101) S N18 [74] A i = Φ i ˆ σ i , i = 1 , , S Y N Y ?19 [74] A i = Φ ˆ σ i , A i = Φ ˆ σ , i = 1 , S Y N Y ?20 IIB φ i = Φ x i , φ = 0 , | x | = 1 , i = 1 , , S Y Y N Y21 IIB φ + iφ = Φ e ikx , φ + iφ = Φ e ikx T Y ? N Y22 M † φ i = Φ x i , | x | = 1 , i = 1 , , S Y Y N Y23
EM d = 3 , A i = Φ i ˆ σ i , i = 1 , , S Y Y ND24
EM d = 3 , A I = L B ε IJK x J d x K , I = 1 , , T Y Y N25
ES d = 3 , , , φ i = Φ x i , | x | = 1 , i = 1 , . . . , d − , µ = 0 S d Y Y ND26
ES d = 3 , φ i = Φ x i , | x | = 1 , i = 1 , . . . , , µ = − S Y Y ND27
ES d = 4 , φ i = Φ x i , | x | = 1 , i = 1 , . . . , , µ = − S Y Y ND28
ES d = 3 , , , φ i + iφ i +1 = Φ e ikx i , i = 1 , . . . , d − , µ = 0 T d Y Y ND29
ES d = 3 , φ i + iφ i +1 = Φ e ikx i , i = 1 , , , µ = − T Y Y ND30
ES d = 4 , φ i + iφ i +1 = Φ e ikx i , i = 1 , , , , µ = − T Y Y ND31
EM d = 4 , F = (cid:63)F S × S Y ? N32
E d = 3 (cid:101) S N33
EM d = 3 , A = Φˆ σ (cid:101) S N34
EM d = 4 CP Y ? N35
EM d = 6 CP Y ? N36
EM d = 3 S × S N Table 1 . The columns are as follows:
Model gives the citation where to find the given supergravitymodel or indicates the model as described above,
Fields lists the supergravity fields taken to be non-trivial, ∂ M gives the boundary manifold, E states whether co-homogeneity one wormholes exist inthe model, WD states whether wormholes ever dominate over the disconnected solution, NM stateswhether the model has field-theoretic negative modes and BN states whether the model suffers frombrane nucleation instabilities. In most cases Y/N indicates yess/no. Question marks (?) indicateissues not investigated or not resolved. In the columns marked by NM and BN the possible entriesare: Y - all wormholes have brane nucleation instabilities/ field theoretical negative modes; YD -all wormholes that dominate over the disconnected solutions have brane nucleation instabilities/field theoretical negative modes; N - all wormholes are free of brane nucleation instabilities/ fieldtheoretical negative modes; ND - all wormholes that dominate over the disconnected solutions arefree of brane nucleation instabilities/ field theoretical negative modes. Boundary manifolds markedwith (cid:101) are squashed at the boundary: e.g. (cid:102) S denotes a boundary squashed S . Entries filled ingrey represent situations for which it does not make sense to fill the respective entry. Model 9 isclosely related to the mass deformation of N = 4 SYM studied in [27]. – 84 – eferences [1] G. V. Lavrelashvili, V. A. Rubakov, and P. G. Tinyakov,
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