JHEP11(2020)167 Springer October 5, 2020 October 27, 2020 : : November 30, 2020 : a,c Received , Accepted degrees of freedom in each Published ) 2 N ( O Super Yang-Mills theory in terms of ) Published for SISSA by https://doi.org/10.1007/JHEP11(2020)167 N and Gábor Sárosi a = 4 SU(2 theory in which [email protected] N , )) N U( extremal D3-branes in the decoupling limit. The solution × ) N Matthew DeCross N a,b (U( . 3 S → 2009.08980 ) The Authors. N c AdS-CFT Correspondence, D-branes
We construct a single-boundary wormhole geometry in type IIB supergravity , SU(2 [email protected] sector are entangled in an approximate thermofield double state at a temperature ) N Pleinlaan 2, B-1050 Brussels, Belgium CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland E-mail: gabor.sarosi@cern.ch David Rittenhouse Laboratory, University of209 Pennsylvania, S.33rd Street, Philadelphia PA,Theoretische 19104, Natuurkunde, U.S.A. Vrije Universiteit Brussel (VUB), and International Solvay Institutes, b c a Open Access Article funded by SCOAP ArXiv ePrint: propriate choice of parameters, andWe comment also on describe mechanisms a for construction generating of traversability. a double wormholeKeywords: between two universes. that gives a globalgeometry monodromy has to a dual some interpretationa coordinates, in Higgsed while preserving orientability.SU( The much colder than the Higgs scale. We argue that the solution can be made long-lived by ap- Abstract: by perturbing two stacksinterpolates of from a two-sided planara AdS-Schwarzschild harmonic geometry in two-center solution theThe interior, in construction through involves the a CPT intermediate twist region, in the to gluing an of the asymptotic wormhole to AdS the space. exterior throats Vijay Balasubramanian, Knitting wormholes by entanglement in supergravity JHEP11(2020)167 15 16 Super Yang-Mills (SYM) theory geometry with a single boundary ) 5 N S ], albeit in a complicated and nonlocal × 5 5 – 10 1 28 18 = 4 SU(2 9 N – 1 – 31 5 14 33 23 21 25 12 13 21 1 29 The simplest connection between wormholes and entanglement involves two copies of 4.1 Instability timescale 4.2 Stabilizing with rotation 3.4 Regions II-III:3.5 linearized regime Joint solution3.6 in regions I-II: Joint monopole solution3.7 contribution in regions I-II-III: Global dipole structure and contribution flux conservation 3.1 Region IV:3.2 two-center harmonic solution Region V:3.3 asymptotics Region I: black brane system should correspond to an asymptoticallyand a AdS long-lived interior wormhole.and The purpose describe of its this properties. paper is to construct this wormhole a CFT entangled in the “thermofield double” (TFD) state, a two-party purification of the 1 Introduction In the AdS/CFT correspondence, the entanglementthe structure geometry of and the topology boundary of CFT the encodes way. bulk We AdS consider space the [ dual ofin a a state Coulomb in phase the where the infrared modes are thermally entangled. We argue that this B Solving the linearized equations 6 A double wormhole between universes 7 Discussion A Perturbative equations of motion 5 Traversing the wormhole 4 Instability of the solution 3 Wormhole geometry in supergravity Contents 1 Introduction 2 Description in Super-Yang Mills JHEP11(2020)167 with (1.1) 0 ]. → 12 0 α , there will be ]. Supergravity Λ 14 in an approximate , L 13 Λ ]. To construct a wormhole ], but only in the case of con- 2 17 . extremal D3-branes in (9+1)D – i ] harmonic function solution as symmetry [ ∗ i N ) 15 10 N we have labeled the different regimes i ⊗ | i | 2 2 . These stacks of D3-branes are separated SU(2 / gauge symmetry has been partially broken i L held fixed decouples the AdS regions from ) 0 βE N . This system is dual to the eternal black hole, α − i e – 2 – i | i SU(2 X = i universe, we will study a state in the Coulomb branch of ] plus corrections due to the multicenter nature of the 2 TFD | . In this region, the perturbative corrections from the nonzero single ]. The procedure for constructing this geometry is illustrated 5 regions as one nears either stack. Taking the limit ], where the background, and therefore they linearly superpose. The lead- . In type IIB supergravity, the low energy effective theory dual S 9 5 7 5 , , )) S 8 6 × S 5 N ], corresponding to two stacks of × × 5 11 5 U( × sector of the Higgsed SYM) up to the Higgs scale ) , corresponding to the Higgs scale. In the limit that ) Λ N geometry outside the region containing the two stacks of branes, which splits N D3-branes [ 5 SYM theory [ SYM, this configuration corresponds to a multicenter solution sourced by two S . We will begin with a two-centered BPS [ (U( indicates the CPT conjugate of N SU( S i × 1 ∗ 5 i = 4 | = 4 Now we heat up the solution by entangling the degrees of freedom living on each brane N N temperature and from theto two throats the can vacuum simultaneously AdS being treated corrections as from linear the corrections symmetry left around throat the right are throat, monopolepole which defines effects corrections region from and II. the do However, left we not throat can break include as spherical multi- linearized corrections, and these will be dominant over wormhole solutions have been previously studiednecting e.g. two in different [ asymptotic spaces.in which In figure a differentregion coordinate I, the patch solution orthe is limit horizon, approximately in will the regions two-sided be IIis non-extremal used and close black to III, to brane. vacuum the AdS describe Far effects from the of non-extremality solution. are small In and the solution will find that thisdirections between gluing the must two introduce throatsten-dimensional in a spacetime order remains global to globally respect monodromy orientable.as flux that finite The conservation. inverts temperature complete However, breaks some solution the the is full tive spatial supersymmetry unstable, potential of for the the BPS scalar solution fields and turns that on break an the effec- deep bulk, the approximateAdS-Schwarzschild thermofield black double brane state [ willexterior be solution. dual to No thenonzero known two-sided temperature, solution planar so exists we forthese solve the corrections for in multicenter these different black coordinate corrections brane patches in geometry glues perturbation together at the theory. wormhole Matching solution. We the ratios of thethe five-sphere asymptotically coordinates flat to space, leaving a geometry which is(in asymptotically each AdS [ thermofield double state. This has the effect that in the IR of the field theory, i.e. the originally found in [ Minkowski space and controlled by aby parameter a distance an AdS into two smaller AdS between distant regions of a the down to to stacks of in figure where i.e., a wormhole between two asymptotically AdS universes [ thermal state on each factor: JHEP11(2020)167 and (1.2) µ k supergravity in four ]. A large body of = 2 20 . Figure inspired by the Λ N . 0 dλ < µν T ν – 3 – k µ k with a larger AdS radius. ∞ 5 ]. −∞ S Z 19 , 1 × ] 5 18 25 such that [ ] argued that this condition could be avoided for supersymmetric traversable worm- 17 λ ], meaning that there exists an infinite null geodesic with tangent 24 – 20 . Perturbative construction of an AdS wormhole solution in a single asymptotic space by Entanglement between disconnected non-interacting boundary theories gives rise to The authors of [ 1 Consequently, in order toof build stress-energy a in traversable the wormhole, bulk. there Several suggestions must for be introducing a thisholes negative negative connecting stress-energy two source asymptoticdimensions. AdS universes in the context of pure gauged ing a traversable wormhole(ANEC) requires [ that oneaffine violate parameter the averaged null energy condition negligible and the solution isV approximately approaches the that multicenter of BPS pure solution, AdS which in region wormholes where the boundariesrecent are work separated has by alsosignals causal been through horizons directed traversable towards [ finding wormholes mechanisms in that the can context create of and AdS/CFT. send In general, support- III simultaneously. These willbrane be geometry. linearized perturbations Thesethe of solutions the causal show finite horizons, that temperature butTherefore leading black the we multipole singularity find effects in thatregion this remain IV they wormhole small is is grow where near not in both of throats the the have interior non-perturbative AdS-Schwarzschild type. effects, towards but the Finally, their singularity. non-extremality is near-horizon limit as depicted in [ nonlinearities up to fourthleft order throat in breaking the the multipoleIII. spherical expansion. symmetry We around will This the also captures right be effects throat, able of which to the defines present region solutions which are valid in regions I, II and parts of Figure 1 taking the decoupling limitcaused by of the two thermofield stacks double of entanglement structure D3-branes below and the scale correcting for the finite temperature JHEP11(2020)167 ]. A 32 , 31 SYM generated = 4 This motivates us to 2 N we explain the pattern of 2 – 4 – ], which, while not an exact duality, has provided 36 – 33 ], although the presence of bulk fermions in the supergrav- 39 ]. 38 , 37 ], including the Casimir energy of bulk fields running in non-contractible cy- ], and nucleating and supporting wormholes via cosmic strings [ ], incorporating the perturbative gravitational back-reaction of bulk quantum 28 . Region I: perturbatively corrected black brane glued with inversion to the rest of the , 26 30 , , 27 Jackiw-Teitelboim gravity [ 25 29 The rest of this paper is organized as follows. In section These constructions use the fact that the eternal AdS-Schwarzschild wormhole is 2 We thank Simon Ross for discussions regarding this point. 2 ity spectrum implies thatdepending, e.g., other on mechanisms the mentioned final above sign may of cancellations also betweensymmetry be Casimir breaking a energies. in the possibility field theory and describe a particular entangled state in the IR. We will find that thesingle-boundary leading geometry classical preserve corrections this marginal comingdescribe non-traversability. from a the mechanism global by which structurethe our of presence single-boundary our of wormhole a may natural becomeby “double-trace” traversable the by type Wilsonian operator RG in flow the [ IR of and provided an experimentalpossible setting in by the lab which [ probing wormhole traversability maymarginally be non-traversable in the sensezons, that the so null arbitrarily energy small vanishes along negative the energy causal perturbations hori- render the wormhole traversable. fields [ cles [ particularly productive setting hasAdS been the correspondence betweenfurther the support SYK that model explicit boundary and couplings may render the bulk geometry traversable throat solution. Region IV:vacuum AdS the with (extremal) larger two-throat radius. solution. Region V: far from bothinclude throats, inserting explicithole double-trace [ couplings between the boundaries of the worm- Figure 2 geometry. Region II: linearized perturbations to vacuum AdS. Region III: perturbatively corrected JHEP11(2020)167 - p ∧ is p ] ...µ · 1 (2.2) (2.1) , ... µ µ ∧ F A we use p [ , σ i dt p 4 r ! + µ 2 · ] 123 j t µ D3-branes in a ∂ . . . g F , φ i 1 i N σ φ = = [ 1 φ · µ δ µ µ g i,j X p D are four adjoint Weyl dx D i ] a antisymmetric represen- φ ...σ ∧ 2 2 YM 1 µ λ 40 6 g σ D p ... − d i ] + ∧ gives rise to a vanishing com- X b ...ν µ i ¯ λ 1 . − , φ ν i dx a gauge field, φ g λ are the Clebsch-Gordon coefficients [ = 1 µ a ) − ¯ λ 1) ab i D N √ µαβγδ 1 − ! µ C 1 iab p d F ϕ ¯ σ ( ¯ C 1 a 5! symmetry to the ... = + ¯ λ SU(2 i 01 R p = ψ X a,b,i − − d – 5 – , by expanding F → ...ν YM µν Λ is an 1 1 g SU(4) ˜ ν F ] φ ) ν µν F ] + b F ∗ ,A we discuss a mechanism for rendering our wormhole ( 2 µ , λ π i 5 θ A [ φ i 16 [ we explain how to use our results to construct a double a (all other components will be zero throughout this paper). + λ + . The indices of all perturbative geometric quantities are raised 6 ) SYM in terms of component fields is [ 5 µ ab i µ µν A C dθ B ν F ν ∂ = 4 ∧ µν A X a,b,i − F N + ... ν representations of the ν We work in “mostly-plus” signature for Lorentzian metrics. The con- YM A 2 YM 1 ∧ g B 4 µ g µ 1 ]. 2 ∂ is the Levi-Civita symbol and + are six adjoint real scalars, and the A − n dθ ( 41 we solve for the metric and five-form of the wormhole solution in perturbation i = 5 1
2! spacetime dimensions is φ 3 ...µ ...θ tr 1 d µν 1 = µ θ F = ) F ν 0 At zero temperature, any diagonal configuration of B L + in the AdS/CFT correspondence map to the positions of D3-branes in ten-dimensional µ ( i In the dual gravityof theory, D3-branes this can is be equivalent superposedWe choose to at a any the vev location statement that insingle that will space transverse correspond an without coordinate simply arbitrary any by to force number distance separating between two them. stacks of φ flat space, while the off-diagonal elementsthe are branes excitations of [ open strings stretching between mutator in the potential for the scalars, so there is a large moduli space of stable vacua. where the covariant derivative onfermions, fields in the adjointthat couple representation, two tation. The diagonal elements of the vacuum expectation value (vev) of the adjoint scalars forms in where 2 Description in Super-YangThe Mills Lagrangian of dr The notation and combinatorial factorsA used in symmetrization ofwith indices the are background for metric. example commute with perturbative In variation. general The action this of means the Hodge index star on raising the components and of lowering does not traversable, and inwormhole section between two asymptotic universes. We concludeand in remarks the Discussion for with future comments directions of study. Conventions. vention for five-form components is that theory in typethe IIB DBI supergravity action and tois describe estimate controlled its the by global instability thethe structure. timescale same SYM of theory. ratio In the of Weby section also wormhole, scales adding show and that rotation. that show governs it that the is In it thermal unlikely section that effective the potential wormhole in could be stabilized In section JHEP11(2020)167 (2.4) (2.5) (2.3) limit, scales 2 1 limit so N ϕ has mass N 1 φ ]) are zero. µ i , which may be ) φ ,A 1 N ] b ϕ in the large ¯ ][ λ : µ . SU(2 ψ, N [ (1) a , in order to set the dimensions AO × ψ, A O ¯ ) λ , 0 . Therefore, the observable M α ) ab N 1 0 2 . Consequently, to make the ]) ,... N ¯ + 2[ 2 i C 0 − 2 πα √ , /N ] , φ 2 0 µ Λ a,b N 1 X (1 expectation in the large ϕ O = ][ i YM ψ, A (1) ! g ([ in the adjoint of O N,..., B 0 ψ, φ ] + , then the connected component of the two- 2 √ YM M b , so its connected two-point function scales M g Λ (1) + 2[ † + (1) B to appropriately normalize the gauge-invariant ψ, λ O 2 N, 1 – 6 – [ times and the vevs of all other ] O 0 for the “antihermiticized off-diagonal” piece of M i a where each block is ϕ √ N λ µ − . N √ ∂ (Λ ab are ψ 1 ] AO
ψ, φ C B of the classical observable µ C 1 ([ 0 M M as a parameter. We introduce M 2 N ϕ 0 diag i † A B ψ a,b 0 α πα √ X X ψ, A M M 2 [ Λ tr 1 πα 2 to canonically normalize its kinetic term, 2 YM YM 2 − YM g g = is repeated µ ] = N ig is required by dimensional analysis since the field = + + A 2 . In detail, we would like the classical value of the observable N M 2 s is strictly negative, which is required to give the correct signs ψ 2 ` given by . But we know from large-N index counting that if the expectation √ YM ) ψ ψ, M tr g [ 2 Λ ψ = 2 AO 0 N + → ( M α 0 µ L should be included in A The effective Lagrangian for fluctuations about this background is, rescaling = SYM does not include N i ∼ O 3 2 1 L √ ϕ with an arbitrary Hermitian matrix = 4 should be treated as a classical source. , since the components of tr ψ ) N 2 1 SYM. ψ ϕ N requires an overall normalization proportional to ( , after proper normalization, to have a finite correctly as a fiducial scale where we expect the SYM description to break down, anticipating the To understand this effective Lagrangian, it is instructive to expand the commutator of The factor of We introduce an explicit factor of tr 2 1 2 1 The = 4 O h 3 . Note that tr ψ φ φ M below. Armed with this knowledge we furtherof rewrite thecorrespondence commutators with in the the supergravity description effective to be discussed later. written in block form as We have labeled the final matrix where the vev dimension one in four spacetimefrom dimensions. open This string should theory, be sincetions understood it of as provides the a the open scale energy stringN coming cutoff endpoints moving such on that the thethe D-brane massless gauge world-volumes excita- field are described by point function of itstr quantum fluctuations should be properly normalized value a factor of tr that there is aas well-defined classical gravitational dual.as Next we observe thatvalue of tr a classical observable is taken to be where the eigenvalue classical observable tr with the background JHEP11(2020)167 2 (2.7) (2.6) ) gauge factors ) i AO ) N φ set by the N ( c i E SU( X SU( 0 2 ] µ Λ πα . The extra terms λ 2 being gauged, but it is ,A )) 1 ) + ϕ N N [ b AO U( ¯ λ SU(2 a × µ,AO , SYM theory, so in fact the ¯ λ ) A R ) i ab N are typical of those that appear ∗ 1 N N Λ i ¯ λ 1 C √ (U( ϕ ⊗ | clear, defining the ’t Hooft coupling S a,b X SU(2 L + 2 refer to each of the two Λ i λ i → | R √ factor should be dual to two AdS throats 2 ) / ) i 2 µ,AO N . N A + Λ βE and 0 factors of the gauge group. This is because, ] − 2 i L e ] where these terms vanish. ) SU(2 Λ SU( – 7 – c πα , φ b,AO λ N 2 43 1 . Therefore, at energies below the Higgs scale, the λ
r i d − , and 2.3 ⇒ 5 = τ i ( 2 S H = 2 r ∧ P → 0 / 0− Z 2 1 0 α α 2 ˜ x − = α 1 κ d Nτ ˜ t, = 2 → 2 4 ∧ i r | = = r H 1 1 = . In this limit the (nondimensionalized) harmonic function t ˜ , − x 2 is subleading, so spherical symmetry about the stack of branes . Q 0 2 ~ Q d Λ 1 L 1 r and 0) L r 0 ± ∧ , 0 ˜ t α ˜ r ~ | 0) , d i , ) 0 r → 0 , ∗ + , 0 2 4 0 , 1 L ˜ r , 0 where the horizons of the two stacks of branes are well-separated. Given 0 , (1 + 1 , 1 0 , the charge of a single stack is, by Stokes’ theorem, − 2 4 , Λ L L ]. By the BPS condition, the charge is equal to the number of branes = ( = = 1 2 45 is the wormhole horizon radius, to be introduced in subsequent sections. This F − . We introduce the parameter 2 0 ds )[ ~ 0 0 ~ 1 0 Λ = (Λ r r α 1 α It is convenient for subsequent sections to expand this solution close to the stack of The flux through the five-sphere jumps discontinuously when the radius of the five- 3.1 is valid in this limit. Furthermore, this expansion is valid around either stack provided the branes at the origin, becomes simply That is, all dependence on where these definitions, the full solution in region IV is Here nondimensionalization will bescale convenient in otherin regions the limit where it removes the length Lastly, we nondimensionalize coordinates by the rescaling We now take the decouplingamounts limit to fixing themetric, we Higgs rescale scale of ( The normalization comesin from ( the normalizationthe of tension of the a kinetic single extremal term brane, for the five-form displacement of the stacks of branes. sphere around one stacksmaller of than branes crosses through the other stack. When the radius is Here JHEP11(2020)167 (3.19) (3.20) (3.21) (3.22) (3.23) (3.16) (3.17) (3.18) . . i 5 5 dθ dθ , by which the 2 5 Λ ∧ ∧ Ω 4 4 d dθ dθ + i ∧ ∧ r 2 2 3 3 ˜ r ˜ r d . The charge is dθ dθ deep within a single throat 4 , ∧ ∧ 1 5 L 1 θ 2 θ 2 4 S ) ˜ r dθ dθ = 2 × ˜ r sin sin ( d ∧ 5 ∧ 1 2 2 4 ∞ 2 , θ θ 1 2 θ 1 1 ) ∧ L 0 4 3 ) dθ dθ α 0 sin sin ˜ sin x 4 4 1 + )( α 1 2 3 θ d θ 3 ( θ θ θ 4 θ 4 N 4 ∧ ) L 4 ∞ r sin sin 2 2 (2 π sin sin sin L sin 3 3 ˜ ) + x s 2 5 ˜ r 1 2 3 4 θ θ 2 4 (2 d d = Ω θ θ θ θ θ ˜ 2 2 x ~ 2 s πg – 12 – d ∧ g d ∧ H 2 1 3 sin sin sin sin sin sin + sin + ˜ x = 4 ˜ x 1 2 3 4 5 2 2 5 2 = 2 d θ θ θ θ θ θ d θ 2 θ ˜ ˜ t r ˜ 4 ∞ r with AdS length 3 3 d d ∧ ∧ Q L 5 ˜ t cos cos cos sin cos cos − 2 sin sin d ( S ˜ x r r r r r r 1 1 ) + d θ θ 2 × 4 ======4 ) ˜ 4 x ~ 4 ∧ ) 5 yields ˜ 3 4 5 6 1 2 r ˜ d r 1 r r r r r r 2 ˜ x ( / + 2( d 1 1 2 2 + sin + sin ± ˜ t ∧ − d − simplifies to ˜ t H 1 d 1 − 3 ( symmetry is unbroken. , taking the decoupling limit, rescaling, and nondimensionalizing, the H 3 ˜ 2 r 4 ˜ ) r 2 ˜ h r L ˜ r 4 ∞ N 4 2 ∞ 2 L L = 2 L L SU(2 = = 4 = = 4 4 ∞ 2 2 L F F 2 2 ds ds 0 0 1 0 1 0 α α 1 1 α α from the flux quantizationthey condition. are At at infinity, the the origin,where flux as the sees expected. both This stacks region of corresponds branes to as the if UV in the field theory This solution is vacuum AdS leading to harmonic function Writing solution in region V is In this form, theare linearized apparent. corrections These to corrections vacuum will AdS be useful3.2 in subsequent sections. Region V:In asymptotics the region far from both stacks of branes, we take the limit and series expanding radial coordinate is defined appropriately. Defining the hyperspherical coordinates JHEP11(2020)167 (3.31) (3.27) (3.28) (3.29) (3.30) (3.24) (3.25) (3.26) ) with . . ] i 3.25 , rescaling 5 5 0 46 , dθ dθ /α 0 of an extremal ∧ ∧ 11 ) r , 4 4 2 5 τ 10 Ω dθ dθ one approaches the d . and 2 ∧ ∧ 8 0 0 r r 0 3 3 1 4 . The solution preserves . → dθ dθ /α ) + i 4 0 + , 4 0 r r ∧ ∧ r r ( 1 L 2 3.1 2 2 ) − 2 /f ) dθ dθ 2 0 1 + 2 5 dH α ∧ ∧ unchanged. Nondimensionalizing dr ( Ω s 1 1 ( ∧ ) d 2 4 times the tension N 3 r . / ) dθ dθ s 0 ( 1 + 4 of section 4 4 ) α N f dx θ θ ) r πg ) ( 0 Λ ( 4 4 ∧ 1 α 4 ˜ ) r (4 keeping fixed ( sin sin 2 H L π dr 2 4 3 r 3 4 0 ) ˜ − r i θ θ (2 dx ∧ L d π r 2 + s 2 2 ) + 3 (1 ∧ g 2 → (2 4 0 2 2 2 s 1 r ˜ r sin sin 0 – 13 – dx ˜ g r d~x 1 2 2 2 α 2 d = dx ∧ + θ θ − + . As the horizon radius 2 3 2 3 ∧ ∧ 4 4 0 F = 2 ˜ r r x ~ 3 = ∗ dx d dt ˜ sin dt sin x Q 5 4 2 ) ) − . The resulting solution has the same form as ( d 1 1 S ∧ r r L ∗ r θ θ Z 0 ( 1 ∧ 4 4 + ˜ f α 2 2 dx 1 κ 2 ⇒ − ˜ x ˜ t sin 2 ) = 1 ( (1 + = → d ∧ d r 2 4 4 4 0 ( / r + sin = ∧ r L 1 L dt f 4 1 1 − 2 ˜ r Nτ Q ) ˜ x ) r + 4 and r d 4 1 + ( − isometries induced by the brane locations. Following the conventions = ( and 0 L ∧ 1 r H s 4 0 H Q 4 4 ˜ t 5 r L α d 2 r = = 3 ˜ r ˜ 2 r ) it can be written as → SO(6) F 4 h − 4 0 4 ds r 0 L × 2 r L , no overall scaling on the five-form, and , L 3.12 1) 4 2 ) = 1 + , 1 + r = = 4 L /r 0 ( 4 2 s α F ], the metric and five-form of the asymptotically flat solution are [ H L 2 ds We now take the decoupling limit = 0 SO(3 1 0 47 → = α 1 α F 2 This is the backgrounddropped solution out in of region thedecoupling I. five-form limit Note after is that the simply the decoupling limit. finite temperature Consequently, factor the has charge in the L H following ( The charge remains equalbrane, to so the number of branes Consequently, the charge is with extremal limit of the braneas solution. The five-form can be written explicitly in coordinates of [ In region I, the solution isrections coming the from geometry the of second the throat two-sideddescribe in black the the brane geometry full with close geometry. perturbative to cor- In onethe this of location region, the of the two solution the stacks will with stack,the the that origin is, of in coordinates the placed limit at 3.3 Region I: black brane JHEP11(2020)167 . , ) 4 0 at 2 r / ) 1 1 , and − (3.33) (3.34) (3.35) (3.32) and ( r, θ δH ( + δH 0 δH ∼ O H ˜ , ) coming from r = , 3.30 H ) , so the leading nonlin- ) ). In this case the 1 8 ) both from the throat ! r ), ( − 2 5 5 ( r, θ ! 0 3.35 Ω ) + 2) ( with S 2 (Λ 1 d H 3.29 θ 2 × O d~x r δH 5 δH 6 + . + + − Λ 2 )) # 1 ) in 1 3.4 4 4 0 dt θ term in ( (3 cos(2 r r # 9 1 2 3.7 4 4 0 4 r − r r ) 4 . Λ r 1 + 2 L / ( " − ) 4 2 and therefore we can keep the multipole 0 0 2 1 L α 8 H + " ( − dr 7 ) ) + 2 cos(3 − N d Λ
1 1 Λ s θ θ
∧ . Since both of these corrections can be treated ∼ ) ) πg 3 1 2 – 14 – ) 5 2 ) ) 1 cos( r ) Λ dx ( ( . Note that the leading nonlinearity from the two r, θ r r = 4 0 4 ( ( 4 (3 cos( O ∧ δH r, θ 4 0 3 ( ( H L 2 /r r L 4 H δH 4 4 0 r dx δH L 1 2 + 4 1 2 ∧ in the linearized regime. 4 4 1 ) + L Λ − ) in 1 + 7 dx 1 − , one defined by expanding ( 2 5 / ∧ ) = 3.29 2 (Λ 1 1 / S ) 1 O dt r ) × − ( r, θ ) ∗ 0 5 ( r ( H and 0 δH + 4 4 H r L ) up to . = = (1 + ) we will describe the perturbative corrections to ( 8 2 F 3.35 ) = ( 2 r ds 0 3.5 O 0 1 ( α 1 0 α H We define region II as the patch where spherical symmetry around the throat is ap- After nondimensionalizing, the multipole expansion is controlled by powers of 9 proximately unbroken and hencemonopole we order. can stop This in the means multipole keeping expansion only of the earity is at This solves the linearizedperturbations equations of of AdS motions simplythe because other it by is expanding thecentered ( sum harmonic of function comes two at linear expansion ( and the Hodge star must be applied so that in the result we linearize both in where and from the wormholewhere the can background is be empty AdS linearized.function and and corrections the to These blackening this factor coming regions are as both are from linearized the and defined harmonic the by equationscan of be motion written are simply linear as in the the linear superposition perturbations, the of full the solution two, 3.4 Regions II-III: linearizedRegions regime II and IIIwithin are a the single intermediate throat regimes such far that the from corrections the to horizon vacuum and AdS sufficiently deep In section the second throat, though we firstthat describe gives the rise general to structure of these the perturbative multipole corrections expansion in section This leads to the extremal quantization condition for the black brane in AdS, JHEP11(2020)167 , i 5 . The (3.38) (3.39) (3.40) (3.41) (3.42) (3.36) (3.37) dθ . . B ∧ i . In terms 5 4 r = 0 ˜ r dθ dθ 0 ˜ r ∧ 00 ΩΩ ∧ δg 4 3 δg 1) ) dθ dθ ˜ r − 2 5 ∧ ∧ 4 Ω − 3 r 2 d 5 r dθ dθ . 4 2(2˜ ) ∧ ∧ ˜ 2 r − 2 # 1 + 5(˜ = 0 = 0 = 0 ˜ t = 0 2 5 ( 0 ˜ t ˜ t ˜ t 00 dθ ˜ t ˜ i t dθ ˜ 0 ˜ t Ω 00 00 ˜ t i ˜ 2 4 δg 0 ΩΩ ΩΩ 2 d ∧ θ δg ˜ x ~ ) δg ˜ δg δg x ~ 1 + ) 1 d ) ) δg δg ˜ d r 1) ˜ ˜ r r ) ˜ ΩΩ r sin ˜ t i dθ + 1) ˜ 0 ˜ t i 4 ˜ d − 4 3 + − − − ) = 0 + 5 + 5 ) 4 δg θ symmetry θ 5 ˜ ˜ 2 r δg r 2 r 4 ∧ δg 5 5 ˜ ˜ r r r 2 4 ˜ 0 ˜ ˜ r r r d r r r ΩΩ 3 ( r (˜ (˜ d sin ˜ x ∧ δg δg δg sin + (˜ 3 d − + 3(˜ − + (˜ − 3 2 (1 + 2 4 θ i ˜ + 6˜ i i 00 − − 1 i ˜ ˜ ˜ ˜ 2 1 x ˜ 0 0 θ ∧ r i i SO(6) ˜ ˜ 2 i ˜ ˜ ˜ t t r − d 0 i 0 ˜ 3 ˜ ˜ t t ) + (1 + 0 0 2 ΩΩ ΩΩ δg δg δg ˜ × 2 r b + ) ˜ x ∧ sin ˜ + ˜ – 15 – r ( δg r δg δg ˜ δg δg r d sin 1) 1) 3 4 2 2 2 1) ) r δg ˜ t 1 θ 1) + ˜ ˜ − − , x ∧ − ˜ 2 r − − θ d 3 i i d ˜ ˜ 2 1 5 ) 0 16˜ i i + 1) ˜ ˜ 4 + 1) − ˜ 4 4 t ( ˜ t r ˜ x ˜ t ∧ r r 4 d 4 4 sin δg δg − d (1 + r 1 r r SO(3 3 3 1 δg ) ˜ x 4 ∧ θ 1 + 4(2˜ 4 5(˜ 1 + 3(˜ 1 )) sin 00 ˜ ), i.e. the non-extremal black brane. In this subsection we ΩΩ d r ˜ − − ˜ r t 4 r 0 d + 3(˜ − + 3(3˜ b 2 + 5(˜ − (˜ 2 ∧ δg a − 1 ˜ ˜ ˜ ˜ t t t − 2 b r (1 + 0 ˜ ΩΩ r ˜ ˜ r r ) for the perturbations to the metric and the five-form in the ˜ ˜ ˜ ˜ t t t t 2 d 0 0 3.30 ˜ ˜ 4 r r 4 d 1 ) − δg ) (1 δg δg δg + + sin 4 ˜ r ˜ 3.6 1 2 r 2 δg δg )) 3 3 3 ˜ r a − ˜ r r r r r 2 ( (˜ 1) 4 2( − + 1) a ˜ r + (1 + + − + 1) − 1 ) and ( 4 + 8˜ + 8˜ + 8˜ − r 4 4 1 i i i ˜ ˜ ˜ ˜ r i i i 1 ˜ ˜ ˜ r r ) and ( 2 (˜ (1 + ˜ r 2 − 3.29 δg δg δg 3 3 ˜ r 3 3 3 3.5 ˜ − + (˜ + 5(˜ r ˜ r − r r r h h − 0 ΩΩ 4 " 4 2 2 L L δg L L + 24˜ + 24˜ + 24˜ 4 a a a = 4 = = = 4 r 3 3 3 5˜ 2 2 r r r F F 2 − ds 2 ds 0 16˜ 16˜ 16˜ 0 0 1 0 1 1 α 1 α 1 α α − − − The geometric equations of motion are: where the metric perturbationsof are the all perturbations, functions the only Maxwell of equations the and radial self-duality coordinate constraint reduce simply to contribution of the farperturbations consistent throat with down the the near throat. We begin with an ansatz for the 3.5 Joint solutionNow, in we regions solve I-II: ( monopolebackgrounds of contribution ( deal with the case when spherical symmetry is intact, that is, we solve for the monopole In this regime,procedure the involves equations several ofconsequence undetermined motion of constants a can and residual be diffeomorphism an freedom. solved undetermined by function hand; as see a appendix solution explicitly reads in nondimensionalized coordinates JHEP11(2020)167 ) r . (˜ a 5 (3.47) (3.45) (3.46) (3.43) (3.44) dθ ∧ 4 . 0 dθ , r 1 ∧ L Λ − 3 ! = dθ 4 4 0 0 L the non-perturbative Λ r ∧ 2 2 5 4 + Λ Ω 2 dθ 4 / 4 d ˜ x ~ . This rescaling puts the ) in the linearized regime, r 1 , L ∧ 2 5 r d 4 1
Ω L Λ ) ! 4 ˜ d r ˜ d 2 r 3.36 ) dθ 2 d ˜ 4 r = ( ··· r ∧ θ d~x and the perturbative corrections 1 2 0 ∧ ( 3 + + r 1 2 3 + 1 sin 1 ˜ , dx x (1) 2 θ − 3 1 + d − ~x 5 ∧ # ). This ansatz turns out to be correct θ dt L Λ 1 ∧ 4 4 0 Λ 2 2 # cos r r ∼ O 2 + 4 4 0 2 r = ˜ dx x 3.18 r r sin ˜ r r − d 0 2 4 ∧ 1 ~x − θ + ˜ ) ∧ + 4 1 " ˜ r 3 2 , 1 1 ˜ 4 t t 2 " ( – 16 – ˜ 1 dx x d L Λ Λ sin 1 2 ) and ( d dr − ∧ 1 4 + ∧ θ
) = 2 ˜ dt t ˜ 4 2 r / 4 3.17 0 1 + ) 1 / 1 d r outside the horizon at all times. We solve the equations t 1 ( ∗ sin 1 − ) 1 2 4 4 4 4 2 4 ) but with a rescaled horizon radius 1 ! ˜ r L Λ − 4 L 4 4 2 (1 + 2 ˜ r > + 1 L Λ ˜ − r + 4 4 0 = ) can be obtained by linearizing in 3.25 4 d r Λ 4 ) 4 + (1 ˜ r r 1 + H L 2 4 4 4 ˜ r 1 3.43 ˜ r r L 1 + 2( + + + −
s − gets large, this solution matches onto ( 1 1 = = ˜ r 2 2 3 F ˜ r ˜ r 2 ds 0 − 0 1 ) in the form ( 4 α 1 α 2 L L 3.45 . When ) = = 4 . The full perturbative solution in region I is 4 ) 2 r F ( (˜ 2 ds b The physical interpretation of the leading monopole contribution from the presence of 0 O 0 1 α 1 α It is interesting to ask ifthe we wormhole. can capture The the above leading solutions effectthe contain of the other spherical monopole symmetry throat. breaking contribution on At from the next presence order, of there is a dipole contribution from the harmonic function is easy to see thatflat this space, solution is in just rescaledsolution a ( single coordinates non-extremal black brane in asymptotically 3.6 Joint solution in regions I-II-III: dipole contribution which is obtained bythe truncating nonlinear the dependence multipole on expansion the at harmonic monopole functions, order as but well keeping as the blackening factor. It are region II. the other throat is tobe create seen a by small noting region thatsolution of ( (in flat dimensionful space coordinates): around the black brane. This can In this regime, we are close to the horizon, so of motion by the ansatz thatof the the perturbations two-throat will be solution equalprovided given to in that the ( near-horizon we corrections includeand an additional contribution to the five-form perturbations Note that we are in the region JHEP11(2020)167 . 5 F r S H → in the (3.53) (3.54) (3.55) (3.48) (3.49) (3.50) (3.51) (3.52) . The ) ) g and 3 4 ( H g Γ πx ) 2 H → ∞ 1 4 ) √ r 2 5 2 − ( ( , g Ω Γ h d 2 x r ∼ ; are proportional to 5 2 , so that the location ) ; ) + ), as well as the only 5 4 r , , ( 1 4 3.4 ) = 0 = 0 → ∞ , 0 /f . Instead, the perturbation . r 2 1 x 5 0 ( ··· ( F ··· g dr 2 ...θ Q h ( + → → ∞ 1 + ) 2 ) = 0 / 9 4 ) 1 rθ r r 1 1 00 g . 3 4 ) θ ( θ ) ( ) Γ 1 ) Γ r rh dF 5 4 ) ( 5 ) as 5 , ( 1 4 g ( + r, θ ) cos Λ 1 ) cos Λ Γ h component of the curvature equation of ( − r 0 g 2 r 4 g − ( 4 ( 0 F / ( 1 / h 3.47 3 r 1 g F H 3Γ x 4 rθ x h h 8 dH − r )(5 4 0 2 4 ∧ ) + + r r 2 + 4 + 4 3 2 – 17 – − 4 4 x d~x 1 1 dx 4 ; Λ Λ r 1 2 . Similarly, we take a general ansatz for + ∧ ) = gives 4 + + r 4 0 2 2 − ( r 5 ; + ( 2Λ 4 4 dt 1 1 F 1 4 Λ dx r r g ) ) due to the cancellation of the branch cuts starting at h / , h r − ∧ 1 3 ( , ! 4 4 r . Requiring the 1 f 5 4 4 4 = 1 0 5 L L = 1 + r r − − dx ( x − = =
2 ( B ∧ / 1 → ∞ g 1 F 0 Q F r r 0 − dt H 2 H ) ) r and 1 ∗ = as 4 r r r, θ ) = ) = /r ( r x 4 0 g ∼ (1 + ( ( r in the five-form are allowed to be different, g H B F Q h − F h = = ) to vanish at order H ∼ 2 F g 3.4 2 ) = 1 ds 0 h 0 1 r α 1 ( that are separately present in the two hypergeometric functions. . It turns out that the latter condition translates into α 0 f It would be interesting to further analyze this solution. It seems like it is not possible to = 1 decays towards the horizon,the i.e. other it throat is appears to decayingbecomes have in a large tortoise significant again effect coordinates. on as the So wesingularity interior, the approach where inside the presence the this perturbation of singularity, wormhole since does is not therefore factorize, not so of the geometry the is AdS-Schwarzschild really type. a The full ten-dimensional wormhole. and is analytic at x have a perturbation that decays towards the singularity of the horizon is not affected by the perturbation. The solution is then This is a second order equationThe with other two initial is conditions. fixed Oner is by < fixed r requiring by the solution to stay real in the interior of the wormhole, Imposing this, itnon-vanishing turns component out all ofthe the Maxwell’s equation remaining equation, components of ( that is, motion ( with whereby both must be asymptotically equal to ( blackening factor exact. We takemetric a and general ansatz where the harmonic functions We look for a solution including dipole effects in all three regions I-III, that is, we keep the JHEP11(2020)167 , where The gluing 4(a) 10 near each throat. N (b) the two throats and the 5 . Our setup is analogous to the circuital law N 2 – 18 – units of flux near infinity and N 2 (a) . . Spacetime wormholes with orientable (a) and non-orientable (b) Cauchy slices. In the 4(b) The way that flux conservation works in our supergravity wormhole is very similar, In the geometry that we have described, both AdS throats have a positive net five-form We thank Juan Maldacena for raising this point. 10 although there are some technicallives differences in because ten the fluxorientable. dimensions. comes from We a In five-form illustrate particular, and on the a Cauchy spacetime slices diagram of in the figure wormhole will remain a loop purely byby modification cutting of open the the geometry,invert wormhole, the without and angular adding gluing coordinate. sources. itis This back This a results to non-orientable in is the surface a achieved ambient that sourcelesson space supports “Klein-bottle” figure while a wormhole, nonzero twisting which circuital to flux at infinity. This is shown flux far away from thefor throats a rather magnetic than field inthere are two dimensions, no sources where for theagain, the closed curl. the line wormhole This integral without is must sources illustratedenclosing be has for both conserved a zero if 2D throats. line wormhole integral in for However, figure the one magnetic may field support on a a non-vanishing loop line integral on such twisting during gluing. flux towards infinity, so that there are This presents a puzzle: ifthat there the are flux no should sources thread in through the the wormhole, wormhole flux and conservation close demands in the outside, giving zero net 3.7 Global structure andHere flux we discuss conservation how regionstwo I-II-III throats (the in wormhole) aprocedure should single leads be spacetime) to glued some so tothe interesting that regions global spacetime IV-V the properties (the has five of a form the moduli flux wormhole. space is We will coming conserved. show from that the freedom to add a certain amount of Figure 4 orientable wormhole, conservation of circuitalsecond flux throat demands relative that tothroats the the points flux the first reverses same direction throat, direction. at while the in the non-orientable wormhole the flux at both JHEP11(2020)167 In 11 which . i g i L x . dθ R = dr i = dθ , dr L differential points out we refer to the term 1 dr and dθ = for the electric part and g at the left throat do not ) dt i L dr θ = , . The L/R i g 1 ( dt dθ dr dx and − ∧ L = magnetic part r 3 g of the five-form we refer to the term i , and and three spatial coordinates at the right throat. This is because the dx dx g dt t i R , ∧ , θ g 2 g dr , dt R dx r = ∧ and by the – 19 – 1 g electric part dt dr dx ∧ near each throat, so the two five-spheres are centered ∧ 3 5 g in the IR, which generates upwards time evolution on S dx dt R × ∧ H 5 2 + dx L ∧ H 1 . Now, both electric and magnetic parts of the five-form are oriented 5 dx S ∧ Vol dt d for the magnetic part. At the left side of the wormhole, we choose the exterior ) . Spacetime diagram of gluing the wormhole to the throat regions. In the middle we have L/R ( 5 S The gluing of the rest of the coordinate directions at the right interface is determined For the following discussion, by the The equalities that describe the “gluing” between the left/right throats and left/right exterior wedges 11 Vol by requiring the five form to be continuous.should be Consider understood to startingpatches be with where specifying they the the are transition five-form defined. functions in on the the wormhole manifold in the coordinate the right wedge, theSchwarzschild radial time coordinate coordinate runs points downward, outwardsHowever, in towards we the the opposite would right direction like throat,both as to and the sides. left glue the wedge. the ThisHamiltonian is throats looks what to like we the expectboth wormhole sides. from so Therefore, the that in field time the theory, points right since wedge, up after we on must Higgsing take the SYM in the same direction inexpression the ambient in spacetime, local so deep coordinates: d in each throat, both haveSchwarzschild the coordinates same in the leftof wedge of local the coordinates Penrose of diagram to the match left the throat: direction geometry only fibers into AdS at different points. proportional to proportional to wormhole before we glue themThe together, left and and the right orientation throats of shareare the a globally coordinate time coordinate differentials. defined withHowever, the the same natural orientation radialcoincide in and with the angular the ambient corresponding coordinates coordinates space outside the throats. Figure 5 the Penrose diagram ofshow the the orientation eternal of black thefrom brane, coordinate the differentials and plane the ofright sides the throat. represent figure The the in three differentials throat the must regions. left always throat form We the and same in right the handed wormhole, system. but it points inwards in the JHEP11(2020)167 , 0 t we = 5 acting g t direction. ≡ symmetry (6) are flipped 1 O are oriented R θ 1 R t (3) 5 of the five form dθ O the electric part 5 dθ ,..., and coordinates (parallel both ∧ R 4 i = 2 x dr dθ i ∧ , i approximately factorizes in 3 . 1 5 dθ subgroup of the dθ S − ∧ coordinates, which is an element 2 = 6 (5) and in the right Schwarzschild wedge. dθ O R 3 1 , ∧ θ 2 1 R − , det , . . . , r 2 dθ , the direction in which the throats are = r 1 = 1 0 r ) i 1 , θ i ( – 20 – fixes dx are flipped in Schwarzschild coordinates relative to and (5) dt t O − actually keeps its orientation throughout the wormhole and = . This way, in the electric part we make up for the sign by 0 the magnetic part ) 0 dr t 1 dr θ ∧ ( and 3 − coordinate by an inversion at the gluing surface. and perform the gluing by twisting the R dx 1 = θ dr ∧ 1 R ∧ 4(b) θ 2 3 twisting is an inversion of the of this unbroken symmetry with dx 1 dx θ and ∧ ∧ i (6) 1 2 x in the gluing function, while in the magnetic part we invert the O 13 − dx dx i . coordinates, where the . ⊂ x 5 ∧ subspace is unbroken by the configuration of two throats, so nothing will depend ∧ i = So a direct gluing would lead to a discontinuous five-form. However, we can follow 1 i r i g x (5) dt x 12 4(b) dx O Now we briefly discuss the moduli space of solutions. Note that the only restrictions Note that this twisted gluing results in a smooth geometry, because the In terms of transition functions between the right wedge and the right throat, this On the other hand, in the right throat, the basis differentials ∧ What we mean here is that tracking the global five-form from the leftThe throat total to spacetime the is right also throat orientable on since the there exists a globally defined “upwards” time, which in the ∼ ≡ ∈ R 12 13 5 i R outside results in five-formsas that in point figure in opposite directions in the leftwormhole and region right is throats just if Kruskal we time. draw them on these coordinatesnot even written in down. theon fully the Similarly, nonlinear there time-dependentseparated. is solution The that an we unbroken R have This makes the completethis five twist form is positive, continuous. thefigure Note Cauchy slice that remains since orientable, as the opposed total to determinant the of 2dof example the of In this properly alignedmagnetic basis, components there of the is five an formthe compared explicit five to sign form the difference right components throat.x in continuous, both Then we in glue the order with electric toinverting make and the transition functions the idea from figure to the brane) and the works as follows.introducing First we new align coordinates the basis of coordinate differentials on the two sides by relative to the left throat.of these However, coordinate the differentials solution in fordt both the throats. five-form So looks the theappear orientation same of reversed in compared terms tothem. the right wedge of the wormhole for the purpose of gluing wedge of the wormhole, both the left exterior wedge.F Since both of theseregion. are The flipped, same the appliesthe electric for wormhole. part the of magnetic the part, five-form, since the exterior wedge, where we mustshow glue the the orientation geometry of back theThe to the coordinate rest right differentials of throat. that the Onthe are coordinate figure same changing differentials, during way this throughoutthe process. the same figure. solution In in the both Schwarzschild the coordinates, left the and five-form right has wedges. But note that in the right exterior left throat and continuing into the left exterior wedge and across the wormhole to the right JHEP11(2020)167 3 dx (4.1) ∧ (3.56) 2 dx correspond ∧ denotes the i 1 ∗ x dx H is the pullback of the . , where 4 ∗ ab is the tension of the stack G ξ C T +1 p SYM factor. The inversion on H ⊗ H d Z . = 4 µ branes at nonzero temperature. We + N N ab SO(5) 2 . Therefore, from the field theory point of G × C det – 21 – − p , one needs to turn the bra vectors into ket vectors ISO(3) charges in the field theory, so it is natural to think ξ × R +1 p R d H ⊗ H , which is an element of 2 Z / 1 T ρ − = S denotes the group of (orientation-preserving) rotations and translations of is the charge density coupling to the five-form, are coordinates on the world-volume of a brane, . This gives a freedom in picking the group element with which we twist the µ i 5 ξ S ]: corresponds to inverting the coordinates. ISO(3) 52 branes collide and form a single stack of Vol ], one may introduce a constant time shift in the identification of Schwarzschild time 5 i d Finally, let us comment on the field theory interpretation of the gluing twisted by S x 51 N form [ Here the of branes, of can compute the timegoverning scale the of dynamics this of instability onefull by action stack examining is of the the branes tree-level Dirac-Born-Infeldbranes in effective (DBI) the and action action background describing their of the coupling the geometric other to dynamics stack. of open the The strings, plus the coupling of the branes to the five- 4.1 Instability timescale The wormhole solution we have describedmass is is not an larger extremal than (BPS)gravitational its solution of charge. (NS-NS) type Consequently, IIB: force it its the is suffers underlying from larger branes, an than instability: so the that the repulsive attractive at five-form late times (R-R) the force wormhole between disappears as the two stacks with an anti-unitary symmetry.symmetry in There any is quantum one field theory, such which anti-unitary is transformation CPT. 4 that is a Instability of the solution of CPT. This islooks natural like for the the thermofield followingthe double reason. state. thermal As density The discussed TFD matrix before,dual state Hilbert the is space, state defined where in bra from thelives vectors the on live. IR square a In root doubled order Hilbert of to space define the thermofield double, which inversions. The inversion of Schwarzschildto time and time the reversal parallel and coordinates the parity (TP) inabout the it right as IR theview, action the of right charge IR conjugation field theory factor is “glued back” to the UV field theory by an action between constant time shifts betweensingle left boundary and wormholes right). is Therefore, the total moduli space of Here, the remains a symmetry of theand solution, (ii) it reversesgluing, the orientation resulting of in both ain [ moduli space.with In the addition global to time the in spatial the twisting, throat, as which pointed gives out an extra real parameter (the difference on the spatial twisting at the right gluing surface are that (i) it is from the subgroup which JHEP11(2020)167 (4.5) (4.2) (4.3) (4.4) from the r . 4 + Λ . 4 0 3 r q dx Q ∧ 2 + , ) ) dx 4 4 0 ii r ∧ g 1 − + Λ = 4 4 , dx ii r r i ( ( ∧ x 4 4 as the effective mass and charge of the Λ L dt = 2 1 i ˙ r − ,G is the pullback of the potential for the five- µV − ! rr 4 4 4 4 4 0 g = C r r L Λ 2 – 22 – r directions which have comparable size, we cannot t, ξ Q − + ), far from both the horizon of the thermal branes 5 + ˙ 1 = 4 4 S r 00 L 0 s and g 3.46 ξ 2
Λ = 4 4 0 ), ( r TV Λ M is the (regularized) brane world-volume, 00 i − G = ξ 3.45 3 1 + 4 d M s R to the brane, and 4 r = + Λ = 4 4 µν r g V C dt . The pullback of the potential to a brane sitting at distance 3 Z , 2 4 2 , L Λ = = 1 i S The extremal probe brane starts at a point in the geometry which can be approximated In order to study the dynamics of one stack of branes as a probe, the backreaction of probe brane is, defining probe brane where In spherical coordinates, the probeof brane moves the only metric in the is radial given direction, by so the pullback and all other components are zero. Consequently, the effective action experienced by the thermal stack is then zations to work in “staticspacetime gauge” coordinates. in which the world-volume coordinates are parallel to the where wormhole is long-lived. by the flat-space region ofand the ( AdS throatfour-potential, of we the use extremal spacetime Lorentz branes. transformations and To world-volume compute reparameteri- the pullback of the metric and one stack slightly andas leave being the separated other from extremal. thethan extremal We a stack. may brane Such think located a about deep branethe in the experiences thermal the a extremal stack. AdS higher probe throat We acceleration will ofchosen show the so that extremal as the stack, to where temperature make it and this is separation lower further of bound from the on branes the can instability be timescale arbitrarily high, i.e. the in the full two-centerthroat. geometry, as Moreover, each each of stack the of stacksthey have are branes a separated field sources that in its is thetreat own the asymptotic the size independent of two AdS the stacks AdS as radius,obtain point-like and a objects since interacting lower via bound weakextremal on fields. probe the brane Nevertheless, timescale located we of may halfway between the the instability two by stacks of considering branes, the where motion we of heat up an form. Evaluating the DBI actionaction using at the tree classical level metric and whereand five-form we the gives have gauge the taken field effective the on backgrounds the for brane the to antisymmetric bethe two-form zero. probe on the background geometry should be negligible. However, this is not the case background metric JHEP11(2020)167 . is as 2 ) 0 (4.9) (4.6) (4.7) (4.8) 4 r # πL (4.10) 2 − r (ˆ , and all = 5 ˆ t O d β M dθ , , but keeping = 0 ! 2 r 1 ˆ r , the Newtonian Q 4 . 1 component of the + , where + 1 ! 2 − 3 00 4 4 − ˆ ˆ t r ... ˆ G r ˆ t ˆ r small. In terms of the d d β 2 + ˆ r/d Λ d ∼ ˆ / t ˆ r 4 1 0 d d 3 r 4 Λ 0 − 1 − r 2 2 = . 4 4 L 1 ˆ r 5 ˆ r θ − " 5 ∼ θ and take 6 4 4 g t 8 ˆ r 2 5 ˙ , θ +1) + − 4 5 6 + r ˆ r 2 1 . In this case, the (ˆ 2 ) 2 1 2 rr t s g ( + − to find at lowest order 2 5 − − 2 r θ 4 = 14 4 – 23 – 4 , + ˙ ˆ r ) ˆ t ˆ r ˆ r 2 t that we imagine to be order one, and dimensionless ˆ t directions. We will not attempt to perturbatively d 2 d ( 00 d − d r i Λ g 1 + r 1 1 2 r/ p = by v u u t
= ˆ 00 t 2 − ˆ d r G 4 4 + 1 π/ ˆ r Z 4 at the scale of the dynamics. Therefore, we introduce the ˆ r = 0 1+ r ˆ t 4 in this expansion, coming from balancing the orders of the leading potential 2 2 d p θ ) (Λ) Λ L 3 Z = ( M 4 2 O Λ ∼ O 4 L = + 1 ... ˆ is 0 r 4 r r M ˆ t S as before. We may now expand in ˆ r = ˆ t = Λ ˆ r/d d 1 as before. Since we take the probe to be extremal, we set / d θ S 0 t 2 Z r 0 L r 4 4 = Λ L = 0 r M ˆ t fixed since The speed = Λ 14 S and kinetic terms and/or from the equation of motion. pullback of the metric is Using the same coordinates asconstrained the to previous the section, equator the is DBI action for the extremal probe around each other inconstruct the such transverse a spinning solutionthe in DBI the analysis present above work.non-extremal for black On branes. the the We other case parameterize hand, thethe where brane we equator trajectory the can in extremal repeat a circular probe orbit brane around rotates around the cold or the stacksarbitrarily of long-lived. branes to be widely separated,4.2 the wormhole solution Stabilizing can withOne be rotation may made wonder whether our wormhole can be stabilized by making the throats spin and so the accelerationoriginal can time be coordinate, made thethe small instability inverse temperature timescale by of is making the black brane. That is, taking the thermal branes to be very This is motionexpected in for a a charged flat object ofwith space codimension what six attractive would in have Coulomb been ten spacetimedynamics found potential dimensions, from are in which the simply agreement tree-level scales closed string as exchange. The resulting where slow-moving approximation for the probe, r/ dimensionless radial coordinate time dimensionful quantities then scale out in front of the action, This expression should be expanded at large radius compared to the horizon JHEP11(2020)167 (4.17) (4.15) (4.16) (4.11) (4.12) (4.13) (4.14) . , = 0 4 ! , the inverse of /β ... 3 + 2 + ! ∼ 4 1 − ω − 4 4 1 2 ˆ r − . leads to stable or unstable or- 6 ) 8 4 3 ! when the rotation speed is / 2 2 1 + c ) ˆ ` r + , ˆ r 2 ω 2 8 2ˆ , 5 . + 6 r √ 2 . ( 3 c 4 2ˆ + c ω r r 3 1 c 2 r 2 2 r 4ˆ − 4 2ˆ 0 3 r = 2ˆ L + 3) r 2 5 √ 2 ˆ t − c 4 + c √ 2 ˆ r d r √ dθ 4 rω 1 4 (ˆ – 24 – ˆ r = 4
= ˆ = ˆ r 2 = ˆ 6 − ω ˆ r ˆ r 2 ω 8 ˆ ˆ ω t 2 1 2 d d 2 ) for circular orbit frequencies directly without series + ) = ω 6 r 2 ( r 4.10 V ˆ t ˆ r 8 d d 2 + 1) (ˆ ω 1 2 2 4 r
( ˆ t − , d + 3ˆ 5 ˆ t 8 s d Z dθ 4 2 0 2 3
ˆ r , is r ω and taking the slow-moving approximation yields 4 2 6 2 ω r r = Λ L + ˆ = ˆ ` M + 4ˆ 5 ˆ t d 8 dθ = 2 S ω We find an additional solution in this case with angular speed at leading order in We can also try to solve ( 10 ˆ r given by reduces to the algebraic equation attractive force, which is theon opposite this of the analysis, situation itunless in is normal nonlinear 4D effects unlikely Kepler conspire that motion. to the stabilize Based wormhole a solution circular orbit. canexpanding by be taking stabilized the by circular rotation, orbit as an ansatz. In that case the equation of motion momentum We see that thethis circular happens orbit is corresponds that to the a centrifugal maximum, piece i.e. in the it effective is potential unstable. dies off The slower reason than the We can check ifbits. the For circular this, orbit we radius examine the effective potential written in terms of conserved angular Therefore, the angular speedthe needed instability to time scale. obtain circular In orbits dimensionful coordinates, is this speed is velocity and therefore we can obtain circular orbits of radius the same result asradial previously equation of with motion, the assuming Newtonian the rotational existence kinetic of a energy solution added. with a The constant rotational Expanding in JHEP11(2020)167 . 0 0 r < pass (4.18) (4.19) (4.20) UU = 0 V dU T , R 2 c ]. Here we wish ˆ r . In the extremal 25 281 + 5 . supports two different 1 4 does not depend on c r . ˆ ω ω . 3 33ˆ c / 1 ˆ r , − ! , null rays along 3 8 c 2 / 4 r 1 , while we saw that the rotation 2 ) 5ˆ Λ Λ = 0 ωL √ 2 / − , and that this occurs at the reasonable ω
0 (ˆ Λ r 2 12 UU c ˆ r + 3) = T √ = This value of 4 c c ˆ L r r = (ˆ 15 c + 4 3 r
1 simply because it is inversely proportional to and unstable when + 1) − – 25 – 4 c 4 c and (which exist for both the eternal black brane and r ˆ r
(ˆ 281 . and 2 2 c 1 ˆ r U, V 2 & 9 ω √ c 3ˆ ˆ − r 2 ω 2 2 4 c √ r = L 2 3 c ˆ r which is certainly small as exceeds light-speed, even if it does not diverge. One can check that the = + 12ˆ 2 c Λ ω 8 c 4 L r r / 2 3 5ˆ √ 3 in dimensionful coordinates. This solution did not appear previously 4 is ) ω = 4 c . r 4 2 2 / +Λ 1 Ω √ . In the absence of any stress-energy, 4 c 3 2 r / ( 4 2 = 0 L Λ = Λ c V r = is One might worry that ω 15 and therefore not perturbatively slow-moving. through the bifurcation surface and asymptoteany to negative infinity perturbation in either will direction. pull Consequently, back the horizons and createmaximum traversability. value of radius that it can be madeto traversable by analyse a if small negative the energytwo-sided null perturbation perturbation Kruskal [ of coordinates thefor geometry our wormhole near geometry) the thealong horizon requirement to spoils violate this the ANEC property. is written In 5 Traversing the wormhole The two throats in ourtraverse through wormhole it. are separated Near the byAdS-Schwarzschild horizons, causal black the brane, horizons, wormhole which looks so is like it a a is marginally perturbation not non-traversable of solution the possible in planar to the sense persists even in theversion extremal of the limit. extremal two-center One solutionis that likely may is not therefore perturbatively possible wonder stable. duelong-lived if to An rotating gravitational exact there binary and solution exists five black form a hole.solution radiation, rotating stabilized This but by would we rotation. really provide just a want starting a point for a wormhole that is, the orbit islimit, stable these circular when orbits rotateThese with orbits a are finite not angular velocity, directlyto as relevant a for non-rotating stabilizing solution. our wormhole, Thisin which the is is circular because a orbits for perturbation to self-consistency be we perturbatively would small want in the rotation We can study the stabilityfinds of that these the circular (dimensionless) orbitsin frequency-squared by of linearizing the around radial the oscillation, solution. to One leading order or in dimensionful coordinates, from perturbing the action around small possible radii, or JHEP11(2020)167 . ) 0 tt r also ) = (5.4) (5.2) (5.3) (5.1) = 3.48 r t dt T ( r g R h one finds 0 ), which is corrections. r ) 6 = 3.45 Λ r / , ) do not affect the (1 O 2 . We have ) as λ 3.43 4 0 around r 3.4 , at the horizon, we need up to − component of the geodesic 0 2 4 = 0 1 r . ) r θ b = 0 ˙ ) µν = x are dynamically determined by 2 that we have described already ˙ 1 a t . µν ˙ ¨ r F I θ 4 0 x g )) F r const is a null geodesic at + µν at 4 mn F N µν − αβγδ r = ( g F 4 ν 4 1 4 plane is no longer decoupled from the ˙ ( U( t r = 0 F L J 2 r − tt ˙ × 5 r tr T ) − Λ ) and σµ ) 2 t µαβγδ N r effective subfactors in the IR from the Wilso- µν L F ( F ) g F ρσ – 26 – h 4! (U( since the location of the zero of the blackening = 0 N F 1 · ) µν S 0 1 1 ˙ F 4 νρ r θ θ ( . The monopole corrections ( SU( F → I = = = ) tr µν sin ( ˙ , so in order to have 3.6 ) we find that vanishes along the horizon. This follows because as noted r r µν F 0 2 N ∝ factors. The couplings T r r of the dipole corrections, spherical symmetry breaking does ) UU ) − IJ 3.48 tr 5 = T N V SU(2 − ∝ r ) = U( I , this correction can be obtained by linearizing ( . Examining the condition (Λ 1 ˙ 0 V θ O r 2 . This can be seen by examining the r 1 and section 3.5 ( = ]. At leading order, these include single-trace interactions of the form θ . Therefore, 1 r ˙ d around dλ θ 3.5 53 ], one mechanism to generate negative contributions to the ANEC that (and the rest of the seven coordinates fixed as well). Therefore, , 0 r is proportional to 1 39 ··· 34 θ I − , are the two V r + 2 25 , ) √ 0 r 2 ∝ − ) vanishes at = 1 r 1 4 plane at fixed ( ˙ ˙ r r 2 I , where − As in [ We will now evaluate the ANEC for the monopole- and dipole-corrected Einstein-Rosen / r ( I 3 angle, so the near-horizon geometry is effectively three-dimensional. Regardless, the Γ V π − I 1 20 associated to the moduli of branesto in traversability the are other the stack(s)). double-trace Of interactionsthe more that interest IR are to factors generated. us with in These respect the directly couple CFT: g and abelian singleton degrees of freedom in the other CFT factors (i.e., the Goldstone modes allow traversability is towormhole by introduce adding a a nonlocaltheory double-trace symmetry coupling type breaking interaction between in thegenerates the couplings two field between throats the theory. two ofnian In RG the fact, flow the [ field Applying this to the solution ( Therefore, at the order not affect the marginal non-traversability of the single-boundary wormhole. that for any fixed affinely parameterizes the nullWe worldlines, identify so the stress-energy the tensor ANEC from quantity the can right-hand be side written of ( where dot indicates derivative with respect to affinethat parameter θ location of the horizonfactor is stays not at affected.t Moreover, the null geodesicsequation, comprising the horizon remain on the bridges of section marginal traversability since at the end ofthe section asymptotically flat black braneis in more rescaled coordinates. complicated The to dipole analyze contribution because ( the JHEP11(2020)167 . , ) ]. ], i ) j N 25 25 φ and i ( SYM t CFT φ SU( are well Tr =1 n i = 4 kl L = Q O N SYM are the ij ij L = ) are dual to the O O N 2 = 4 F 2 N ) as generated by the SU(2 ) are structurally of the . This is because the ) N 5.4 factors) ] take the deformation to 12 ) and tr V t, ~x 25 SU( N 2 − ( ) are dual to the bulk dilaton F 2 1 . These are marginal to leading φ SU( ) 5.4 20 t, ~x × ( 1 20 φ ) individual subfactors CFT t ( n h different asymptotic universes. In this case, n ], and they all have either vanishing or negative – 27 – 56 . Since our solution is perturbatively unstable, we -symmetry. They are of the form 0 t R SYM theory. Furthermore, [ therefore furnish kl R SO(6) = 4 O that comprise the operator ( factorization. In order to work out the effects of deforming by ij L N 2 O ) N F of the N generated by the RG flow is an irrelevant deformation; nonetheless, I ]. In our setting, the components tr 12 20 V ] takes the boundaries to be connected with the same time orientation 55 SU(2 , corrections to their dimension. The corresponding single-sided double- 25 54 [ /N I required to generate negative contributions to the ANEC as shown in [ 1 due to large 2 φ ], the double-trace interactions are taken to be relevant deformations of the 1 /N φ 1 25 scalars in the 12 ). A full analysis should understand the net effect of all such RG-generated double- h In addition to the terms that are naturally generated by RG-flow, we can try, like [ In [ 5.4 flow and the trace operators (i.e. anunderstood operator in in the strong one coupling of regimeanomalous [ the dimensions. low The energy negativebinding anomalous energies dimensions coming are intuitively from understood the as attractive nature of the bulk interaction between two should be a relevanttheory. operator The in lightest order single for trace∆ it operators = not in 2 a to single destroyThe factor the possible of UV deformations order in these operators (with either sign of the coefficient) we would need to understand their RG on the other boundary.have taken In time our to setting, runwe the upwards need on wormhole not both resides flip sides, in the so time a there orientation single is between universe a theto and unique two add asymptotic we throats. by time hand some deformation that generates traversability in the IR wormhole. This be a quench, turned on afteralso some expect time the coupling strength toin be detail. time-dependent, although Lastly, [ we haveby not analyzed taking this the deformationasymptotic time to on be one structurally boundary of the eternal black hole runs in the opposite direction Hamiltonian so thatboundary. they The are term renormalizablethis and is there not is ascale concern it no flows as backreaction to we the at know the that the AdS theory is UV-complete, since above the Higgs therefore be interesting to determineWilsonian the RG, sign of at the least coefficientother in of double-trace operators ( perturbation that theory. canof be In ( generated fact, by the in supersymmetrytrace our transformations operators setting, on there the are sign of various the null stress-energy. gauge theories holds. Therefore,form the double-trace interactions This indicates that theIR of natural the operators symmetry-broken that theorypossibly are arise of weak from the traversability. correct theresults form Wilsonian However, in to a in RG generate traversable traversability, wormhole, flow the albeit while in the Gao-Jafferis-Wall opposite the protocol, sign lengthens only the wormhole. one It sign would the dual bulk geometry generallythe consists single-trace of terms tr in component bulk dilaton incorrespond the vicinity to of the IR the of first the and CFT second where throats, the as approximate these factorization deep into bulk two regions When the full UV CFT is genuinely a product of JHEP11(2020)167 (6.2) (6.1) , but the 3 SYM, and we ) N ). This spacetime is 6 we explained how to join = 4 SU(2 3.7 N . , , while the low energy factors are B,R B,R ; B to be approximately dual in the IR , which suggests that these operators A,R kl R ⊗ H 6 i and O ij L B,L A TFD O ⊗ H ⊗ | – 28 – B,L A,R ; capture certain effects of two non-extremal throats factors) to get interesting new wormhole configurations. For 3 ) A,L ⊗ H 3 i N A,L SU( TFD H | . Then, the low energy Hilbert space is R . In this case, the global time runs in opposite way in the two asymptotic regions . A double wormhole between two asymptotically AdS universes. This geometry, which and 6 L In the dual field theory we now start with two copies of and we expect ato wormhole a configuration tensor like product figure of two thermofield double states patch-wise described by thepatches same are solutions glued together that differently. we have discussed inHiggs section each copy. Letcalled us label the two theories these throats so that weways end to up join with the a solutions wormholeexample, of in one section a could duplicate single the universe.in spacetime There with figure are two throats also and other joinand them no in a twisting way is shown required to enforce flux conservation (see figure 6 A double wormholeThe between solutions universes we discussed inliving section in a single asymptotically AdS spacetime, and in section particles. The same intuitiveconnecting reasoning the applies two to low the energy should two-sided operator be (i.e. marginally an relevant operatormake at our strong single coupling, boundary and wormhole one traversable. should be able to use them to Figure 6 can be constructed fromwith the IR solutions factors in entangled the pairwise text, between is them. dual to a pair of Higgsed Yang-Mills theories, JHEP11(2020)167 , that SO(6) small. 2 of locally Λ . We have / 0 3 r SO(5) × subgroup of the SYM to the scalar vevs ], gluing the two sides of 51 ISO(3) single boundary wormhole = 4 × 5 SO(5) as explained in section N S R B × 5 ⊗ H Our wormhole is non-extremal, and A SYM, where the gauge group is Higgsed H SYM theory. It is tempting to speculate ) = 4 charge. This should lead to long-lived states As emphasized in [ N R N charge. – 29 – R = 4 SU(2 N . We argued that the solution is dual to an approximate Λ / 0 ] for a recent discussion in the case of global symmetries. It r we found that an extremal probe brane can be put on a stable , which are entangled. In the field theory the small parameter is 57 ) N 4.2 . Therefore, there is a moduli space 5 SU( S Another possibility is to stabilize the wormhole by making the throats rotate around introducing a global monodromy consistingdirections of to rotating the and brane, translatingsymmetry the and of parallel also spatial the rotating byequivalent but the unbroken globally differentterpretation solutions. of It this would in be the interesting dual to understand the in- Global monodromy and moduli space. a wormhole to a singleof asymptotic region the breaks eternal the black two-sidedof brane boost-like wormholes Killing geometry labeled symmetry and by correspondingly, thetwo there “monodromy” throats is of on a Schwarzschild the one-parameter time outside. family as one In goes addition between to the this, we have found that there is a freedom of with finite temperature symmetry breaking.supergravity picture, The we states expect are them onlyThis to long-lived, decay is due since consistent to from with gravitational the the andhigh expectation five temperatures: that form all radiation. symmetries see mustwould [ be be restored interesting at to sufficiently understand this effect better. are unstable in more thanaction. four The dimensions, stable and orbit is wepossible find possible to has here add finite this due angular effect to velocity perturbativelythat in the to the in our five-form extremal solution. inter- the limit, Nevertheless, so dualscalar this it theory finding vevs is suggests away not one from can the origin create by a adding local minimum in the effective potential of the consequently has to be unstable.developing an This effective instability is potential dual atargued in finite that the temperature, wormhole as can illustrated be in made figure parametrically longeach lived other. by In making section circular orbit around a non-extremal black brane. This is surprising since planetary orbits thermofield double state in ainto single two copy copies of of the ratio of the thermal scale toThermal the Higgs effective scale. potential and solution by matching a two-centerbrane extremal in black perturbation brane theory. solution Preserving to continuitya of a the global two-sided five-form AdS in monodromy the black orientable. solution in required some The of smallseparation the parameter of coordinates, in the the although throats, problem the is total the geometry horizon remains radius compared to the is, we must take theHiggs temperatures scale. of the thermofield doubles to be much smaller7 than the Discussion In this paper, we constructed an asymptotically AdS We can embed this state in the UV Hilbert space JHEP11(2020)167 in the geometry, . It would require 5 ) S N ) into the UV theory, 2.7 SU(2 SYM theory. = 4 As we discussed, continuity of ]. This requires a double trace N 25 SYM in the UV (or to a wormhole in a = 4 – 30 – We have showed that corrections coming from the N ) N factors are part of the total ) factors in the Higgsed N ) SU(2 N SU( have negative anomalous dimensions. SU( kl R ]. To use these branes to probe the wormhole in the field theory, O 62 ij L – O 60 . ]. These brane states are created by determinant and subdeterminant operators scalar operators of the theory, though it would also be useful to check that the ] negative contributions to the ANEC were generated by negative Casimir-like 1 remains zero along the causal horizons. It would thus be interesting to see if the ). θ 59 29 , 2.7 UU 58 In [ One may also try to make the wormhole traversable by adding a double trace coupling We have pointed out that such couplings are naturally generated in RG due to the ∆ = 2 dUT inverted flux [ in the field theory [ we would want toOn construct the such gravitational operatorsgiant side, gravitons in we as the they could light are explicitly infrared moved through test factors the what after wormhole, happens expecting Higgsing. them to to the emerge with corresponding Probing the monodromythe through five-form the requires wormhole. atwo-center twisted ambient gluing spacetime, of although theinteresting interior the way of AdS-Schwarzschild complete geometry probing spacetime the to resultingwormhole. remains the monodromy orientable. is Giant to gravitons send An a areand giant spherical graviton are through D3-branes supported the localized by on their the angular momentum and by interactions with the five-form win out. There is potentiallyvanishing total the Casimir possibility energy. that In themechanism any underlying case, supersymmetry for the enforces vacuum traversability a energies in providehave another discussed. competition potential or collusion with the other effects that we cycle threading their wormhole solution.spectrum In of our type solution, IIB theretheir are classical supergravity various backgrounds which fermions to have in vanish.similarly vacuum the provide These fluctuations, Casimir-like fermions, vacuum though and energies weenergy the in have is bulk our set important, setup. bosonic as fields, The before; should sign so of the it total is Casimir important to check which contributions ultimately will no longer flowsingle to spacetime). a We single havethe argued that suchtwo-sided relevant operators double traces can be formed from vacuum energies coming from the lowest Landau levels of the bulk fermion running in a a careful analysis to accountsee for if the the net resulting effectpresent sign of paper makes all but the these is wormhole double certainly traversable. trace an interactions This interesting and is problem. to beyondby the hand. scope of This the would have to be a relevant double trace operator, otherwise the theory wormhole can be madecoupling traversable between using the two the ideas in [ fact that in the UV the two which possibly includes anstate ( ambiguity in the implementation ofMaking the the wormhole energy traversable. cutofftwo in throats being the innot the spoil same theR spacetime marginal in traversability of the the first wormhole, few in the orders sense in that perturbation the theory ANEC quantity do that it is related to some ambiguity in embedding the IR state ( JHEP11(2020)167 = µν (A.6) (A.2) (A.3) (A.4) (A.5) (A.1) δg one finds: ). In general, δR ) is . ) 3.6 3.2 . ) αβγδ ν . αβγδ ) and ( ) ν δF , , 3.5 δF µν perturbatively, so we have . αβγδ h ν µαβγδ λ αβγδ ¯ F ∇ = 0 µαβγδ δF = 0 F λ ¯ + F ∇ by tracing both sides, noting that δR + 1 2 ): αβγδ µαβγδ αβγδ αβγδ ¯ F − F ν 3.5 ν δR ¯ F αβγδ h F ∼ + ν ν ) ¯ ∂ F µ in terms of the metric perturbation δF µαβγδ αβγδ ∇ µαβγδ ν µν F 1 2 + ¯ µαβγδ F δF µν – 31 – ( − δR F g ( δF ) µν ( λ ν ¯ g δ h µαβγδ 4! ∧ ∗ µ 4! 1 · 4! ( ) 1 · δF · 4 1 ( ∇ 4 δF λ 4! 16 = 1 · ∇ = + = 4 µν = F µν ¯ g = δR δg µν , δR µν 1 2 µν ). Since we impose self-duality at all orders, µν 0 = ( δR ¯ R − δR δg 3.2 − µν µν ¯ with its background equation of motion and rearranging for R δR δR 4 − µν ¯ − vanishes to all orders as a consequence of the self-duality constraint. The R δR δR = at background order, but it is not obvious that µν We now make use of the formula for , to first order in the perturbation, to arrive at ( δR = 0 µν µν ¯ ¯ h That is, perturbative equation of motion for the metric is therefore simply Substituting R retained this term forg now. Now we can constrain where the last expression is to all orders. Consequently, the variation of ( A Perturbative equations of motion In this appendix, we derive thebars perturbative will equations of indicate motion background-order ( quantities.equation We of begin motion by ( variation of the full geometric (Grant No. 38559),and and QuantISED by DE-SC0020360. the VBsupported Department also by of thanks National the Science Energywork Aspen Foundation through was grant Center in PHY-1607611, grants for progress. for Physics, DE-SC0013528, hospitality which while is this We are gratefulLampros to Lamprou, Alexandre Juan Belin, Maldacena,for Vishnu Onkar useful Parrikar, Jejjala, conversations. Simon Arjun Ross, MDResearch Kar, and is Fellowship Tomonori supported under Guram Ugajin by Grant Kartvelishvili, was the No. supported National in DGE-1845298. Science part Foundation by The the Graduate research Simons Foundation of through VB, the It MD, From and Qubit Collaboration GS Acknowledgments JHEP11(2020)167 as (A.7) (A.8) (A.9) g (A.14) (A.10) (A.11) (A.12) (A.13) s cb Γ δ − c sb sb Γ g ) bs δ . g s ca ac ) s ac Γ h s ac Γ c δ Γ . δ bs − − g µνρστ a sd − ∇ c sb cs s ac Γ ) + c a g Γ Γ h s ba bs . δ ¯ s ba gδF s ac c bs g Γ . Γ Γ 2 h − + ∇ ) s ac δ δ δg ) ¯ g − p bc c sb sc + s ac (Γ + − − h g a sa . We review the derivation of this Γ δ Γ + a h ¯ p g Γ c sc sa s ab a s ac − − g Γ Γ s bd ∇ = δ δ s bc (Γ ( sc sc − ∇ b δ Γ s ab bs g µνρστ a b µν δ + Γ g δg ∇ ¯ s ab h − ) + F c s ac 1 2 δg s ab Γ + g ) a ba + Γ bs Γ δ Γ ∇ h , Γ g c sc µν − δ − bs b d c sc ¯ g Γ g s ac − ∂ ) + ∂ √ ¯ Γ g + a ) + δ Γ ) s ab s ac a c ab – 32 – ( − s ab . , − sc ∇ bc Γ h sc h µ − ) (Γ Γ b g c δ g 1 2 p bs ∂ a bd δ δ δg . c ac g ∇ sc s ab 1 2 s ab ( Γ + ( − Γ g = a + + a ) sc Γ s ac − δ 2 1 (Γ − ∇ ∂ ∂ g δ sc s ab ( Γ ab δ c ac ) c ac c ac g b c a δ Γ = s ab h Γ = = Γ Γ h s ab 2 − Γ δ b δ δ g ) + ) = + − ( δ b b Γ ) a bc with this formula, one finds ∇ b − bc bc ∇ − ∇ δ ∂ ∂ a bad Γ sc bc bc µνρστ 1 2 g ) δ g g g √ ab , from Sylvester’s formula: R a δg + − − = = 2 a a δ ( g c ab − s ab c b gF − ∇ ∂ ∂ ∇ a = Γ ) δR c ab c ab ( ( ( Γ − h ab ) − c b δ δ δ δ δ ∇ a Γ Γ h ( h √ bd c ab c δ δ √ c a ∇ c c ( = = = = = Γ ( R ( µ ∂ ∂ ∇ δ c ( ∇ ∂ bc c c − ∇ ∇ = = = h 1 2 a δ ∇ ∇ ca ab ∇ h = = = b δR ∇ ab Γ + δ δR bc h a Now we must consider the variation of Maxwell’s equations: ∇ Recall the variation of which was the claimed formula for the variation of the Ricci tensor. Expanding the variation Rearranging and permuting the indices gives using metric compatibility. Now cyclically permuting and adding a convenient sign gives metric perturbations To compute the variation of the Christoffel symbols, expand the covariant derivative of the formula below. The Ricci tensor is Relabeling indices and varyingvariation each term independently gives a formula in terms of the where the covariant derivative is taken with respect to JHEP11(2020)167 . 5 S (B.1) (B.3) (B.2) × (A.19) (A.15) (A.16) (A.17) (A.18) 5 , 5 . dθ 5 ∧ ...θ 4 1 . r θ ) ¯ dθ F r 123 ) 2 h 2 5 t ∧ 5 ¯ 123 F Ω t 3 + d 2 h ...θ ) r 1 dθ δF θ + ∧ ΩΩ 123 + . 5 t δF 2 r b , δg F ...θ ) + dθ 1 = 0 123 θ 5 rr t = 2 ∧ ¯ F h F ˜ ...θ ) 1 r ( ˜ 1 r 5 33 θ θ rr + (1 + ¯ dθ g 5 δg F 4 g θ 2 ( 22 θ ˜ r . To begin, we evaluate the self-duality ˜ 5 r 33 ¯ − g θ ˜ r d µνρστ d g 5 11 ) θ sin . . . h ˜ r 22 ∧ . In terms of the components of the metric ¯ g ΩΩ 3 δF ˜ r 1 5 g 3 tt θ θ δg 1 ˜ g x δg + 11 2 ...θ θ . . . g d 1 g g 1 + ¯ θ tt θ + 5 sin ∧ r F 1 – 33 – + ¯ ˜ t 2 θ 2 ˜ t (1 + gg ... θ µνρστ ˜ x 123 5 2 t gg − 3 ) on the background metric of vacuum AdS ¯ δg d 1 ... F ˜ + r and − √ ...θ 2 h ∧ r − δF + sin 1 rr + θ √ − i 1 ˜ 5 ¯ 1 g i rr ˜ A.19 2 θ ˜ x ¯ 123 g θ ¯ g t 5 ˜ = = 33 x ~ δF d δg 4 θ − ¯ 5 F g d 33 r 3 5 ¯ g θ ∧ ¯ g ) 2 5 p 22 i ˜ ˜ t ˜ t θ i − ...θ ˜ ¯ 123 g 22 d ¯ 1 d ... g t ¯ g )) sin a θ µ ) ) and ( δg 11 1 r ˜ t 2 ∂ )) θ ¯ ˜ g t 11 δF (˜ 1 ... r δF b ¯ g θ tt (˜ 1 δg + ): tt h h θ a ) by hand in the linearized regime, region II. In this regime the + A.18 (1 + ( ( 1 ¯ g r θ 5 ¯ ¯ g g 2 ¯ g ¯ 3.6 g 3.6 r − − 123 ¯ g − ...θ (1 + t (1 + 1 + (1 + + ˜ 2 − p p θ F p 3 ˜ r ˜ F r − p − + − 4 = = 2 ) and ( L r 5 L 3.5 ...θ 123 = 4 = 1 t θ 2 F δF 2 Lastly, the perturbation to the five-form must leave it to be self-dual. However, one ds δF 0 0 1 α 1 α where all metric perturbations are functionsconstraint only of equations ( They reduce to only one independent equation, motion ( ansatz for the metric and five-form perturbations takes the form of interest. B Solving the linearizedIn equations this appendix, we demonstrate the procedure to solve the perturbative equations of To proceed further, one requires more details about the background metric and five-form Now removing the background-order equations and expanding perturbatively, we find components of the five-form are and five-form the constraint can be written explicitly as must be careful because theperturbatively. Hodge Let us dual assume involves the factors metric of is the diagonal and metric that that the also only nonzero contribute independent Therefore, we find ( JHEP11(2020)167 , 2 5 = 0 (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) 0 Ω where (B.15) (B.10) (B.11) (B.12) (B.13) (B.14) b d 4 1 r c 4˜ 2 8 ˜ r a − 2 + ˜ r . c = 0 4 d ˜ = 0 r ˜ t . ˜ t 1 ∧ ˜ t ˜ t = 0 ) = a 3 δg 5 r ˜ t ˜ ˜ x t (˜ δg + dθ d f : . δg b 2 + 8 ) ∧ ∧ , b + 8 i ˜ 8 i ˜ 2 4 2 i ˜ ˜ r i . ˜ + 8 ˜ x / δg , c 1 + , 4 i dθ ˜ 2 d i 1 δg ˜ ˜ r a 2 . 1 ∧ ∧ , c ˜ 3 t δg ˜ 0 x ˜ ~ a t + 2 3 = 0 1 + 24 d − 2 ˜ x + 24 ˜ ˜ r r , , a dθ ˜ r 0 ˜ 0 ˜ r r d 4 rδg ˜ + 5 r via 1 ˜ t 8 d + 24 ) = 0 ˜ 0 ˜ . r r ˜ t ∧ a ˜ ˜ t ˜ r t ∧ 2 1 δg 8 ˜ ), we find an equation determining t 00 ˜ ˜ t r ˜ t ˜ t / 2 ˜ ˜ 0 ˜ r rδg a r a ˜ t δg + 3˜ 4 2 ˜ − rδg d δg δg ˜ is independent of the others and may i r ˜ δg a dθ 0 i − ˜ B.8 15 + 1 rδg which is solved by − − a ˜ t ∧ + + to 0 4 ΩΩ 4˜ 0 ˜ t i − ˜ t 1 ˜ ΩΩ )–( 00 0 r i ΩΩ rδg ˜ t + 20( 1 ˜ i 4 4 ˜ c i ˜ 1 δg 4˜ 0 4 1 ΩΩ ˜ − ˜ t δg r r + 5 δg 0 ˜ t c ˜ δg c r δg 1 dθ ˜ = 0 4˜ r 2 δg ( B.7 rδg 4 + 9˜ 2 a 8 − 3 4 r − 0 θ ΩΩ + 5 rδg ˜ r a 2 − rδg + i ˜ ˜ t + – 34 – i c ˜ 0 ˜ t + ˜ t 00 ΩΩ 2 + 5˜ 0 ΩΩ ˜ ) for t 15 + ˜ ˜ sin c t b rδg 2 ) + + 5˜ ˜ g t 0 δg ˜ t 0 ˜ δg ˜ t t i δg ˜ 3 0 ˜ t ˜ t 0 i δg ˜ − ) + 3( 0 ˜ 2 t = 1 + θ = B.5 − r + 5˜ 4 1 + 5 i δg 2 ˜ rδg and five constants rδg i i ˜ ) = ˜ ˜ c r 0 rδg i + 2 ˜ 2 rδg B.6 ˜ t r − ΩΩ ˜ t + ˜ 3 4 r (˜ δg sin + 5˜ 00 2 ΩΩ 8 δg i ˜ + 6˜ + ˜ 0 f i ˜ t ˜ 2 ˜ r b a 2 δg 3 δg + 3˜ 00 ˜ t + ˜ i + i ˜ ˜ θ rg ˜ 0 0 i i ˜ ˜ b δg ˜ t 2 3 10 − 0 δg = ˜ t ˜ t ˜ r 0 2 5( d ˜ r rδg rδg r − 1 2 a ˜ r sin = 2 ) ) reduces it to 2 rδg ˜ t ). Combining Maxwell’s equations and self-duality gives 2 1 ˜ t ˜ r + ˜ c δg − θ ˜ r + 3˜ + 8˜ i + ˜ ˜ 3 2 ) 00 i δg 4 ˜ ˜ t i ˜ 00 00 B.10 ˜ t i δg b ˜ 2 3.41 δg 4 ΩΩ 2 δg δg 1 + ) sin . Now examine the three remaining geometric equations of motion. By r 2 2 δg (1 + b 2 r r 1 + 2 ). Plugging into Maxwell’s equations we also find only one independent 2 3 ˜ r , a + ˜ + ˜ + 3˜ ˜ r − ) into ( 1 2 − 1 a a a b ˜ a r 4 3.40 2 L 8( 16 16 16 + + (1 + B.12 L − − − − = 4 = 2 F are constants. Therefore we can relate in terms of the others: 2 ds 0 2 0 1 ˜ r α 1 ˜ r α , c 1 Notice that the equationbe of solved motion directly, yielding: ( We have consequently fixedterms the of general one perturbative arbitrary solution function in the linearized regime in c Plugging ( Plugging this back into all three equations we find that all threeThis are is a solved differential as equation long in as: for constants taking the linear combinationδg of ( equation which is identical to ( meaning that the five-formmotion charge are is conserved. The independent geometric equations of identical to ( JHEP11(2020)167 . , 42 . = 0 2 61 (B.16) c Int. J. Phys. (1999) , ]. , = (1991) 197 Nucl. Phys. 2 60 , a (2000) 086003 , (1991) 409 brane 360 2 SPIRE 3 / 62 IN 4 D 273 ][ Gen. Rel. Grav. hep-th/9809055 Multiboundary Fortsch. Phys. , , = , 1 a Phys. Rev. D (2014) 185015 , (2003) 021 31 Nucl. Phys. B Phys. Rev. D 04 , , Phys. Lett. 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