JHEP11(2016)024 Springer June 27, 2016 : October 16, 2016 November 7, 2016 September 1, 2016 : : : Received 10.1007/JHEP11(2016)024 Revised Accepted Published doi: + 2)-dimensional anti-de Sitter d a,b Published for SISSA by ) Gross-Neveu model provides the holographic + 1)-dimensional anti-de Sitter space. All mas- N ( d O and Soo-Jong Rey a . 3 Jaewon Kim 1605.06526 a The Authors. ) Heisenberg magnet or c Higher Spin Gravity, Gauge-gravity correspondence, Higher Spin Symmetry N

( Understanding Higgs mechanism for higher-spin gauge fields is an outstanding , O [email protected] Fields, Gravity & Strings,Institute Center for for Basic Theoretical Physics Sciences, Daejeon of 34047, the Korea Universe, E-mail: School of Physics andSeoul Astronomy 08826, & Korea Center for Theoretical Physics, Seoul National University, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: of varying depth. We present intuitiveyield picture non-unitary which system higher-derivative boundary in conditions termsterfaces of in boundary action. Wedual argue that conformal isotropic field Lifshitz theory in- for and massive and propose partially experimental massless test higher-spin of fields. (inverse) Higgs mechanism sive higher-spin fields are describedspectrum depends gauge on invariantly the in boundary termsserved conditions of that imposed Stueckelberg at higher-derivative fields. both boundary ends condition The For of some is the higher-derivative inevitable wedges. boundary for We conditions, ob- spin equivalently,ary spectrum-dependent greater conditions, bound- than we three. get a non-unitary representation of partially-massless higher-spin fields Abstract: open problem. We investigatetion. this Starting problem from in aspace the and free context compactifying massless of over higher-spin Kaluza-Klein alight field compactifica- finite and angular in massless wedge, ( higher-spin we fields obtain an in infinite ( tower of heavy, Seungho Gwak, Massless and massive higherspace spins waveguide from anti-de Sitter JHEP11(2016)024 57 39 56 26 19 5 52 10 13 34 29 45 33 21 13 21 44 44 – 1 – 10 48 6 field 49 s 7 9 51 +1 d 53 to (A)dS k + 2 d 56 8.2 Kaluza-Klein modes8.3 and ground modes Waveguide boundary8.4 conditions More on8.5 boundary conditions Decompactification limit 7.1 Case 1:7.2 open string Spin-two in waveguide harmonic with potential higher-derivative boundary conditions 8.1 Stueckelberg gauge transformations 5.1 Mode5.2 functions of spin-two waveguide Waveguide boundary conditions for spin-two field 4.1 Mode4.2 functions of spin-one waveguide Waveguide boundary conditions for spin-one field 2.1 Kalauza-Klein mode2.2 expansion Vector boundary2.3 condition Scalar boundary2.4 condition D3-branes ending on five-branes and S-duality B Verma module and partially massless(PM)C field From (A)dS 9 Holographic dual: isotropic Lifshitz10 interface Discussions and outlooks A AdS space 7 Higher-derivative boundary condition 8 Waveguide spectrum of spin- 6 Waveguide spectrum of spin-three field 4 AdS waveguide spectrum of spin-one field 5 Waveguide spectrum of spin-two field 3 Waveguide in anti-de Sitter space Contents 1 Introduction 2 Flat space waveguide and boundary conditions JHEP11(2016)024 ] ]) 4 1 , 3 ] for 6 , 5 bertrand russell This is now known to be a mistake. ]. In flat space, compactification of 8 , 7 – 2 – and that when a thing moves, it is in a state of motion. ]). As for their lower spin counterpart, one expects that their masses 2 People used to think that when a thing changes, it must be in a state of change, The Kaluza-Klein compactification provides an elegant geometric approach for dy- In this work, we lay down a concrete framework for addressing this question and, using A novel feature in (A)dS background, which opens up a wealth of the Higgs mechanism, namically generating masses.compact Compactifying internal space, a one massless obtainsa field in tower in lower of dimensions massive higher not fields.one dimensions only In and a on (A)dS massless spin-two a space, field field a but theories version also were of compactification studied of in spin-zero, [ spin- underlies the mass spectrum.compactifying higher-dimensional The (A)dS idea is spacesystematically to to study utilize lower-dimensional mass the (A)dS spectrummanner. Kaluza-Klein space of approach This and compactified [ Kaluza-Klein to higher-spin setupconformal also field field permits theory in from a gaugeterms which concrete invariant of the realization conventional above of global Higgs holographic symmetry mechanism breaking. dual can also be understood in mechanism of massive higher-spin fields are(if not any) yet of fully understood, partiallywhat the massless are Higgs mechanism origins higher-spin and fields patterns remains of mysterious. massive higher-spin For fields? it, both to situations, analyze the pattern of the massive higher-spin fields as well as Higgs mechanism that its mass is varied.in The all (A)dS possible extension polarizations.higher-spin of fields. massive They In higher-spin have (A)dS background, field arbitrarypart there of in values are possible flat of also polarizations massive space massvalues higher-spin is can of and fields eliminated mass-squared be for are and by which are partial called called gauge partially massive massless invariance. higher-spin They fields. have Just special as the Higgs in curved background suchdescription of as massive (anti)-de higher-spin fields Sitterdynamical and description space their of Higgs ((A)dS). the mechanism (inverse) As would Higgs necessitate such, mechanism any formulated gauge inis invariant (A)dS that background. a massive higher-spin field may have different number of possible polarizations as were generated by a sortto of Higgs massless mechanism, higher combining higher-spin spinmassive Goldstone gauge higher-spin fields fields fields. [ undergo inverse Conversely, Higgsirreducibly one mechanism and into expects split its that, massless polarization in higher-spinthe states the other gauge massless hand, fields limit, it and is higher-spin well-known Goldstone that the fields. higher-spin gauge On invariance is consistent only 1 Introduction Massive particles of spinas higher in than string two are theory— not and but only experimentally also a a as possibility(See, necessity in for — for hadronic example, both consistent resonances [ dynamics theoretically (See, of lower for spin example, gauge [ fields they interact to JHEP11(2016)024 ). 7 and in 8.2 1 ] may be viewed as projectively 8 . The modes independent of , 7 2) modules. It turns out that d, ( ], we show that presence of bound- so α, α − independent scales: the scale of waveguide field on an (A)dS waveguide whose internal ]. The Kaluza-Klein mass spectrum depends s two 10 – 3 – , Kaluza-Klein modes 9 ] and its (A)dS counterparts [ . 10 , 9 ]. The partially massless fields arises only if the internal compact ground modes 13 , 12 ]. Starting from non-unitary conformal gravity, this work showed that projective reduction 11 is reduced to zero. We call them ). These spectra reveal several interesting features: . The modes that depend on the size is the counterpart of massive Kaluza-Klein All massive higher-spin fields,tured both by fully the massive Stueckelbergby and mechanism, a partially in tower massless, which of are thederstood higher-spin struc- Goldstone fields in modes of terms are varying of spins. provided branching rules The of Higgs Verma mechanism can be un- size and theof scale the of resulting (A)dS masshigher-spin curvature spectra. fields radius. (thus While the This themodes same comprise Kaluza-Klein as entails of for modes an full the are variety interestingmassive of flat all higher-spin pattern mass space fields. fully spectra: compactification), massive massless, the partially ground massless and fully states in flat spaceα compactification, so they allthe become wedge angle infinitely is heavy theWe as counterpart call of the them zero-mode size states in flat spaceThe compactification. (A)dS compactification features tially massless fields in (A)dSboundary space degrees can of be freedom intuitively and understood (non)unitarity by ofThe the spectra their presence split dynamics of into (see two classes, section α distinguished by dependence on the waveguide size, The spectra contain notless only fields [ massless andspace massive has fields boundaries and but specific also boundary partially conditions. mass- The (non)unitarity of par- 7 Compactifying a unitary, massless spin- Formally, the compactifications [ • • • • 1 reducing on a conformalsystem hypersurface. in This [ viewpointyields was partially further massless spin-two studied system, which fordifferent is partially also route non-unitary. massless of Here, spin-two physics. wespace stress we with We are start boundaries taking an from to entirely added unitary, obtain benefit Einstein of non-unitary, physics gravity partially that and massless Higgs compactify spin-two mechanism on system. can a be Our triggered suitable setup by internal dialing has choices another of boundary conditions. fields in lower-dimensional (A)dSfigure space (as summarized at the end of section and specify suitable boundarypactification conditions of at higher-dimensional each (A)dS boundaries. spacelower-dimensional (A)dS over We an space shall as internal refer space “(A)dS to with waveguide” compactification. boundaries the to com- space is a one-dimensional angulararies wedge and of rich choices size of [ boundary condition permit a variety of mass spectra of higher-spin on specifics of themassless compact higher-spin field internal in space. aify higher-dimensional Here, to (A)dS the a space and idea lower-dimensionalonly Kaluza-Klein (A)dS is compact- massive space. that but we One also startently of partially from from our massless unitary, the main higher-spin above results fields situations, is upon that, we compactification, to must differ- produce choose not the internal space to have boundaries higher-spin field theories was studied in [ JHEP11(2016)024 gauge s 2) in terms of , + 1 boundary degrees of d ( so ) Heisenberg system or space to foliation leaves N -dimensional (A)dS space- ( +2 d d O limit. Both are realizable in heavy is equivalent to the Stueckelberg gauge N -dimensional isotropic Lifshitz interfaces ]. +2 d space. Both theories are well-defined. The d 15 [ +1 d – 4 – +1 d field are uniquely determined by the spin- s 2), viz. decomposing (A)dS d, on (A)dS ( s so ] proposed so-called ‘radial reduction’ that reduces a higher-spin 18 gauge symmetry on (A)dS – ] for spin- s 16 14 . + 1)-dimensional Minkowski spacetime to that in + 1)-dimensional conformal field theories such as d space and compactifies it to (A)dS d ] proposes to decompose the higher-spin representations of +2 8 d We associate the originproblems with of non-unitary partially boundaryWe massless conditions obtain higher-spin self-adjointness involving fields higher-order ofPhysically, to derivatives. the we Sturm-Liouville spectral interpret analysis thefreedom by newly extending introduced the Hilbert Hilbert space as space. We recast the spectral analysisThey in play terms the of role a ofproblem pair raising of is and first-order lowering factorized operators. differential into operators. symmetric The quadratic quantum associated product mechanics Sturm-Liouville Hamiltonian. of these operators, akin to the super- For spectral analysis, weInstead, we further extract mass bypass spectra working andtransformations. Stueckelberg on structure This from the is the because linearized linearized the gauge part linearized field field of equations. equation action and hence for thesymmetries. massless quadratic spin- As our focus isto on non-interacting higher-spin mass fields. spectrainstead Moreover, and we of their analyze quadratic linearized action. Higgs field Itfar mechanism, equations is as we known the limit that gauge the our transformations two analysis are approaches also are kept equivalent in to so the linear order. fermion magnetic materials andspectively. in This multi-stack suggests graphene anrealizations/tests sheets exciting of at (inverse) possibility Higgs Dirac for mechanism point, condensed-matter for re- experimental higher-spin gauge fields. symmetries [ The ground mode spectrawith the and critical (inverse) behavior Higgsin we mechanism expect ( therein from matchGross-Neveu perfectly fermion system in the large massless spin- To highlight novelty and originality of our approach, we compare it with previous In obtaining these results, we utilized several technicalities that are worth of • • • • • tion vertices but does not guaranteetheory consistency of is reduced theory not as known interacting(A)dS higher-spin in flat space.work Our [ approach starts fromhigher-spin higher-spin representations gauge of theory in works. There havefields. been various The approaches work fortheory [ in higher-dimensional ( origin oftime. higher-spin This approach describes one-to-one correspondence between flat and (A)dS interac- highlighting. JHEP11(2016)024 d to 7 . 8.5 discusses to (A)dS k 10 + space times an d +1 . The non-abelian d B , we start with the spin-one space. In our approach, we 2 space to AdS +1 d summarizes our conventions for +2 d A space, and refer to it as AdS waveg- . The choice of boundary condition +1 d +1 d , we argue that isotropic Lifshitz interface of – 5 – 9 . C , we explain how the Kaluza-Klein compactification works space down to (A)dS . In section 3 8 +2 2)-modules is briefly reviewed in appendix , respectively. We explain in detail how the spectral analysis of d 6 d, ( so and 5 in section , s 4 . While it takes an advantage of the discrete spectrum of these unitary repre- in our approach) that specifies compactification size or with a set of boundary +1 α d 2 is demonstrated in appendix ≥ ) Heisenberg magnet or Gross-Neveu model is the simplest dual conformal field theory The rest of this paper is organized as follows. In section k N ( open internal manifold withrequires boundaries. to A impose complete a specificationuniquely suitable of determine set the of the compactification boundaryprovides mass a condition new, spectrum at tunable parameter in eachthe in boundary, AdS addition conventional which to compactification, in the triggering size turn the of (inverse) internal Higgs manifold mechanism. that features for 2 Flat space waveguide andThe boundary salient feature conditions of our approach is to compactify AdS O which exhibits the (inverse) breaking ofvarious global open higher-spin symmetries. issues for Section furtherthe investigation. AdS Appendix space. The AdS waveguide method via the Kaluza-Klein compactification from (A)dS they arise from higher-derivativeconditions boundary can conditions arise (HDBCs) for and spinprovide that two the or such physical higher. meaning boundary of Athe the simple intuitive higher example picture derivative is boundary why considered non-unitary conditionsextend in and representation to section we appears spin- provide by reduction. All procedure that, in lower dimensionsto and uniquely at fix linearized level, theare the equations possible gauge of transformations and motion. arethem. rich We sufficient In pattern show particular, that of we aand show Higgs variety that, partially mechanism of in massless(PM) and addition boundary fields to mass conditions on fully (A)dS spectra massive can higher-spin are fields, be obtained massless realized. from For the latter, we show that for (A)dS space.compactification We of demonstrate (A)dS that theuides. so-called Janus On geometry thisfields provides in conformal geometry, section we studyequations of mass motion spectra and for of gauge spin-one, transformations spin-two fit consistently and each spin-three other, and confirm waveguide in flat space.the We Stueckelberg emphasize structure that and theization consequent Kaluza-Klein components. Higgs compactification mechanism We also manifests by showtransformations combining that various restricts the polar- possible consistency of setof equations of spin-one of boundary motion field. or conditions of In among gauge section various components sentations, this approach is rather limited(such for as not having a tunableconditions Kaluza-Klein parameter that yields themassless requisite or mass partially spectrum. masslesshave In higher-spin both particular, of fields it them. in does We (A)dS not summarize give more rise specifics to of these comparisons in section of (A)dS JHEP11(2016)024 . )) (2.4) (2.5) (2.6) (2.1) (2.2) (2.3) . We + 1)- , such } x, z d ( φ L φ ≤ and z µ ≤ A 0 , is excited along the  ≡ { φ M µ L A ∂ I , − . Combining the two gauge µ . M A ) = 0 Λ ν A m λ . z 2 m , A ∂  = z ∂ = µ δ φ − ) = 0 µ φ m A ν A , where interval ∂ (mode function of spin-zero field + z L ( ∂ I z φ ∝ ∂ µ − and × )) – 6 – − φ , d φ ∂ ). The equations of motions are decomposed as , δ φ µ 1 µν µ x, z ∂ R ( Λ ∂ F ( , φ λ µ µ µ µ µ 2 1 ]. Recall that the Stueckelberg Lagrangian of massive µ ∂ A ∂ ∂ A with a prescribed boundary condition that ensures the ∂ 14 + + 2)-dimensional spin-one field 2 = = = ) = = ( d z µ µν µ ∂ F Mz M ( Mν + 2)-dimensional flat spacetime with two boundaries, paying δ A F F A µν δ A − d F M M + 2)-dimensional spin-one field to a spin-one field and a spin-zero ∂ ∂ 1 4 d − = + 2)-dimensional coordinates into parallel and perpendicular directions, ), we learn that the mode functions ought to be related to each other as L d ), one imposes the same boundary conditions for both 2.3 0, the Stueckelberg system dissociate into a spin-one gauge system and a 2.6 is referred to as the Stueckelberg spin-zero field. This field is redundant for ), and the ( + 1) dimensions, φ , z → µ d x m It would be instructive to understand what might go wrong if, instead of the re- Inside the waveguide, the mode functions of the gauge parameter Λ should be chosen Inside the waveguide, the ( = ( (mode function of spin-one field 6= 0 because it can be eliminated by a suitable gauge transformation. In the massless z M -direction. The field can be mode-expanded, and expansion coefficients are ( ∂ Being a local expression, this relation must hold atquired each eq. boundaries as ( well. the eigenfunctions of ∆ := self-adjointness. compatible with the modevariations functions eq. of ( the spin-one field massless spin-zero system. z dimensional spin-one and spin-zerocan fields be of chosen various masses. from any Importantly, complete mode set functions of basis functions. It is natural to choose them by The field m limit, spin-one vector field is given by which is invariant under the Stueckelberg gauge transformations while the gauge transformations are decomposed as We note that bothstructure the of equations Stueckelberg system of [ motion and the gauge transformations manifest the x field in ( To gain physics intuition, wespin-one first warm — up waveguide ourselves in withparticular ( the attention electromagnetic — to massless different relations spins. between boundary Thedecompose flat conditions the spacetime and ( is spectra for fields of 2.1 Kalauza-Klein mode expansion JHEP11(2016)024 ) = 0 x, z with (2.8) (2.9) (2.7) ( (2.10) ,L 2 φ =0 ) restrict z | , φ 2.6  ) = 0 — imply b z , . + ,L , L z ), it follows that n π =0 = 0 = 0  z a∂ | = 0 2.2 )  ) z i sin ( ,L n ) := ( ( ) µ n ( n φ =0 ) vanishes everywhere as ( A ,L z φ , eq. ( µ | z φ z ν ) ( φ ∂ reads =0 ∂ z ∂ φ z ( | φ =1 + ∞ µ φ + X n ) ) ) n A z ( µ n z ∂ ( ν and A ) imposes the boundary condition ( ) = ∂ A z µ µ = 0 and ( M 2.6 ∂ A L φ L n π nπ ,L form a complete set of the orthogonal −  satisfies a first-order differential equation =0 = 6= 0 corresponding to the vector boundary µ z   | ). φ ∂ ) ,... ,L , b z L  , so individual coefficient in the above equa- 1 n π ( = 0 is special, as only the first equation is 2.6 and z , L =0 µ ) are also expanded in a suggestive form  z – 7 – I = 0

n A L  − a 2.2 n π = 0. Modulo higher-derivative generalizations, we = 0 )  z satisfies second-order partial differential equation, so µν  ( n ,L ) z µ µ n sin ( A =0 A z L F z | n π =1 ∞ ∂ µ µ ) for X n  ∂ µ A ) and eq. ( ∂ z = 0 and from the field equation of ∂  h cos − 2.1 z ) ,L ) = 0. But nπz/L n M ( µ z L ( =0 = 0 corresponding to the scalar boundary condition. Hereafter, we ,L n π A z φ |  ) =0 µ , b z z =0 ∞ ∂ | ( X n ) µ cos φ z 6= 0 ∂ ]. ( =0  µ ∞ a ) = X n 10 z , A ( = 0, 9 1, satisfies the Stueckelberg equation of motion for massive spin-one field z µ . The corresponding mode expansion for are arbitrary constants, the relation eq. ( ∂ ) must vanish everywhere. Likewise, ,L ≥ A z ,L ( n ), so the two sets of boundary conditions imply that =0 ) and zero-derivative (Dirichlet) boundary condition on the spin-zero field a, b µ z | ) A 2.1 = 0 The most general boundary conditions compatible with the relation eq. ( x, z z For Kaluza-Klein compactification of flat spacetime, the Stueckelberg structure of higher-spin fields was ( ( z 2 µ µ tions ought tononempty vanish. and gives The themodes, equation zero-mode of motion for massless spin-one field. Allfirst noted Kaluza-Klein in [ The mode functions sin( basis for square-integrable functions over so the field equations eq. ( We may imposeA one-derivative (Neumann)at boundary condition on the spin-one field for the spin-one fieldhave as two possible boundary conditions: condition and analyze each of them explicitly. 2.2 Vector boundary condition conditions do not preserve the relation eq. ( the form of boundary conditions forthe spin-one Robin and boundary spin-zero fields. condition for Forwhere example, the if spin-zero we field, impose these two sets of boundarythat conditions — eq. ( well. One concludes thatconditions. there is We no remind nontrivial that field excitations this satisfying conclusion such follows boundary from the fact that these boundary pose one adopts theA zero-derivative (Dirichlet) boundary condition for both fields. From and hence as zero-derivative (Dirichlet) or one-derivative (Neumann) boundary conditions. Sup- JHEP11(2016)024 , , (0) µ (0) A φ 0, all (2.13) (2.12) (2.11) n π/L → = L n m . Denote the ). The gauge is not needed. . L I 1) 2.3 (0) φ ≥ -th mode. It follows n n ( i ) n M ( Λ δA | and their completeness relation L n i . n π is replaced by the most general n ih |  − 2 n z z z | h ∂ = M − (2) ) L dictates the spectrum of equations of as is replaced by the covariant counterpart n π n ( 2 z δA  z δL M ∂ h δ φ − n δA cos X . ) modes take precisely the form of Stueckelberg n – 8 – ( = ,L Λ i ··· ) belonging to the Hilbert space of M , = 0 =0 z ∞ 2 , we have ( X n z , δA L | I M + 1)-dimensional fields read A M ∈ (2) = 1 Λ = 0) and d and the measure d z n δA δL ≥ 2 z h n ). Varying the quadratic part of action with respect to the gauge ( 0 −∇ z 0 = − ) z n ( ( δ = 0 mode is present only for the gauge transformation of spin-one field. Λ with associated gauge invariance. Also, there is no spin-zero field . The converse also follows straightforwardly. Note that this argument is µ . The second equation follows from divergence of the first equation, so n = ∂ M i (0) µ 0 = A z | ) ). /δA n n π/L n ( µ ih (2) ··· = n , | δ A n δL 2 . We will practice this elementary fact repeatedly throughout this paper. z -th mode, the equations of motion is projected to the same , h m zz n n g The fact that normal modes and their mass spectra are extractible equally well from The key observation crucial for foregoing discussion is that the same result is obtain- = 1 P √ z n universal in the sense thatfrom it holds action for or any linear energyfor Sturm-Liouville functional. system curved which is internal In derivable Sturm-Liouville space particular, operator for it which holdsd for the linearized operator higher-spins and It is elementary to concludeonto from this equation that,that projecting the the spectrum gaugemotion transformation of gauge transformation normal modes of the Sturm-Liouvilleas operator variation and integrating over of action as the( Stueckelberg action for a tower of Proca fieldsthe with linearized masses equations ofelementary consequence motion of and Fourier from analysis.this trivial the At fact. linearized the risk Consider gauge of fields transformations being is pedantic, an here we recall We note that the This is the gauge transformationtransformations of of a all massless gauge higher gauge vector transformations. field. Importantly, We the also Stueckelberg note gauge invariance that fixes gauge quadratic part The gauge transformations of ( Moreover, the spectrum isno consistent energy with flow the across the fact boundary that this boundaryable condition from ensures Kaluza-Klein compactificationtransformations of that gauge preserve transformationsFourier the modes: eq. vector ( boundary conditions can be expanded by the just confirms consistency ofStueckelberg the fields prescribed become boundary infinitelyspin-one conditions. massive. field In the Asan limit such, important there result that only follows remainsremains from the the massless prescribed and massless boundary gauge conditions. invariant, Intuitively, so Stueckelberg spin-zero field mass JHEP11(2016)024 is (0) µ (2.16) (2.17) (2.14) (2.15) A . In this φ . Clearly, φ only ( . , . 0) (0) φ ≥ = 0 = 0 n (  ) i n ) ( n ) ( φ n . This and the divergence φ ( µ µ ν Λ ∂ ∂ . A − L  − ) n π z ) n equation of motion, in turn, put ( µ n ( ν = µ A 0, these Stueckelberg field becomes L ) A n π A n 6= 0 are again Stueckelberg massive (  → L , there only remains the massless spin- L n π n nπ L δ φ  /L ) sin µ   x ∂ ( ) L  n π n is invariant under the gauge transformations. z ( – 9 –  Λ L (0) − n π φ =1 ∞  X n µν . . In the limit ) n 1) and ( cos ) = F ≥ = 0 consists of massless spin-zero field =0 ∞ µ nπ/L X n n π/L n ∂ x, z n ( = = Λ(  h n ) n z n ( . This and the divergence of m m + 1) dimensions. This then singles out one-derivative (Neumann) φ Λ L d n π µ to zero-derivative (Dirichlet) boundary condition.  ∂ µ in ( = A sin ) (0) n ( µ =1 = 0 zero-mode for the gauge transformation, and so no massless spin-one ∞ φ X n . Once again, this is consistent with the fact that this boundary condition and one-derivative (Neumann) boundary condition to the spin-zero n δ A (0) µ φ + 1)-dimensions, along with associated gauge invariance. This requirement then A d , in turn, single out zero-derivative (Dirichlet) boundary condition for µ in ( Once again, we can turn the argument around. Suppose we want to retain massless Once again, the above results are also obtainable from the Kaluza-Klein compactifi- We can turn the argument around. Suppose we want to retain massless spin-one field A µ (0) symmetries with masses spin-zero field boundary condition for the spin-one field There is no gauge field. TheWe spin-zero also zero-mode note that the gauge transformations take the form of the Stueckelberg gauge cation of gauge transformations.boundary condition, For the the gauge gauge function transformation can that be preserves expanded as the scalar With these modes, the gauge transformations of fields are spin-one fields with mass infinitely massive. Below the Kaluza-Kleinzero scale 1 field ensures no energy flow across the boundary. Consequently, the zero-mode absent from the outset). All Kaluza-Klein modes case, the equations ofabove motion, except that when the mode-expanded, mode take functions exactly are interchanged: the same form as the boundary conditions for linearizedgauge transformations. field equations can be extracted2.3 just from Scalar linearized boundaryAlternatively, condition one might impose no-derivativeone (Dirichlet) field boundary condition to the spin- A singles out one-derivative (Neumann) boundaryfor condition for the massless fieldsunder are inhomogeneous associated local or with rigid gauge transformations). or So, this global argument shows symmetries that (namely proper invariances JHEP11(2016)024 . ]. φ d = 4 20 , and N 19 . Here, µ H A M = 4 hypermul- = 4 vector multi- N N )). = 4 hypermultiplet. If 2.6 N = 3 case. waveguide — a waveguide which d +1 d 2) isometry and which has a tunable size of , ) duality symmetry, under which the two brane – 10 – Z + 1 , and the hypermultiplet moduli space d ( = 4 vector multiplet and V so N M = 4 vector multiplet and hypermultiplet are interchanged with each = 4 vector multiplet. In the presence of five-branes, half of sixteen N N ) are correlated each other (for example as in eq. ( M 2) sub-isometry within A d, ( space, we can construct a “tunable” AdS = 3 and adjoined with maximal supersymmetry, the two boundary conditions are so d Boundary conditions of lower-dimensional componentfrom fields (for example, Kaluza-Klein spectrum is obtainablelinearized gauge either transformations. from linearized fieldStueckelberg equations structure or naturally from arises fromfor Kaluza-Klein flat compactification space not but only also for (A)dS space. +2 The Type IIB string theory has SL(2 Consider a D3 brane ending on parallel five-branes (D5 or NS5 brane). From the Summarizing, d • • • internal space. 3 Waveguide in anti-deWe Sitter now space move toAdS waveguide in (A)dSretains space. Here, we first explain how, starting from other. Thisthree-dimensional is gauge yet theory, which another exchangesthe way two vector hyperK¨ahlermanifolds of multiplet provided by moduli demonstratingfollowing space the our approach, well-known we mirrorof see gauge symmetry that and they in global can symmetries also of be component derived fields. entirely from the viewpoint sets “NS5-type” boundary condition:vector Neumann multiplet boundary and condition Dirichlet on boundary three condition dimensional on three-dimensional hypermultiplet. configurations are rotated eachthree-dimensional other. In terms of D3-brane world-volume dynamics, the the five-brane were D5tiplet. brane, If the the zero-mode five-branesvector is were multiplet. the NS5 three-dimensional brane, Inboundary the terms condition: zero-mode of is Dirichlet D3 the boundaryand brane three-dimensional Neumann condition world-volume boundary on condition theory, three on D5 dimensional three-dimensional brane vector hypermultiplet, multiplet sets while “D5-type” NS5 brane viewpoint of world-volume dynamics,tions the to flat stack space offour-dimensional waveguide. five-branes provides The boundary originalsupersymmetries low-energy condi- is degree broken. ofplet freedom At is of the split D3 boundary, into the brane three-dimensional four-dimensional is The two possible boundaryWhen conditions discussed above arerelated universal each for other all by dimensions context the of electromagnetic brane duality. configurations in This Type feature IIB can string theory, be studied neatly most recently seen in in [ the Before concluding this section, we commentin how terms these of features the are realized brane in configurations string and2.4 theory S-duality for D3-branes ending on five-branes and S-duality JHEP11(2016)024 . 3 ∈ . φ +1 θ hy- d ) µν (3.2) (3.3) (3.1) + 1)- sin g +1
2 ρ d d 2 1 ` , y, z θ direction 1 -direction = θ d . y θ − 2 y 2 d . Note that ρ y , x 1 θ d -direction, the + +tan y t, yy 2 cos g µν ρ ρ + θ h 2 = + d ) 2 . z tan  ~x +1 2 d − . See figure , and remaining ( θ ν 2 π y is dimensionally reduced to d + d space and change bulk radial A 2 2 2) is part of the original isom- ` t νy µ (AdS d s +2 h d, ∇ + α < d ( µ − 2 ) so ∇ = d θ 2 +1 2 d y 2 where d + 1)-dimensional tensors. Consider, for ` + 2)-dimensional tensors continue to be cos 2 2 into a semi-direct product of AdS d becomes α 2 ` d z . ρ (AdS C νθ ≤ . From this foliation, we can construct the s to polar coordinates, + +2 h d 2 = π d
θ y  µ 2
 z 2 θ – 11 – ≤ ∇ , the isometry 2 z = y space can be represented as a fibration of AdS 1 ” does not play a special role. It can be chosen from any of and Kaluza-Klein compactify along the α + d y θ . cos θ − + d 1 +2 ) give rise to ( d 2 − 2 . The second term is a manifestation of non-tensorial = 2 d y θ ]: 2 x d µy 2 π ) not 2 h , . Moreover, when compactifying along the + d ). In particular, ( ` 2 π +2 + d 2 d ≡ ∞ + 2 − 3.1 ~x [ t µ 2 = space in the Poincar´epatch with coordinates ( space is at d ∈ A z +1 − + d (AdS d θ +2 s +2 2 s d d t d 2 2 d + 1) dimensions. ` (d z θ d − 2 = 1 , where 2 2 2 ν 1) coset space. The semi-direct product structure can be straightforwardly generalized ) ` z cos A , under mild assumptions, we prove that the semi-direct product waveguide − 2). However, globally, this does not hold in the Poincar´epatch. The reason is +2 and another spatial coordinate 1 z d = = , d C z 2 space. We start from the of Poincar´epatch AdS : µz 2) isometry transformation does not commute with translation along SO( δ +1 + +2 / + 2)-dimensional Poincar´emetric is independent of d, d ) d +2 ) is the unique compactification that preserves covariance of tensors. R (AdS ( + ( d d s s d d ν so × d so 3.2 We can explicitly construct the semi-direct product metric from appropriate foliations In fact, any attempt of compactifying along an isometry direction faces the same diffi- Consider the AdS The choice of spatial Poincar´edirection “ A 3 + 2)-dimensional tensor does , d +1)-dimensional tensors. For instance, µ 1 d d The boundary of AdS AdS waveguide by taking the wedge the SO( to other descriptions of the (A)dS space. See appendix space over the interval, eq. ( of AdS coordinate With this parametrization, the AdS Here, the conformal factor arisestion because which we is compactified globally thein non-isometric. internal space the This along compactification compactification a eq. bypasses direc- ( ( the issues that arose In appendix culties. As such, wepersurface shall and instead an foliate angular AdS coordinate over a finite interval: ( example, a small fluctuation of∇ the metric. The tensor transformation in ( AdS metric would work wellThe for reason is the as AdS follows. compactificationetry Locally we at look each for.that Actually, it is not. at the Poincar´ehorizon, R The ( dimensional space is again Poincar´epatch. Therefore, it appears that this foliation of JHEP11(2016)024 . n ) α (3.4) (3.5) (3.6) ≤ θ ≤ are adjoint α . n − − ( s 2 − d = 0). The waveguide +2 L d z ) θ . 2( θ d n 2, as in the middle figure. ] spanned by the above and π/ − )  n ’s to express Kaluza-Klein (cos θ component that arise from α, α n = s − θ [ , so acting on inner product of ),L α < π/ 2 θ ] := (cos field will carry the Weyl scaling θ θ ∂ L [ 3.4 s ∂ for n derivatives in the (A)dS wavegude is not a Killing vector, so although +2 ) α d θ θ θ + µ ∂ ∂ ≤ (cos θ ` 1 ≤ = α
− – 12 – θ . One might try an alternative compactification ] where d are equal, their first derivatives differ each other. field upon waveguide compactification θ θ 2 tan [ + 1)-dimensional spin- 1 α S d n 4 is given in the right figure. +2 in the Hilbert space cos  d + π/ µ n = θ = ∂ ) d θ α ), the AdS boundary is located at ` 1 +1 d . We also introduce the first-order differential operatorsL ρ, θ = ,S n L ··· , ]. 1 , 21 . This is not possible. The vector α ). The differential operator is quadratic in + 2)-dimensional spin- = 0 ) = dVol(AdS s -th mode function for (  d 2 n ( s +2 = d − . Anti-de Sitter waveguide: the left depicts a slice of AdS space in Poincar´ecoordinates θ d ) = ) of Weyl scaling weight fields defined by the integration measure in eq. ( θ Z ( s A comment is in order. In this paper, we mainly concentrate on AdS waveguide. How- Before concluding this section, we introduce the notations that will be extensively used An important consequence of compactifying along non-isometry direction is the ap- S ). In polar coordinates ( | ∈ n s Θ n y, z dVol(AdS weight ( spin- each other. ever, we can straightforwardly convert the results to the dS waveguide which we introduce From the general covariance,are it in follows the that combination allnormal-mode of of equations, these gauge transformations operators.the and Sturm-Liouville boundary So differential conditions. we operator will acting As on use weL spin- will see, Evidently, ( mode functions: in later sections. We introduce the mode functions of AdS waveguide as follows: aries at the metric at hyper-surfaces We reiterate that theAnother AdS consequence is waveguide that is the the integration unique measure of choice the for waveguide tunable is nontrivial compactification. this waveguide is embeddable toIIB string supergravity theory: for such nontrivial geometryJanus dilaton arises geometry and as [ axion a field solution configurations of and Type is knownpearance as the of thescheme conformal of factor AdS tube by putting periodic boundary condition that identifies the two bound- Figure 1 ( is constructed by takingFor the example, angular the domain waveguide for JHEP11(2016)024 (4.1) (4.2) . The } µ, θ { ), as can be , we learned θ 2 = , M tanh − ) = 0 : they run from 0 to φ +2 µ ) to ( d ∂ θ space: they run from 0 to , − µ +1 . Therefore, d θ A 0 ) = 0 φ ( L ( µ 2 ∂ − d − L µ − A space, while unbarred quantities are tensors 0 µν . +2 ( L ( F ` d – 13 – ν µ ∇ ∇ = = are the indices of AdS θN µN ¯ ¯ F F ··· M M ). ∇ ∇ µ, ν, 3.6 MN MN ¯ g ¯ g θ θ 2 2 will be used to represent the indices of AdS sec sec . The spectral differential operators in dS are obtainable by changing the . The only technical difference is to replace (tan ··· 5 C . The AdS radius is denoted by ] to the same extent that the higher-spin gauge symmetries fix the equations of space decomposes into two polarization components: +1 M,N, d 15 +2 For foregoing analysis, we use the following notations and conventions. The capital Through lower-spin examples, spin-one in this section and spin-two and spin-three in d + 1. The greek letters . For the waveguide, index for the internal direction is We first consider the methodAdS using the equation of motion. The spin-one field equation in d barred quantities represent tensors inof AdS AdS 4.1 Mode functions of spin-one waveguide fields [ motion of higher-spin gauge fields. Therefore,obtaining it information suffices to about use the the gauge mass transformations spectra for of dimensionally reducedletters higher-spin fields. d method has one more important advantage:can the be dimensionally derived reduced by equations ofbecome the motion Stueckelberg second transformation. method. Afteras The compactification, restrictive point the as is gauge the thatwill transformations gauge the show symmetries Stueckelberg below), (because so symmetries it the are completely latter fixes follows the from equations the of former, motion for as all we massive higher-spin quadratic part of theother action. method is Oneincreasingly using method complex the is structure gauge for usingmethod higher transformations. the spin is equations field less As of (even practicalrelatively the at motions, for easier the equation adopting while to linearized to the of deal level), general with the motion higher-spin and first contains fields. can be The applied second to method general is higher spin fields. The second this will be extendedplications to (some higher-spin of fields whichboundary in actually conditions later are open sections, identifiable. new modulo physics), some we technical will com- later explain sections, in detail we how shall the compare two alternative but equivalent methods at the level of work out Kaluza-Klein compactification onthat the boundary Janus conditions waveguide. of Incilitate different section the polarization Stueckelberg fields mechanism.product are In structure combined is our one the AdS key another feature, compactification,a to the where priori. choice fa- of the boundary Here, semi-direct conditions were we left develop unspecified methods for identifying the proper boundary conditions. As seen in table tangent functions in eq. ( 4 AdS waveguide spectrumIn of this spin-one section, we field focus on the lowest spin field, spin-one, in AdS space and systematically in appendix JHEP11(2016)024 ) ) ) s ( 4.1 3.4 ∆ (4.5) (4.7) (4.8) (4.9) (4.3) (4.4) − ) obey to each 2 4.2 , − 1 1 d | | 1 0 ) (4.6) θ Θ Θ can be mode- ( pure imaginary, ) and ( 11 00 B φ andL c c 10 . )) , 0 4.1 ) − − + 1) dimensions is θ µ , c θ ( d ( A . = = , equations of motion 01 A 1 1 | | c 0 1 1 0 n 2 0 n | | 1 0 ), labelled by the mode ’s are in general complex- ( L ( Θ Θ θ 2 ) ( 10 ) 10 n )Θ )Θ 1 ( c θ | 0 2 , c s n . ActingL φ 2 L − . ! = α d 01 1 1 2 2 | | c L 1  =0 1 0 | − ∞ λ X n 1 d 0 ](cos Θ Θ = θ Θ [ ( L ( L ( := = 01 10 0 θ c c − − ]: φ +2 00

d c µ := := − = d α, α 1 1 | | are conjugate to each other and hence the = − 1 0 ! [ Z 2 1 1 + 1)-dimensional fields | | Θ Θ andL 01 − 1 0 ∈ c d d (1) (0) – 14 – Θ Θ θ 10 )) = c θ ( 1 ) and − | ! θ 1 2 andL B ( = 2 − 1 Θ 0 0 | d − 1 n ). The ( d 01 11 is that boundary conditions, equivalently, mode functions c c Θ 0 2 , φ are adjoint each other with respect to the measure eq. ( ) 0L )( L )( µ − L = n 2 ( µ θ A 1 ( −

| A d 0 + 2)-dimensional spin-one field strength: A 2 Θ d , on the interval ) =0 2 ∞ θ X n − 2 ) := ( ··· d andL θ = , ), respectively, one obtains two Sturm-Liouville systems for spin-one L ¯ 0 2 A µ , , 4.5 1 ](cos A µ , must be related each other such that each term of eqs. ( θ ¯ [ A φ is positive semi-definite. This also puts the coefficients +2 = 0 d ) must to be expanded by the same set of mode functions. We find that this 2 = ( µ n λ ) that determines the mass spectra of spin-one field in ( d we impose Dirichlet boundary conditions at 4.2 M 4.1 ¯ Z A and for µ spin-one Sturm-Liouville : ∆ What we learn from section spin-zero Sturm-Liouville : ∆ A with the property that the eigenvalues of respective spins are paired up Here, we took intoeigenvalue account thatL equations of eq. ( and spin-zero modes, for the field strength of ( provided in eq. ( factorized to a product of two first-order elliptic differential operators, We first note thatL among the spin-one modes and thevalued spin-zero coefficients. modes. These Here, equations reveals that the Sturm-Liouville (SL) operator for the same boundary condition.are Otherwise, accompanied as with we independent learned boundaryno in conditions degree section for of each freedom fieldand left and eq. there after ( would dimensional be consistency reduction. condition leads Therefore, to each the term relations of eq. ( Mode functions are determined onceabove, proper our boundary conditions key are strategy prescribed.first is As require not stated gauge to specifyboundary invariance conditions of boundary that conditions various are at higher-spin compatible the with fields outset. such and gauge then Rather, invariances. we classify all possible expanded in termsharmonics of a complete set of mode functions Θ where JHEP11(2016)024 . for 1 | (4.11) (4.12) (4.14) (4.10) 1 n ≤ · · · 2 2 λ ≤ 2 1 there ought to . λ , # # m 2 ≤ 1 commute with the . | − 2 0 0 n d 0 † λ 01 n L Θ 0 c 0 Q are not only conjugate " ≤ L 10 n 0 . By the aforementioned . c 01 n 1 − | # for some and 1 m ic ,L 2 = 0 = 0 (4.13) 2 1 − Θ L | = − d Q } 0 1 m 2 00 n d † L c , viz. 0 − # 0 i d 1 Q | = , L = 2 spacetime supersymmetry of ten- 0 n 0 0 0 0 ··· † , " − Θ 11 n 2 Q N " ). Moreover, the double degeneracy " c ) is nothing but ) as “spectrum generating complex” , { = † 1 † = = , 3.4 = Q are conjugate to each other with respect Q 2 n 4.11 4.10 } H λ 0 = 0 Q , − n Q proportionalL to { and in eq. ( andL – 15 – n and where 2 H − there ought to be present a spin-one mode Θ d , , with # } m # # 1 † 1 1 | | | 0 n 1 0 1 n 0 n Q ) is the statement that for some , and for a spin-one mode Θ , 0 Θ 1 Θ Θ 1 | 0 0 | L " 0 m Q 4.4 ) thatL 0 n 0 i { ), equivalently, eq. ( space. Θ L " #" 10 n pure real. Hereafter, we label the eigenmodes in the ascending 2 4.6 = 1 1 for some 00 n ic 0 − | | = 4.4 c +2 . d n 1 n 0 11 1  d | = H Q Θ Θ 0 m , c 11 n 0 To see this, let us combine the two Sturm-Liouville problems for ··· # c 2 ) are then the statement that the SL spectrum is doubly degenerate: 00 , 1 | " c − 2 4 1 n 0 d , 4.4 ). = L 1 Θ , " # 1 1 . | | Q 4.10 1 n 0 n = 0 H Θ Θ n " is a consequence of the fact that the supercharges = H proportional toL 00 n m c n In fact, we can attribute such double-degeneracy to a hidden supersymmetry of the Stated differently, the two first-order elliptic operatorsL = Note that the hidden supersymmetry is unrelated to the 4 11 n c SL operator dimensional Type IIB supergravityof in the which supersymmetry. the Janus geometry is a classical solution that preserves half reinforcing the fact in eq.to ( the inner product defined by the measure eq. ( Then, the two-component SL operator and the spectral relation eq. ( Let us also introduce two supercharges complex eq. ( spin-one and spin-zero modesmodes into one Sturm-Liouville problem acting on two-component As such, we referfor to spin-one eq. field ( in AdS ordering of eigenmodes, weindex labeled the paired spin-zero and spin-zero modes byeach the other same but alsomodes act with as doubly raising degenerate and spectra lowering operators between spin-one and spin-zero order of their eigenvalues andThe relations label eq. them ( by for a spin-zero modesome Θ be present a spin-zero mode Θ and the eigenvalues JHEP11(2016)024 is = = for ··· are 2 n n , 11 0 1 λ | 2 c (4.15) (4.16) (4.17) (4.18) 1 n , ··· 1 for , , 2 1 | , 0 n are nonzero = 0 are nonzero. become zero. n 11 n = 1 ), none of the ) allow to de- 10 0 c 10 0 . c n , c 6= 0 = 0 4.5 4.4 , respectively. By for 01 0 2 1 1 c | | 1 = 0 | − 0 0 0 0 for 0 n and d 1 | . Θ Θ 1 space. This is actually 0 0 | 2 2 01 0 1 n c − − Θ d d +1 = 0 = 0 d L L andL 1 1 and the eigenvalue for spin- | | 0 1 n 0 n , and that the eigenvalue 00 n Θ Θ space with mass c i i 0 from the above situation (1), , , ) ) 01 n − . All higher modes, Θ = 0 ) n c 2 +1 ( n 0 from the situation (1), leading to 1 d → ( is the ground mode with vanishing | − φ 6= 0 = 0 1 0 d , it may so happen that there exist φ µ 1 1 1 | L → 01 ν Θ ∂ | | and 2 0 0 0 c 1 0 1 0 and spin-one eigenvalue 0 ∂ . 10 0 − Θ Θ 10 n c 2 n 00 n ) 0 0 − c c λ n ) ( µ n − ( ν A . All higher modes, Θ A = 0 10 n – 16 – L c are annihilated byL 2 n 10 n 2 h c 1 is the ground mode with vanishing eigenvalue | − ( M µ 1 n d 1 | 01 n ∇ 1 0 c 0. By the double degeneracy property, they are paired n . There is one spin-one ground mode of zero eigenvalue. + > X ) and Θ n ··· 00 n ( µν 1 , c | 0 n F 2 , µ . By the spectrum generating relations eq. ( = 0. This means that Θ ∇ h 1 = 1 ··· | , 0 0 n 2 n Θ , X spin-one ground mode :L of the spin-zero SL operatorL 2 1 + 1)-dimensional field equations into spin-one and spin-zero modes as , for − d d 00 0 1 | c double ground modes:L 1 n are positive definite, and are paired up. The spectrum consists of doubly = 0 are necessarily massive. On the other hand, spin-one eigenmodes Θ double Kaluza-Klein modes:L n = 0. This means that Θ 11 n c 1 | ··· − 1 0 all , Θ 2 0 , L leading toL eigenvalue 1 0 of the spin-onenecessarily SL massive. operator OnL the otherhave hand, positive spin-zero eigenvalue eigenmodes Θ up with Θ one degenerate Kaluza-Klein modes. A specialIn case this is when case, both thewe spectrum shall includes distinguish doubly these degenerate two ground cases. modes. Nevertheless, This implies that the spin-zerofor eigenvalue corresponding modes Θ the double degeneracy, the eigenvalue for spin-zero Spin-Zero Ground Mode: this case is when Spin-One Ground Mode: this case is when Doubly Degenerate Kaluza-Klein Modes: this case is when both of Recalling the flat space counterpart in section Returning to the Kaluza-Klein compactification, the relations eq. ( (3) (2) (1) massless — thus unHiggsed — spin-onemore or interesting spin-zero situation, fields so in we AdS wouldRecalling like that to understand the when SL andpositive how eigenvalue semidefinite, this is there comes product about. are three of possible situations for the lowest eigenvalue: We see that thesethe equations Higgs take mechanism for precisely massive the spin-one form field of in Stueckelberg AdS coupling, triggering compose the the ( JHEP11(2016)024 (4.20) (4.19) coefficients = 0 subject 10 0 c 1 ] is finite, the | . 0 0 0. The spin-zero α Θ + and 2 → = 0 − ) 1 α, d 01 0 | 1 10 (c) | c 0 0 c − n 1 Θ 2 − d L = 0 andL 1 | , 1 0 Θ Multiplicity(Θ 0 0 = 0 are taken to zero, the spectrum is again n 1 | 0. By the double degeneracy property, X 1 0 . There is one spin-zero ground mode of 10 0 c > − ) ··· 11 n 1 , | c 2 and n 0 , (a) dim KerL 01 0 – 17 – c − = 1 . 2 − 2 n d for 0. The spin-one zero modes have no ground mode, while the 1 | n 0 ] is nonsingular and as the interval [ → θ Multiplicity(Θ 01 n c X [cos µ ≡ = dim KerL )=1 have positive eigenvalue s ( spin-zero ground mode : Θ (b) I ··· , 2 , 1 , . Various situations of double degeneracy between spin-zero (red color) and spin-one (blue = 0 n they are paired up withzero Θ eigenvalue. We can classify the pattern of spectral pair-up by the elliptic index defined by Remarks are in order. First, for the doubly degenerate modes of nonzero eigenvalues, only their eigenvalues butare also solutions their of multiplicities theto also first-order the differential pair Dirichlet equationsL up. boundary conditionother. Second, that render the As the the zero two differential measureexistence modes operators of d adjoint normalizable each zero modes is always guaranteed. These situations are depicted in figure if one of them is normalizable then the other is normalizable automatically. Thus, not are nonzero. The spin-zero andhave spin-one no modes ground are mode. degeneratemodes and The have have left nonzero no spectrum eigenvalues ground and (b) mode,depicts so depicts while the the the situation situation spin-one (2) (3) modesspin-one that that have modes ground mode. have ground Thedoubly right mode. degenerate, spectrum but (c) If now both starting from the ground mode. Figure 2 color) modes. The middle spectrum (a) depicts the situation (1) that both JHEP11(2016)024 (4.21) (4.22) (4.23) (4.24) . The 2 is nonzero. -th Kaluza- n )=1 . s ( ) θ I ( = 0 = 1 1 | 2 2 0 n − − = 1 = 0 d d Θ ) 2 2 . n − ( − ) ) d d n Λ n ( ( Λ 10 n Λ c µ 10 n 0 are paired up and so do not ∂ c n X = = ] = 0 ] = 0 (4.25) ) ) , n > ) and the Sturm-Liouville equations, ) ) n n ) n n ( µ ( ) = ( ( θ θ ( φ φ ( δφ 4.24 1 δA 1 µ | ν | = 0= 1 dim KerL dim KerL 1 n ∂ 1 n ∂ 0 0 Θ Θ − − ) ) ) = 1= 0 dim KerL dim KerL n ) n ( n ( n ( µ ( ν 0 0 Λ – 18 – Λ A A µ 0 ∂ L 10 n 10 n c c [ [ n n µ X 01 n X 0. c ∇ ), which was obtained by the method using the equation of + ) = ) = ) θ of supersymmetric system preserves the supersymmetry if n > θ n ( 4.10 ( ( µν 1 i 1 = 0. We see that the situations (2) and (3) preserves the | | = 0 : dimL Ker 6= 0 : dimL Ker F 1 n ’s with the Stueckelberg coupling, viz. the mass of the Proca 0 n i n µ Θ vac Θ c 0 01 0 01 | ) ) for all ∇ n n vac ( µ ( | , c , c n = 0 := 0 : dim KerL dim KerL † δA δφ M Q 10 0 01 0 = 0 6= 0 c c = n n X X 0 10 0 10 c c ). 01 n c = = 4.8 − µ δφ = δA space. Comparing them with the standard form of Stueckelberg system, we also ), ( = 0 or if 10 n i c 4.7 +1 The double degeneracy, as seen above, between spin-one and spin-zero fields is at the Putting together the field equations and the gauge transformations of Once again, the above results have close parallels to the supersymmetric quantum We can also determine the spectrum from the gauge invariances. In the waveguide, only d vac | AdS identify the coefficients field, core of the Higgs mechanism, much as in the flat space counterpart in section We recognize these equations as preciselyelberg the gauge Stueckelberg equations transformations of that motion describe and Stueck- a massive spin-one gauge field (Proca field) in eqs. ( Klein modes, we have We see that the relations eq. ( motion, can now be derived by the variations eq. ( those gauge transformations that doviz. not change gauge the fields boundary and conditionconditions would gauge make and transformation sense, hence parameters the ought same to mode obey functions. the So, same we boundary have mechanics. A vacuum Q supersymmetry, while the situationmodes (1) are present. preserves the supersymmetry only if the ground So we see that spectralFor asymmetry the is situation (1), present whenever the the index is elliptic zero. index For the situations (2) and (3), the index is nonzero. contribute to the index. In the situation (1), we have In the situations (2) and (3), we have Here, we used the fact that all Kaluza-Klein modes JHEP11(2016)024 . . 0 α and  ) = 0 2 (4.26) space. = α − d θ  +1 ( d 1 | 1 0 , we discuss in 7 ) in flat space, we see 2.7 is a product of two first-order ), which are valid everywhere ) s ( In this case, all possible boundary 4.19 = 0 Dirichlet 5 is a zero mode belonging to KerL α ), ( 1 |  1 = = 0 Neumann θ 4.18 | is a zero mode belonging to KerL α 1 | are the Stueckelberg coupling parameters.  1 0 | = 0 θ Θ 10 n | implies that the spectrum of spin-one mode is 2 1 – 19 – | , c ). We emphasize again that the above choice space, we can choose the boundary conditions as − 2 0 d or both is set to zero and so the corresponding 01 n − L Θ c d 4.8 +1 d 10 0 c , , andL 01 0 , c = 0 0 ], trivially satisfy the zero-derivative (Dirichlet) boundary ) does not give rise to massless fields and that the situations α ) follows from the first equation by consistency condition. = 0  = α α, α θ 4.17  4.15 | − = 1 | ), we see that θ 1 | 1 = [ Θ | 1 ) do give rise to massless spin-one and spin-zero fields, respectively. 0 4.15 θ L Θ ( 4.19 ), ( 2, as we shall show in next section, boundary conditions necessarily involve higher derivative ≥ 4.18 s For each of the above two boundary conditions, the mass spectrum is determined So, to have massless fields in AdS The double degeneracy and hence the Higgs mechanism breaks down for the ground For 5 Higgsed spin-one fields. Thehas ground vanishing mode mass of for Dirichlet spin-zero boundary field condition, and Θ there isterms no in massless order spin-one to field.detail accommodate origin The all and possible ground physical interpretation mass of spectra higher-derivative of boundary higher-spin conditions fields. (HDBC). In section that give rise toThe massless first spin-zero corresponds field to and thethe spin-one situation second field, case that respectively, corresponds Θ in to AdS the situation that Θ by the Sturm-Liouvilleof problem boundary eq. conditions ( put all Kaluza-Klein modes to Stueckelberg coupling, leading to in the waveguide condition and the one-derivative (Neumann)Moreover, boundary by condition, an respectively, argument at that similar the to situation the in reasoning eq.in around ( eqs. eq. ( ( we are now ready tosimplify examine and boundary systematize conditions thewhich these do analysis, mode not we contain functions derivatives shall higher must thanconditions satisfy. first first-order. can To concentrate be on related boundarymodes. to conditions all This possible is because choice of the ground mode modes functions eqs. with ( nontrivial ground Stueckelberg couplings vanish. In termsof of the the Higgs hidden mechanism supersymmetry, takes we place see whenever that the inverse 4.2 supersymmetry is unbroken. Waveguide boundary conditionsHaving identified for the spin-one mode functions field as well as raising and lowering operators relating them, to the massive spin-onecontinues (Proca) to hold field in when AdS the space.being mass a A of novelty Goldstone for mode, spin-one the is is AdS space massive. zero, is and that the thismode. scalar picture field, despite From eq.For ( the ground modes, either First, Kaluza-Klein modes ofsecond spin-one equation and in spin-zeroSecond, are eq. the coupled ( factorization together, property that suchelliptic the that differential SL operator the operatorsL ∆ equal to the spectrum of spin-zero mode. The spin-zero mode provides the Goldstone mode Stueckelberg coupling that realizes the Higgs mechanism follows from two ingredients. JHEP11(2016)024 ). (4.28) (4.27) space, 4.10 +2 d . Consider an . B 6 1) space. The set of − d +1 d 1), viz. , ) + ( 2)-module. 1 d . d, − } − ( 0) d ( so field by d, {z ( V ) that each of them corresponds s . We summarize the spectrum of Dirichlet | 2 1), respectively. Moreover, both are space are related to each other, which = ∆ (∆ 4.27 , ⊕ D 2 ) = 0 has vanishing mass for spin-one 1 } ` +1 1) α d − 1 ,  − 1 d ( ( 1 2)-module | {z − 1 0 2 spin D d, d 2). The conformal weight ∆ (the Casimir of Θ ( ( Neumann 0 – 20 – d, so D | ( 2)-module and representations in appendix so d, 0) and , m ( . 1) = ) so 3 d, d , ( 2) representation theory of AdS ) of 1 − ) for spin-one field in AdS waveguide. The left is for Dirichlet , s D d, − ( , s d (∆ n so ( D M V = ∆ (∆ 2 ` 0 ) are precisely the AdS counterparts of “vector” and “scalar” boundary − 2 spin 4.26 m 2) representations, d, . Mass spectrum ( ( so The mode functions of different spinspermits in AdS Stueckelberg structure. For spin-one, this relation is shown in eq. ( Summarizing, from Kaluza-Klein compactification of spin-one field in AdS Our result for the ground modes, which comprises of massless spin-one or spin-zero We summarize our convention of the • 6 (2) subalgebra) is related to the mass-squared of spin- we take following lessons. The pattern thatVerma ground module mode continues comes toour from hold main for the results higher-spin irreducible in fields this representations paper. as of well, reducible and is an integral part of The ground modes ofand Dirichlet and spin-one Neumann fields, boundary respectively.to conditions the are We massless see spin-zero fromirreducible parts eq. of ( the reducible Verma irreducible representation so conditions for flat spaceeach waveguide boundary studied conditions in in section figure fields, fits perfectlynormalizable to solutions the to the free field equation form a system of massive spin-one field. mode of Neumann boundaryfield condition,L and there isconditions no eq. ( massless spin-zero field. So, we see that the two possible boundary Figure 3 boundary condition to spin-onespin-zero component. component, The and red theKlein squares right are modes. the is Circles ground for modes, inside Dirichlet while the the boundary same blue rectangle condition circles have to are the the same Kaluza- eigenvalue and form a Stueckelberg JHEP11(2016)024 ¯ h ,  (5.2) (5.3) (5.4) (5.5) (5.6) (5.1) + 2)- 7 . d ) is the MN θθ ¯ ¯ h g ¯ h ( − , , , ¯ h MN and , N K = 0 = 0 = 0 ∇ µν h φ φ ¯ h M 2 3 ν ) = 0 φ . ∇ − − ¯ h ∇ d L = 3 + L 1 2 − θθ MN − d − ¯ g ¯ h L d KL space, and ¯ h L − 1 + L + ) L µν +2 d h µ − space: ∇ d g , A d MN 1 K space decomposes into equations µν µ µ ¯ h . This linear combination removes such g +1 d ( ∇ − − d φ A ∇ 2 +2 2 ) − d d 1 = ∇ h MN − M µν ν g µθ h ) + − ( 2 L 2 and ¯ h ρ + 1)-dimensional spin-two, spin-one, and ) 2 ) + ¯ ¯ h A − ∇ . After the compactification, the ( ¯ h ) in AdS − is given by d − ρ L φ NL µν ∇ , φ 2 +2 MN , ¯ h h MN µ  d − ¯ h µν – 21 – ¯ g µ d M θθ g 2 ,A ¯ h ∇ − + 1 ∇ ( MN is the metric of AdS L d µν − g µν 2 h )L + g ∇ µ − MN + h 1 L ¯ h A 3 MN + ν is defined by the linear combination of g − µν 1 − − d g ∇ . d ν L ML µν − + 1 ¯ h h µν + g − ν + 1) (2 d A N µν L 1) A 2 d µν ∇ h − ( µ 1 ¯ h L d − ( (2 ∇ − + 1 − ∇ ( φ/ = ( d ) µν e d d 2 ¯ h F − − − µν ( =  d µ ) h L µν − h ∇ ( g MN − MN φ ¯ h K µν   is the mass-squared, ¯ K 2 denotes for the trace part, ¯ ) = ¯ h M ¯ h mined by Stueckelberg gauge transformations.dimensional Therefore, equations we could of derivegauge motion the transformations. just lower- from consideration of the lower-dimensional It is known that free part of Stueckelberg equation and action are uniquely deter- ( The massless spin-two equation of motion in AdS The equations of motion have cross terms between • 7 MN cross terms. This specific combinationcompactification, is ¯ also the linear part of canonical metric in the original Kaluza-Klein Note that the spin-two field of motion for the component fields ( where dimensional spin-two field isspin-zero decomposed component to fields, respectively: ( where spin-two Lichnerowicz operator: K 5.1 Mode functions ofWe begin spin-two with waveguide the methodmotion for based a on massive the spin-two equation field of in motion. AdS The Pauli-Fierz equation of In this section, we extendbasically the the analysis same to as spin-twoground field the modes. in spin-one the We case, AdS shall but waveguide. presentcase the the The and analysis result idea highlight as is salient turns closely differences out parallel that more as begin interesting possible to for to show the the up spin-one for spin two and higher. 5 Waveguide spectrum of spin-two field JHEP11(2016)024 is +1 d 2 | (5.7) (5.8) (5.9) 1 n + 1)- (5.10) (5.11) (5.12) (5.13) d . ) θ ( 2 | 0 n Θ ) n ( φ n X , . = 2 2 | | . . 1 n 1 n Θ Θ ! ! 1) 2 2 2 2 2 2 n n , . | | | | 2 n 1 n 1 n 0 n − 2 2 . | | M M , φ Θ Θ Θ Θ 1 n 0 n m 1) − − ) Θ Θ 01 n 10 n 12 n 21 n θ c c c c − ( − = = 01 n 10 n 2 |

d n c c 2 2 1 n | | ( 2 n 1 n = = space. The mode expansion of ( Θ 10 n 01 n − Θ Θ = ( c c µ ! ! ) 12 n 21 n 2 2 2 2 12 n +1 n = = | | | | +1 ( c c c d 1 n 0 n 2 n 1 n 2 2 | | m A 21 n Θ Θ Θ Θ 21 n 12 n 1 n 0 n – 22 – L c ) leads to the first set of Sturm-Liouville problems c c Θ Θ n 1 = X 1 3 − = = 5.8 ! ! n − − 2 3 2 2 d = | | 01 n L L − − 2 n 1 n c L 0 0 , in AdS d d µ 3 − Θ Θ 1 10 n − µν c 2 2 d − n h 2 1 − − L L L − − d 0L 0L L µν L L L m g 2 ,A L 2 ), we again expect relations among mode-functions which can

− ) d − θ L L 5.6 ( 2 | 2 n ), ( Θ 5.5 µν ) n ( ), ( h ) leads to the second set of Sturm-Liouville problems for spin-one and spin-zero, 5.4 n X is the trace part, 5.9 ’s are pure imaginary coefficients. Compared to the previous section, we now have = n h c µν h It should be noted thatues the is difference linearly between proportional spin-twocurvature eigenvalues to scale). and the spin-one eigenval- spacetime dimensions (measued in unit of the AdS the relations This also leads tosets relations of to Sturm-Liouville the problems. two sets of eigenvalues that appear in the two separate The two sets ofthe equations eigenmode appear that participate overdetermined, in asbe two shown the separate that spin-one Sturm-Liouville the two problems. mode Sturm-Liouville However, problems function it are Θ can actually related each other upon using The eq. ( respectively: of Sturm-Liouville problems. The eq.for ( spin-two and spin-one, respectively: Here, two sets of raisinganother and for lowering connecting operators, spin-one one and for spin-two, respectively. connecting Accordingly, spin-zero we and have two spin-one pairs and be summarized by the following two first-order coupled differential equations dimensional spin-two, spin-one and spin-zero component fields reads From eqs. ( where JHEP11(2016)024 and 2 (5.15) (5.16) (5.14) | 2 n, M . As these are fields that di- ). It is only that 2 | ), indicates that 1 with examples. 5.5 , are decomposed +1 were nonzero, the } n, d θ 01 n 2 | 5.14 5.2 M c 1 . , ξ ) 10 n µ n, c θ ξ ( { M 2 | 1 n = , 1) = Θ 12 n and θ ) M c ξ ¯ − n ξ 2 ( θ | 2 21 n d ξ 2 c − or vanishing d n, + 2 n = L | X 2 n 2 , M 2 n λ µν µ n, λ ( = If ξ g θ − − 2 M 1 8 − = = 1 L − 2 2 | | d 1 2 ). 2 1 2 2 n, n, + + – 23 – 5.4 . ) θ , ξ M M θ ν ξ ) ξ ξ − − θ µ 1 µ ( ∂ ( 2 − | 1 2 2 n ∇ : : ), which is already reflected in eq. ( Θ = = =L ) from the method based on gauge invariances. The gauge ) n 2 1 µ 5.13 ( µ µν − − δφ ξ 5.14 L L δA δh space, with the gauge parameter n 2 2 2 | | | X 1 n 0 n 2 n   +1 = Θ Θ Θ d is not related to the mass-like term of spin-one field in eq. ( µ 2 3 2 | ξ 1 − − d d n, L L M + 1)-dimensional fields of adjacent spins, are precisely the structure required is the mass of spin-two field in eq. ( 2 d | 2 n, . This case will lead to massless limit of all component spin fields. M 2 | To retain the gauge invariances, the mode functions of gauge parameter must be set We can also obtain eq. ( Again, there are two special cases, vanishing We can summarize the coupled Sturm-Liouville problems by the following spectrum 1 Note, however, = 8 n, n the same as the mode functions of fields: M We should stress thatagonalize these them component at fields linearizedthis are basis. level. in The the gauge basis transformations of given AdS above are those in massless spin-two fields, weThere will is analyze also themM a separately special in case section of these two,transformations namely, in simultaneously AdS vanishing into components the Stueckelberg mechanism actually involvethe the present whole case, tower this of means that componentin the spin the spin-two, fields. Higgs spin-one mechanism. and In spin-zero fields are all involved important exceptional situations, leading to a new phenomenon involving so-called partially corresponding modes amongspin-two different system. spin From fields theseanticipated complexes, combine from we and the can flat become draw space twotwo the intuition, pieces adjacent the Stueckelberg spin of Stueckelberg fields, mechanism information. would onezero. First, work for between Second, as spin-two the and relation spin-one eq. and ( another for spin-one and spin- As in the spin-one counterpart,between these ( complexes, defined by raisingby and the lowering Stueckelberg operators mechanism of spin-two field. generating complexes JHEP11(2016)024 . ), 1) − 5.14 (5.21) (5.19) (5.20) (5.22) (5.17) (5.18) d + 2 . n ) , , . n M ( ( ξ ). = 0 = 0 = 0 1) 10 n ξ . ) ) ) c − 5.6 d n n n ( ( ( d 1) = h φ φ 2( ), ( ) − ν µν 10 n n ’s as ( d c g ∇ r ( 5.5 n 1 i c 1 2 d 21 n ) = c n d − d ), ( ( − r . 10 n d h , δφ + d ` 1 ) 5.4 µ 1) 12 n µν n ), coincides precisely with the , c ) ( 01 n c µ g = 2 n c ξ − ( ` 1) 01 n − c d A − 5.20 − 21 n 2 ( c µ d i µν + ) − ) ∇ + + n ), ( i n ) ( ( 2 ρ = n ) h 10 n n h ( n c h 2 n ξ ( ξ , δ φ 5.19 M µν 2 λ µ A ( g µ 12 n ∂ ρ c − ∂ ), ( 1) − 1 2 ) ∇ 1 2 21 n n − – 24 – d ( c µν space, once we redefine µν = d φ = ) g 5.18 ) + n ), we can fully reconstruct the linearized field equa- 2  2( µ ( n +1 i ( µ h d ) − ), ( s n h + 1 ( µ 5.20 − ) µ h d ) and comparing mode expansion in the gauge variations, n 5.17 ( ∇ = , δA µν + A = 0 and ) g ). In practice, the gauge transformations are much simpler = 0, the gauge transformations are 21 n , δ A ν 01 n n 5.15 c 10 n ( 2 n ) n − c ∇ ξ ν λ λ − ξ , c 5.19 01 n + µν ν µν µ n 2 c ) ( ) g ν n n ) 1 √ ( ), ( M ( ∇ 1 n h ( ), corresponding to the unitary gauge fixing. However, such gauge 12 n + 1 − − = = 2 c A d A d d d h 5.18 µ 2  21 n d µν 5.20 + ∇ ) − h ) ), ( − n − ( δ h ν , c ) ) ξ n 12 n n ) µν ( c ) µ n 5.17 ( ( n M φ − ( 2 h ∇ (  F √ µ = µν − ) ∇ K For the situation that Before classifying possible boundary conditions, we summarize the Stueckelberg spin- Time and again, the linearized gauge invariances uniquely fix the linearized field equa- After the mode expansion, the component field equations read n = ( µν δh 12 n c At these special masscan values, be the deduced Stueckelberg just system from breaks the gauge into transformations. two subsystems We now which elaborate on this. is because thesymmetries spin-one eq. and ( spin-zerofixing fields is can not be possible were algebraically if removed the by masses take the special gauge values: undergo the Higgs mechanism for massive spin-two fields. two system and the Goldstonemasses, mode the decomposition Stueckelberg pattern spin-two of system it.a describes the massive For general same spin-two values physical field, of degree the having of freedom the as maximal number of longitudinal polarizations. This earized gauge transformations eq. ( tions eqs. ( to handle than theily field using equations. the From linearized nowraising gauge on, operators invariances. we or Note shall nonempty that analyze image the the of modes spectrum lowering primar- with operators nonempty always image combine of together and tions or equivalently the quadratic part of action. Therefore, from the knowledge of lin- Their gauge transformations are We see that thisspin-two Stueckelberg system, system eqs. in AdS ( we see we canwhich recover was precisely previously the derived same from the raising field and equations lowering eqs. operators ( as in eq. ( By substituting these to eq. ( JHEP11(2016)024 and (5.24) (5.23) (5.25) (5.26) 12 n . This c 2 . The gauge d/` , 10 . ) = 0 µ = 2 d 2 − ξ . , the Stueckelberg i A 2 m 1  , δφ /` ). 2 − . = µ 1) ξ d } µ . ` 0) ˜ 5.19 − 1 A = ∆ (∆ B r , 2 =0 ` d s 2 , ( 1 ` } = 2 − {z + 1 2 − 1) , ) implies that one of the two 0 λ d , =1 d = ( s − 5.8 massive 2 n r | {z + 1 2 spin M ⊕ D + d ( m ξ massive =2 and the fields | µ s } ∂ 2 2) where | , as given in eq. ( ⊕ D 1 2 1 n , 2 1 Θ , a subtlety arises as the coefficients /` = 2 i } =2 {z − s 2) µ  /` – 25 – d ( λ , d, + 1) 1) {z = ( D | d µν 2 | − g D | 1) and 1 n massless partially massless d 2 ˜ = ( 1 ( Θ ξ , δA ` − 2 − d ]. We can always gauge-fix the spin-one field to zero, and µν − m g 2) = 2) = = λ 15 2) are the irreducible representations of massless and par- , 2) for spin-two field. For the special values of conformal ` 1 1, the Verma module becomes reducible and break into ν 2 n , 1 d, , ) + ( 1 ( λ and corresponding field become pure imaginary. We are thus 1 d − ∇ − 2 (∆ V − 2 µ d | − − 9 d V 2 n d ( ∇ d ( V r = ,Θ D 2 | + 1 n µν 11 ) ]: ν = ∆ (∆ and ∆ = δh ξ 2 13 µ ` d , ( 2)-module 1 2) and ∇ 12 − d, ( d, = ( 2 spin so D µν m This spectral decomposition pattern perfectly fits to the reducibility structure of the For the situation that are pure imaginary. Specifically, the relation eq. ( δh Note that the normalizationIn of the each path integral field formulation, is thisHere, amounts not to we the choosing that define standard the form. the integration contour mass-squared purely imaginary. equal to the mass-squared in flat space limit. Therefore, it differs 9 10 11 21 n from the mass-squared dictated by the Fierz-Pauli equations. See appendix Here, tially massless states, respectively.conformal Using weights the relation between the mass-squared and the Verma weights, ∆ = Therefore, as the mass-squared hitssystem the breaks special into a value spin-twospin-zero partially-massless field (PM) of Stueckelberg mass-squared system and a massive massless (PM) spin-two field [ the remanent gauge symmetrysymmetry coincides [ with the partially-massless (PM) spin-two gauge We see that the above redefinitionmations does and not spin-one alter gauge the transformationsthat fact are these that are coupled the precisely each spin-two gauge the other. transfor- gauge In transformations fact, for we the recognize Stueckelberg system of partially c mode functions Θ led to redefine the modetransformations functions now become field as it has theequations spin-two precisely diffeomorphism invariance. constitute We theimplies also spin-one see that that Stueckelberg the the system Goldstone remainingsystem, with field two which of in the massive turnmassless spin-two was spin-zero is fields. formed given by by the the massive Stueckelberg spin-one system of massless spin-one and We see from the first transformation that the spin-two field ought to be massless gauge JHEP11(2016)024 ). 2 2 /` /` 5.25 2 , and (5.27) (5.28) 2 1) d/` − , d/` + 1) = 0 d } } } ( d 2 = 2 − m 2 = 2 = 0. Then, the = ( = 0 = m α 2 2 | mass-squared 2  2 = 0 | m m = 0 | m θ 2 . This result exactly | = 0 | Θ 2 2 0 | | 3 2 1 | /` Θ − 0 d 3 , . L Θ ,Θ − d 2 + 1) +1 L − = 0 = 0 d d d α α L   12 = ( = = θ θ 2 | | , , 2 2 | | m 2 0 , Θ Θ = 0 = 0 2 3 | | 2 − − = 0 2 | . | d 1 | L field 1 L 2 | Θ Θ 1 5.2 2 1 , , – 26 – Θ − − d 2 2 | | L L 1 n 1 n massless spin-two ) immediately imposes unique boundary conditions Θ Θ , but can be made self-adjoint in a suitably extended functional massive spin-zero field 2 ∼ ∼ 5.14 , L + 1 = 3 component fields (spin-two, spin-one and spin- 2 2 . | | massive Stueckelberg spin-one , s 2 n 0 n 7 = 0 , Θ Θ | 2 3 = 0 2 ) uniquely fixes the boundary conditions of all other component | partially-massless Stueckelberg spin-two − − | 2 d = 0 2 L | 1) corresponds to the spin-one field with Θ | L 2 , 2 2) 5.14 2 | Θ 2 2) 0) 1) d, − ( 2 , , , L Θ + 1 2 ) 1 − so , we tabulate the four types of fields that appear at four special values 1 ) d L ( − d, − + 1 + 1 1 , s ( L d d d D ( ( ( D { { { (∆ D D D D 0) corresponds to the spin-zero field with , . The types of field involved in the inverse Higgs mechanism when a spin-two Stueckelberg + 1 Here, in table B.C. 2: B.C. 3: B.C. 1: Note that the first and the third conditions are higher-derivative boundary conditions(HD BC). HD BC type d ( 12 type I type IV type III type II is not self-adjoint in thespace. functional space We will explain this in section generating complex eq. ( fields. The simplestcomponent choice fields. is As to therezero), impose there are the are Dirichlet then three boundary possible condition boundary to conditions: one of the Likewise, if we impose a boundary condition to one of the component fields, the spectrum ponent fields (spin-one andcontinues spin-zero in to that hold case) forboundary were the condition related for spin-two one the another. situation. spin-onespectrum This component generating For feature field complex instance, in eq. AdS suppose ( for we other impose component fields: Dirichlet They will be shownappropriate to boundary conditions arise in as section the ground5.2 modes of the Waveguide Sturm-Liouville boundary problems conditionsWith with for the spectrum spin-two field generatingboundary complex at conditions. hand, In we the now spin-one move counterpart, to boundary classification conditions of for possible different com- one finds that D matches with the spectral decomposition patterns we analyzed above. of masses, corresponding to the four irreducible representations that appear in eqs. ( Table 1 systems decompose into spin-two gauge fieldof and spin-zero Goldstone and field. spin-one. Typespin-two I In gauge and AdS field. II space, are Type Goldstone these IV fields Goldstone is fields partially are massless, spin-two massive. gauge Type field. III is massless, JHEP11(2016)024 = 0. . We ). (5.32) (5.33) (5.29) (5.30) (5.31) α n z  = 2, viz. = 1. θ 5.27 | d − . 2 n . z 2 2 | | 1 1 = 1 and = 2) | n 2) 2 | n z (0 (0 , M α as 2 | odd Θ even Θ 0 odd parity even parity and odd Θ and even Θ 2 | 2 | 2 , , , , 2 α α , depend on the waveguide size n   )) )) z θ θ = = θ θ n n | | z z odd Θ even Θ , this spin-zero boundary condition is )) )) ). n θ θ z sin( n n cos( , , z z n n 5.28 α α z z = 0 in this case. The spin-one and spin-zero   ). This is not accidental but a consequence ) and the boundary condition eq. ( sin( = = 2 cos( − | θ θ 2 | | ) n ) + n 5.32 θ – 27 – z odd parity even parity odd parity even parity θ z 5.14 )) )) . Substituting the above mode functions to the n n θ θ z − z n n , , ) , , ) + z z , so the modes are also classifiable as either odd or θ ) ) θ ) ) θ θ θ n cos( sin( θ θ n is sent to zero. They also correspond to the “Kaluza- z n n ), are elementary functions: z n n times the Janus wedge, where the mode solutions of the sin( θ θ cos( z z z z B.C. 1 α 3 n n → − z cos( sin( z 5.14 sin( cos( sin( cos( (tan (tan θ θ θ − ). We now examine mass spectra and mode functions for each θ θ θ θ θ θ ) ) + θ θ to AdS n n 5.14 sec sec sec sec sec sec (tan (tan 4 z z θ θ ( ( ( cos( sin( = = = sec sec 2 2 2 θ θ | | | n n 1 0 2 z z Θ Θ Θ ( would blow up as (tan (tan + 1. Note that the Sturm-Liouville equation and the boundary condition are . They are the AdS-counterpart of flat space compactification volume, and n θ θ 2 n α λ λ is a shorthand notation for the boundary values of mode functions, Θ ) at which the unitary gauge-fixing ceases to work and the Stueckelberg system sec sec | = , we get the same expression for spin-two and spin-one component fields except the = 1, the spin-two field is absent as Θ 2 ( n and z n 5.21 In general, solutions for each boundary condition, There are, however, two special limits that are independent of We begin our analysis with To deliver our exposition clear and explicit, we shall analyze in detail for z n z They correspond to “groundthe modes” Kaluza-Klein and modes. have First, interesting masses featureseq. of that ( the are ground modes not are shareddecomposes equal by into to subsystems. the special Second, masses For mode function of some spin components are absent. parameter, so Klein modes”. For thesetogether modes, and mode form functions spin-two of Stueckelberg system different with spin mass-squared, component fields couple We note that, modulothe the same overall spectral as the factor of boundary the condition spectrum eq. generating ( complex eq. ( We also get the boundary condition for spin-zero component Θ even under the parity. B.C. 1 condition that the parity of mode functions must take opposite values: with symmetric under the parity Sturm-Liouville problem, eq. ( reiterate that the boundary conditionsgenerating on complex each eq. set ( are automaticallyof fixed the by the three spectrum types of boundary conditions, eq. ( compactification of AdS where Θ JHEP11(2016)024 from (5.34) (5.35) . space. In = 0, only B.C.3 n +1 d z and , type II . For B.C.2 . , 7 are depicted. Each point type III type IV type I ··· , B.C.1 2 , 1 deserve further elaboration, as θ , and hence negative-definite for all type II type III type IV = 0 tan 2 5 4 n N N α B.C. 3 2 and and and = tan 2 | 1 0 − = 0 is not a solution of boundary conditions Θ n z – 28 – and 2 3 θ , , and offer intuitive explanations of them in terms of α N 2 tan 7.2 θ − θ 2 sec sec are 2 3 5 | B.C. 1:B.C. type 2: I B.C. type 3: II type III have a wrong sign for their kinetic term. We will further confirm 2 1 N N . = = 4 2 2 | | 2 0 2 1 and Θ B.C.3 Θ Θ 2 | 2 0 ( . Such negative norms indicate that the higher-spin fields associated with these α . Spectral pattern for three types of Dirichlet conditions, , we will explain Kaluza-Klein origin of this non-unitarity. Second, norms of some ) or corresponding mode function is 0. This spin-zero field is of 7 5.32 The ground mode spectra associated with Completing the analysis for all possible boundary conditions, we find the following choices of ground modes in this more directly inextra subsection boundary degrees ofdefinite freedom Hilbert needed space. for extending the indefinite Hilbert space to a mechanically to negative probability. Explicitly, for the mode functions the norms of Θ they in factsquared describe is a below non-unitary the Breitenlohner-Freedmansection system. bound of spin-two First, fieldmode it in AdS functions is are non-unitary negative-definite,of because implying indefinite the that metric, mass- the leading Hilbert classically to space unbound has energy the and structure instability and quantum We see that B.C.1 keepsspin-zero mostly spin-zero, and B.C.3 spin-two. keeps mostlysummarized spin-two, in The while figure B.C.2 complete keeps spectrum of each set of boundary conditions is spectrums of ground modes: worth comparing this with the pattern for arbitrary higher-spin infields figure combine and formmassive spin-zero the field Stueckelberg is spin-oneeq. present system ( because of Figure 4 left to right.represents The one spin mode: contents squares ofPoints are inside each from excitation the ground level same modes, rectangle while have circles the are from same Kaluza-Klein eigenvalues modes. and form Stueckelberg system. It is JHEP11(2016)024 (6.2) (6.3) (6.1) ) are ]. In (5.36) 8 B.C.1 . 5.28 λν λ of higher-spin w . s ν space is α ∇ , ) 3 1 +1 µ d = 0 ∇ 2 tan λ 2 ) and spin 3 µ r 1 d µ λ µ λ ( = space: the normalized mode λ in AdS space. In other words, the ) g w 2 3 3 2 1 2 µ +2 µ µ d +2 2 1 at the special mass values. w d µ − µ are related to each other by the 2 1 ( 1 ,N λ µ µ g ) α 3 w +1 ∇ 2 µ d 1 1 λ µ √ ( 2 + 1) w ` 2 d ∇ = µ 1 1 + 2 ( ), whose structure is uniquely fixed by the µ ( N ν ) g + space. In this limit, though, the mass spectrum 3 2, the boundary surface of the waveguide ap- ν – 29 – 3 µ 5.14 2 µ , ∇ π/ 2 +2 µ , d µ ν w 1 µ 1 µ w ( − ∇ ∇ νλ ν ) type II type III . It contains the massless spin-two ground mode as well as tends to 2 3 + 1) = 0. As we will demonstrate below, this complication forces ` µ d vanishes in the decompactification limit. This explains why α w − ∇ µν 2 2 4 ( 3 θ , ζ µ µ θ , N 2 1 2 − µ B.C. 2 µν µ ( ) 1 g g sec sec µ w λ 1 2 ( w 0 ∇ ν N N K ν ∇ space. The reason is that some of the boundary conditions we choose are = = ∇ ν 2 2 | | +2 ∇ 0 0 2 0 + d ) is the spin-three Lichnerowicz operator: Θ Θ , we will show that, for arbitrary spacetime dimension w ( ) = ( 0 w 8.5 K ( At special values of masses,ducible the representations of Stueckelberg spin-two massless systemGoldstone or decomposes fields. partially into massless The irre- spin-two groundprecisely fields modes these and of irreducible massive Dirichlet representations boundary in conditions table eq. ( The mode functions ofspectrum different generating spins complex inconsideration eq. of AdS ( Kaluza-Klein compactification of higher-spin gauge transformations. 0 The field equation for a massless spin-three field Summarizing, As the waveguide size K • • where needs to be traceless,us ¯ to consider appropriate linear combinations of gauge parameters. 6 Waveguide spectrum of spin-threeIn this field section, we furtherextrapolation extend of our analysis lower to spins.higher, spin-three field. Compared a new to This is technical the not complication situation a begins of mechanical to lower spins, show up: for spin-three the or spin-three gauge parameter in the decompactification limit. These ground-mode functions are notfunctions normalizable disappears in AdS as there is no masslesssection spin-two field in thefield, “dimensional the degression” mass spectrum studied of in “dimensional ref. degression” [ spectrum is the spectrum of the mass spectrum for spin-zero ground mode whose normalized mode functions are proaches the timelikeJanus asymptotic wedge boundary decompactifies to of thefor AdS the each boundary AdS conditions does notfield necessarily in gets AdS to the spectrumsingular of in massless this spin-two limit as the associated mode function becomes ill-defined. Take for instance JHEP11(2016)024 .  λ θθθ ) (6.7) (6.8) (6.9) (6.4) (6.5) (6.6) 3 (6.10) w µ λ = w 2 µ 1 µ , φ , ( g + 1)-dimensional µθθ . − = 0 w d 3 θθ φ µ ¯ ζ 2 = 4 3
µ − 1 µ d µ ≡
L + 1 d w 3  d − h ,A 4 d 2 + + 2 − L µ 2 2  1 θθθ L n n ) , ξ 1 µ λ ρµ 3 2 2 n w g ρ λ M M µ − ) + 2 − µθ 2 2 M 2 d w ¯ ζ µν µ d d µ − , 4 g g − − − d 3 2 , decomposes to ( w − 1 L 3 + L +  1 = = = L 1 ≡ 4 ) + 1 = 0 µ M − λ 3 µ 3 3 3 ( ρ 2 ) − | | | ( µ d 3 − φ 3 2 1 λ L A M µ 2 2 2 1 µ n, n, n, h ρ + λ 3 2 ) can also be derived from the Kaluza-Klein − ∇ M λ ∇ + L + w ∇ M M M 2 , 2 4 − w ∇ µ µ µνθ − − − µ 6.9 – 30 – , ξ − 1 µν 1 4 h A d w ), to remove cross terms in field equations between g µ 2 θθ = 0 ( λµ L . φ − 2 ¯ µ ζ g = ) 1 w 1 5.3 3 h µ − : : : λ µ + 1) µν 1 g µν = 0 + 1 − g µ µ + 2) 2 4 3 d A ∇ ( + 1) λ d d A − − − 2 d ∇ 3 ( µ d A L L L ν 1 1 + 4) − + 1 , h  + 2 2 ( λ µ 3 3 3 3 ∇ − ) | | | | ( d 2 d ν θθ n 0 n 3 n 2 n 1 ∇ h g ) − µ −    + ρ + 2 ( 2 d 3 λν µ Θ Θ Θ Θ + A φ ν 3 − w − w 3 4 4 2 3 µ + (3 d A µν 2 λ − − − − − L ¯ 2 µν ζ µ 3 L 3 µ d d d d d − ∇ ( µ 1 +L 3 2 ∇ g L L L L L 1 µ ∇ ≡ ) − 2 − µ ρµ ( 3 2 3 d µ w h h h µ 3 1 − − L 1 µν 2 +1 µ 3 ρ L L − ξ µ d g d − ∇ h  L L + 1) ( + 1) 1 + 1 4 + 2) 2 3 + 3 + 1 µ d d + 1 − ( + 2) − − d − d d L + 2 + 1 d 4 ( d ∇ µνρ 4 ( d 2 d d 3 ( 1 2 L 2 3 − w − 2 ( + 2)-dimensional spin-three field, − ) − +L 3 − + − − ) ρµ = d − h w F We now show that the complex eq. ( Repeating the spectral analysis as in the lower spin counterparts, from the structure ( ( φ ρ 0 0 µνρ  ∇ K K w compactification of spin-three gauge transformations.nize After the the component compactification, gauge we parameters orga- as of the above fieldfollowing equations, spin-three we spectrum find generating that complex: mode functions are related one another by the parity-odd spin-three and spin-one,each respectively, component parity-even fields, spin-two the and field spin-zero. equations read For component fields of different spins: We defined component fields ascounterpart linear of combinations spin-two situation of eq. different ( polarizations, which is the The ( JHEP11(2016)024 . } } } } 7 = 0 and = 0 α (6.11) (6.12) (6.13) |  3 = 0 | = | 0 θ 6) can be 3 = 0 | | Θ 3 | 1 B.C.1 | 4 − 3 = 0 k | Θ − | 0 d d 4 3 | L Θ − ) then read 0 d 3 4 L Θ − − d d 3 6.4 L L − , d 2 space. 3 ξ , L | − 1 d ]. The complex in turn 3 of index ( +1 L − Θ d 4 , which involved just the 15 d 3 − L , , , , − L d 2 µν L g − = 0 d B.C.2 ∼ | . 3 = 0 1) | = 0 3 3 | | | 1 | 3 1 + 2) 3 = 0 − 3 | Θ | | 1 Θ Θ , d 2 3 1 3 ) 4 2 | d Θ − ρ Θ 1 ( 2 ( d 3 − − ξ 2 L Θ L L − 2 − d 2 + L − L − ∼ ∼ d d µν L 3 3 . In all cases above, one of the four component L | | ξ 2 0 α 4 µν – 31 –  Θ Θ ( − , , , , µ = g L ξ θ | 3 1 3 = 0 − + 3 | , = 0 = 0 + , 3 ) 3 | | ξ . | | ) 3 2 3 = 0 2 | 3 | ν νρ + L + 2 | 2 | 2 ξ ξ Θ 2 − Θ ξ 3 4 | Θ µ µ Θ 2 Θ 2 ( µ ( − 2 3 3 − ∂ ∂ ∂ L Θ − d − − ) automatically imposes boundary conditions involving two or d 3 L L L =L 3 = = = L − 6.9 L µ ∼ ∼ µν δ φ µνρ 3 3 | | , , , , δ A 3 1 δ h δ w Θ Θ = 0 ). Moreover, after the mode expansion, these gauge transformations | = 0 3 | | 6.9 3 = 0 0, we automatically get boundary conditions for other component fields from 3 | , | Θ 3 = 0 3 1 4 | | , Θ , there was one choice of boundary condition, 3 − 3 2 4 | L Θ implies the boundary value: Θ , 3 − 4 3 | ): L for spin-two field involved two derivatives to some other component fields. For exam- Θ 5.2 − − 3 = 3 L L − 6.9 The pattern of mass spectra is similar to the spin-two situation. All Kaluza-Klein In identifying possible boundary conditions, another new feature shows up for the spin 2 k L − L { { { { Here, Θ fields is subject to higher-derivativegeneral boundary pattern condition but (HD relegate BC). reasons We and claim intuitive that understandings this exclusively ismodes to form a section spin-three Stueckelberg system. The ground modes for each of the four possible three derivatives to some otherB.C.3 component fields, just like boundaryple, conditions if we impose Dirichletfor boundary condition for theeq. field ( associated with Θ starting from three:any higher-derivative choice boundary of conditions boundary (HDBC)section are conditions. unavoidable for Recallone that, derivative and hence for standard the Robincondition boundary spin-two condition. is situation imposed Once the discussed to Dirichletgenerating any in boundary complex of eq. the ( component fields, the structure of spin-three spectrum can be identified withprovides spin-three for Stueckelberg all gauge information needed symmetry for [ For extracting instance, mass spectrum the after eigenvalue the of compactification. identified Sturm-Liouville with operator minusL the mass-squared of spin-three field in AdS We then see that thesecomplex, relations eq. give ( rise precisely to the spin-three spectrum generating Mode expanding and equatingtions terms among of the the mode same functions mode as functions, we readily extract rela- later simplicity. The gauge transformations of component fields in eq. ( such that the spin-two gauge parameter is traceless. The numerical factors are chosen for JHEP11(2016)024 2 2 2 2 and /` /` /` /` 2 n d/` + 1) + 2) + 1) + 2) 2 = 0 d d d d − ( 2 = − m 2 = 2 ( = 3 ( = 2 ( = mass-squared m 2 2 2 2 m m m m 2) 0) (6.14) , , 1) + 2 + 2 , . d d ( ( 2 + 2 d ( ⊕ D ⊕ D 3) 3) , ⊕ D , 1 3) + 1 − Field types d d, d – 32 – ( ( ( massive spin-zero massless spin-three D D D Stueckelberg spin-one system Stueckelberg spin-two system 3) = 3) = 3) = , , 1 d, ( . Each point represents a single mode function. Points in the same = 3, there are three Verma modules: + 1 − V s s d d ( ( Stueckelberg PM spin-three of depth-one system Stueckelberg PM spin-three of depth-two system V V 2) . , 3) 3) 0) 1) 2) (d 3 , , , , , 3) 1 s0 ) d, − + 1 + 2 + 2 + 2 ( , s d d d d d D ( ( ( ( ( (∆ D D D D D 2)-module. As D d, ( . Mass spectra for spin-three higher-spin field. The horizontal line labels the mode . The type of irreducible representation fields upon inverse Higgs mechanism of spin-three and in table so 5 We see that the pattern already observed for the spin-two field repeats in spin-three The patterns of mass spectra for each possible boundary conditions are depicted in The first three types are Goldstone modes, while the latter three types are massless type type I type V type II type VI type IV type III field. For theyielding a Kaluza-Klein tower of modes, spin-threefield all Stueckelberg with system. Dirichlet boundary boundary For condition the conditions isfields ground absent yield fill modes, from up the the a massive spectrum component universal fields while other and pattern, component massless or partially massless spin-three gauge fields. classes of partially massless fields.depth-one If gauge the field. Stueckelberg If systemgauge extends the field. to Stueckelberg spin-two, system it extends is to the spin-one, it is thefigure depth-two From the relation betweentype mass-squared of fields and that conformal are dimension, present as we ground identify modesor possible in partially table massless spin-three gauge fields. As the spin is three, there are two possible Verma Figure 5 the vertical line labels therectangle spin form the Stueckelberg system. boundary conditions comprise of the irreducible representations of reducible spin-three Table 2 Stueckelberg systems. JHEP11(2016)024 (7.1) adjoined ] D N R ⊕ ) D ( . 2 2 L q [ type V G type IV type VI T 1 type III 2) module (though, in dS q ) class of square-integrable ∈ H d, and D ) + ( and ( and i 2 1)). Is it possible to trace the and so , 2 ∂D L ( space decomposed into component defined and, lacking orthogonality q , ψ + 1 1 S ⊗ ψ | d +2 s n ( ) h d type IV type V D so type III type II ( := Ground modes , , i ψ i , , 2 Ψ ) = , – 33 – 1 D Ψ h type I class of square-integrable functions over Ψ( type IV type III type II N h R space become partially massless and have kinetic terms ⊕ space? ) −→ −→ +1 d D ( +1 = 0 = 0 = 0 = 0 d 2 2 )] L α α α α ψ     1 D ( to = = = = ψ 2 θ θ θ θ | | | | L D D 3 3 3 3 [ | | | | Z 0 1 2 3 Θ Θ Θ Θ := ∈ H i Boundary Condition 2 ) D , ψ ( 1 ψ . The contents of ground modes for each of the four possible boundary conditions. ψ h -dimensional vectors of finite norm: N In this section, we answer these questions affirmatively positive by providing mathe- There is one more issue regarding the HD BC. We observed that, for HD BC’s, some Table 3 with matical foundation and elementary physical setupstandings. from which The we idea can is buildeigenvalue-dependent intuitive this. boundary under- In conditions), Sturm-Liouville we problemextending can with the always HD inner ensure BC product the (andfunctions defining self-adjointness over closely domain by Hilbert related space from origin of thisconditions non-unitarity are from tradable the withyield boundary field interactions the equation such boundary and that conditions. variationpartially HD of massless Is BC? the fields it interactions In from thenthe general, non-unitarity boundaries? possible of boundary to the treat boundary the action origin of of modes non-unitarity localized of at of the component fields inof AdS wrong sign.AdS This space ties belong well tospace, with they non-unitary the belong representations fact to of that unitary the representations partially of massless higher-spin fields in of Sturm-Liouville eigenfunctions aremake neither a orthogonal sense nor ofof complete. the previous Kaluza-Klein If sections, compactification so, how withand how are HD completeness, do the BC? how we mode are Stated higher-spin functions infields fields Θ the of in various approach AdS spins in AdS derivatives of higher-order (higherSturm-Liouville than problem, first such higher-derivative order, boundarynon-standard Robin-type). condition but (HD From also BC) the potentially isis viewpoint problematic. not second-order of only The in Sturm-LiouvilleHilbert derivatives, differential space so operator of the square-integrable operator functions if in HD general BCs are fails imposed. to In be this self-adjoint case, in the the set 7 Higher-derivative boundary condition In the previous section, we discovered anary emerging pattern conditions that for the Janus higher-spin waveguide bound- fields unavoidably involve boundary conditions containing JHEP11(2016)024 , . 2 , ` ∂D 2) , k , (7.2) (7.3) ) = 0 t ( 2 x . = 1 , y a 2 ∂D ( M
2 a and y 1 a k , k ) t − ( , 2 a 1 ˙ y
2 , y a , 1 M M we introduced for the extension and vibrating with vertical am- 1 2 ` N particle = R L ≤ , which are direct counterpart of the a + , x 1 , ≤ B.C.3 particle From the viewpoint of the open string, its L particle and 13 L – 34 – ]. + -dimensional vector space. We shall refer to the as “extended inner product”. The point is that 24 and i N -direction 0 i x B.C.1 , string
h 2 L h ) y t x d ∂ ( Z − = 2 . The system is described by the action ) I y 6 t ∂ ( , stretched along x T . Thus, defining appropriately chosen extended inner product, one can d l Here, we will study this system in three difference ways and draw physical 0 ∂D ). String’s end points are attached to harmonic oscillator particles at ] that the Sturm-Liouville problem with HD BC can always be made self- Z ]. 14 2 T 25 x, t 23 specifies the metric on the ( , = y 22 G As the first approach, we start with boundary degrees of freedom, integrate them out, Utilizing this idea, we shall show below how partially massless higher-spin fields can This example was consideredWhat in is detail lesser at known [ is that this is completely equivalent to the statement that the particle dynamics string 13 14 L involves higher derivatives ingrounds time. in [ This is carefully discussed for both flat and curved spacetime back- where the Lagrangians of open string and massive particles are and convert their dynamicsstring to of HD tension BCplitude for the open string.whose masses, Consider vertical positions a andrespectively. relativistic Hooke’s See open constants figure are zero, the string endpoints move freely.that What endpoint is particles less with obvious a and finiteconditions. also mass less put known, the however, open is stringinterpretations to in higher-derivative each boundary case. We willfor then constructing construct a boundary refined action inner product from and HD a BC procedure and vice versa. The first case werelativistic study massive particles is at themotion each is classical subject ends. field to boundary theoryexert conditions. boundary of It conditions an is that intuitively openmasses clear interpolate are that string between the infinite, Neumann attached endpoint the and particles to string Dirichlet non- endpoints types. are If pinned the to a fixed position. If the masses are condition. We will thentwo-derivative apply boundary the conditions understanding toabove the mechanical spin-two system. fields in AdS space with 7.1 Case 1: open string in harmonic potential adjoint. Intuitively, the finite-dimensional vector space corresponds to adding nontrivial degrees of freedom localized atbe the spectrally boundary decomposed inaction which a render unique the manneran spectral and problem elementary also well-posed. classical how We field shall to theory first determine illustrate consisting the this of boundary idea a by string with two-derivative boundary newly introduced innerfailure product of self-adjointness of theand Sturm-Liouville so operator the all self-adjointness arise can fromcoming be contribution from restored at by appropriate modificationshow of [ the contribution Here, JHEP11(2016)024 ). )) t ( 2 x, t (7.8) (7.5) (7.7) (7.4) ( , y y ) t ( 1   . . ) , y a ) y a = 0 = 0 k ` x, t ), we obtain the ` ( t = = + ( x y x 2

a

y ¨ that are provided by y . y ) to the endpoints of δy ) 2 a t x 2 t , ( k ( 1 2 M 2 , ( + T ∂ y ) = a , y ) y + ) (7.6) ) and express them in terms t = x δy t l, t ` ( 2 ( ( 2 ` 1 , y 2 particle . Our goal is to derive these y = ≤ T ∂ 2 δy x =1 , ` , y X

tL k x a ) ) + d t + ( ≤ y − R 1 2 = 0 x 2 x, t l 0 y ¨ ). We obtain the sought-for boundary ( ] ∂ y ) and y y x t 2 2 ( x 7.6 1 M M δy δy ∂ ) at ) and particle actions turn into some bound- [ T x, t 7.4 ) = ) = 0 (0 ( – 35 – y + , t  x, t (0 ) with eq. ( ( y ) and 2 y x t δy ∂ (  7.4 1 2 x ), y − ∂ = 0 and = 0 and y = − 2 7.4 t 2 ∂ t =0 =0  x x =0 ∂

x 

y y ) x 1 x δy k d x, t T ∂ ( + y Z y − x T 1 . Open string connected to massive particles in harmonic potential. y − 1 T ∂ k   ): t − + d 7.2 y 1 2 x ¨ y Figure 6 Z ∂ 1 ). In doing so, the constraints eq. ( 1 = M M x, t To extract the boundary condition, we derive the field equation of the string from the δI ( y Integrating out the endpointstring particles amplitude amount by combining to eq. relating conditions ( and equations of motion for each particles Imposing the constraints eq.string ( field equation of motion boundary conditions, starting fromthe boundary actions endpoint particle actions.Born-Oppenheimer approximation. This procedure is nothing but theaction classical eq. counterpart ( of Thus, one should beThis able is to achieved by describe eliminating the theof system particle variables in terms ofary the conditions string to amplitude the string amplitude The system is closed, so thecompletely action determines without dynamics of specifying the any variables ( particles, boundary the conditions. string amplitude As is the related string to is particle attached positions to by the JHEP11(2016)024 . .  ) is )   y (7.9) 2 2 2 2 2 ˙ (7.11) (7.12) (7.10) ˙ y y y k 7.6 t 2 t k d d + : y − Z is precisely Z 2 x 2 2 1 2 ˙ e are zero and I , we combine y 2 , 2 e T ∂ I ,M 2 M M 2 1 ,  1 +  M  2 M ( k ) of open string can δ δ y t y 2 x 2 x x d = ∂ = ∂ ∂ 2 ( y y Z − 2 t 2 t M x 1 2  d ` . 0 δy  tδy∂ tδy∂ ) + Z ) y d d t 2 1 t 2 x ( ` y d ∂ 2 1 whose variation is given by =0 = k x x zero, the boundary conditions are Z e − tλ I Z Z d 2 − y 2 1 2 T ` 2 t 2 1 ˙ = ∂ is constant-valued at the boundaries,  M M y ,M x  δ 1 1 − − ), we find that the action y Z x : − M M = = − ∂ ( t δy 7.4 T =    t d y y y d ` ` 0 1 2 2 x x
= k ∂ ∂ y x Z – 36 – x at both boundaries, it is always possible to put Z + 1 2 − − 2 t infinite, regularity of boundary conditions require y y y  + x  2 2 M t t δy ∂ 2 y  ∂ ∂ 2 2   x − t , the boundary conditions are second order in normal T ∂ ) ∂ 2 d ,M  y 1 y − x tδy tδy M − 2 x Z ∂ d d y M ∂ ( y 2 ` x T 2 t ∂ = =0 ). This is the same situation as we have for the higher-spin − − ∂ 1 x and x −  2 y Z Z ) 1 2 y M 7.8 t 2 1  2 y x ∂ t M  x δy M M ∂ δy ( d  ) − ` − t t δy 0 ( x δy ∂ y x y d Z 1 2 x 2 d x d l ` T tλ 0 =0 0 d x − Z Z t ). tδy∂ tδy∂ Z t  d d d =0 1 ) are Lagrange multipliers that imposes the HD BCs. The last line is redundant, d t ` x t 7.3 d ( vanishes at the boundaries. In turn, Z Z M = =0 Z 2 x , x y T Z 1 − − Z Z 2 2 T x λ 2 1 ∂ = We still need to understand how the Sturm-Liouville operator Conversely, we can always reinterpret HD BCs on open string as attaching massive = M e M I e δ I − − By renaming the endpointthe positions action as for in an eq.ends, open ( eq. string ( coupled to dynamical harmonic oscillatorbe particles made self-adjoint at for each HD BC. It is useful to recall implication of self-adjointness for the Combining with other terms in the variation, we get and also combine derivative terms that depend on the mass parameters them to constant values,derivative which terms we that set depend to on unity. string To tension reconstruct the action Here, since they vanish automaticallyobeyed. when the By open reparametrization string of field time equation from the first line is and so the boundary conditions are again reduced toparticles Dirichlet at boundary the conditions. subject endpoints. to Start HD BCs withfield in eq. an the ( open AdS waveguide. stringconditions Solving is whose the the open field same string as equation field extremizing equation eq. subject a ( modified to the action boundary derivatives, so they are indeedthe HD BCs. most Were general if Robinreduced boundary to conditions. Neumanninfinite, and The respectively. Robin Dirichlet boundary boundary Were conditionthat if conditions is in further the limit We see that, for finite JHEP11(2016)024 ) ), ). 2 x ∂ 7.1 ··· − , (7.16) (7.18) (7.17) (7.13) (7.15) 3 ) which , x 2 ( . , f 1 ) . , 2 x ( g 2 g f ), together with ) = 0 x x 22 ( ( n , ( G f ) l . The metric of this ( x f n + R g d 1 ) X ` l . g ⊕ ) for 0 ( 1  Z i f f R i 7.8 2 ) as y 11 . T ) 2 ) is self-adjoint for the Robin x : M i i ≡ 2 2 G x x ∂  i ∂ 2 ∂ n , − h T − − f, g ) + ) (7.14) h ( n h x x λ (0) + ( ( y, h g g n − h ) X 2 n x (0) ˙ ) − ( i T f t i (  1 where y n t x f t T M T correspond to boundary values of d d  ` 2 are HD BC in eq. ( , i y, ∂ 0 n Z 1 t that renders the Sturm-Liouville operator ( 2 y – 37 – X Z ) + f i ) ∂ h 2 R T x i 2 x h ( = , ∂ ) is precisely the extended inner product eq. (  g ⊕ h ) = n g h t ) − 2 X · ( d 7.8 x L ( x, t f = ( y, as Z y x f 2 T d i − h ` , , a general element and its inner product with respect to HD string string = 0 y 2 2 I t I i Z ) as R R i , h h ⊕ ⊕ ≡ y, ∂ ). With these boundary conditions, we now define the “extended i I x, t t l 2 2 i ( ( ∂ . As the Sturm-Liouville operator ( for the open string with HD BC as y h L f n i  i f, g ∈ L X h t = h , n d h 2 λ    h f ) Z = 1 2 x ( 2 f f T n ) would take the form f X = )    7.8 2 x (0) and ∂ = f − f string Motivated by this line of reasonings, we ask if the combined action of open string with I = 1 boundary conditions on element of f inner product” Roughly speaking, two new realare numbers left undetermined by the Sturm-Liouville differential equation and the HD BC. The two-dimensional vector space is givenFor a by function masses space (measured inBC eq. unit ( of the string tension). We now prove that the innerself-adjoint product under the HDIn BC the eq. present case, ( theparticles additional attached vector space at is the provided string by endpoints. the positions Therefore, of it two massive spans HD BCs can be written in terms of some inner product and the open string action viz. ( boundary condition, the normal modeplete functions set can be of basis made orthonormal ofthe and the form string Hilbert a amplitude space com- of square-integrable functions. So, we can decompose inner product for square-integrable functions Denote the square-integrable normal mode functions of ( Robin boundary condition. In this case, we can rewrite the open string action in terms of JHEP11(2016)024 2 0 m ) of 2 x − (7.20) (7.22) (7.21) (7.19) ∂ -space 2 = − L 0 λ ` = =0 x x ,

, as seen from g g ` ). With respect ) ) 0 2 M f f 2 1 7.8 m k k 0 case. In this case, + + can become negative. . We reiterate the key i . f f ) is precisely the action tanh i x x 0 T > T M < y, y T ∂ T ∂ 7.16 h h −  + have negative signs. Take, for − boundary ` I = f f 0 and 0 2 2 . 2 x x 0 + /T m ∂ ∂ 1 2  , 2 1 in eq. ( m M 2 1 2 0 i M M i T M > , M ( ( h string 0 h I sinh 1 λ 1 T T = I = 2 − with − T + M 2 I 0 ` M ˙ . With the extended inner product, we shall T = =0 + g x x  =

` 2  ) t – 38 – ) 1 ` g 2 g d − 2 1 M and  k k − 2 Z = 0. There always exists at least one mode f space. We observe that, after renaming the boundary ) ensures positivity of the norm. However, there are 0 ) in terms of the original inner product over 0, x + + 2 2 i T 2 N k i ) develops an instability to grow exponentially large.  g g ) t R x x ( 0 7.18 = − 7.16 = 0 k < ) for ( i f, g m 1 T T ∂ i ) T ∂  0 k = 2 x 7.8 + ∂ − 2 ,X g ), the proposed action k g − 0 2 x ( 2 x sinh and h ∂ ∂ = X 7.4 h 0 2 h 1 is chosen by the parameters in the HD BCs, eq. ( h 1 M N − 2 M k i M ( i ( R = f g f ) 2 ) = 1 1 T 2 x T x , M ∂ ( ) has the kinetic term with wrong sign, − ) as eq. ( 0 + − = ( X = = 0 1 x, t 7.15 f, ( h M h y )) and so the variable 7.20 There is another example demonstrating the utility of boundary degrees of freedom The “extended inner product” we introduced poses a new issue originating from the is negative definite (whicheq. is ( again a consequence of negativeviewpoint. value of Consider the extended inner product eq. ( This causes negative energy(and of hence the lack open of unitarity) string at at quantum classical level. level Moreover, and the negative mode probability eigenvalue This mode is problematicaction as, eq. upon ( mode expansion, the corresponding component in the instance, whose extended norm is negative for negative value of point here is thatsame extended information inner and dictate product, their HD structures BCs, one another. and boundaryHD BC, actions equivalently, are the bearthe endpoint the extended particle Hilbert dynamics. spaceThis For can happen some be precisely choices when indefinite, of the viz. the metric the HD components norm BCs, dition) and the HDexpand BC the eq. proposed ( action eq.and ( additional inner product over values of of open string attached to endpoint particles, where we arranged the harmonic force term (zero derivative terms in the boundary con- to this extended inneropen product, string is we indeed now self-adjoint: find that the Sturm-Liouville operator ( where the metric of JHEP11(2016)024 4. ) = − x inner d ( (7.23) (7.24) (7.26) o 2 = X L s 2 − d ) mode function ` 2 . HD BC implies = e b = X 2 + x λ = 0 (7.25) a − α , )   = θ = λ ( θ e `

2 n n X  ) Θ k )Θ 2 x 0, the left-hand side is starting θ a ∂ − ( L = 2 and so ≥ − m tanh c = s λ ,( Θ k  4
 − . Recall that spin-two field is the first λ Again, these negative eigenvalue modes − ` 2 ` d 2 = ) . 15  − θ 5.2 negative, it is hard to recognize the above . +1 k x d T λ ` 2 (sec M whereL tanh λ + – 39 – − θ  d  α α λ T λ − ` 2 T < n Z +  Θ 2 contain two-derivative boundary conditions to some of = n ) = cosh i λ x n Mλ ( tanh − e Θ 2 , . From free action of the spin-two field, we get an X = a m n B.C. 3 ). As we deal with spin-two, Θ M λ Θ h a 3.2 . Note that the Sturm-Liouville equation and the boundary condition L and b L implies a, b, c
 ` 2 B.C. 1 − x λ  We first construct the extended inner product for spin-two fields in AdS space. The So, by relating HD BCs to boundary action of extra degrees of freedom, we gain better In terms of boundary degrees of freedom, this inequality means that repulsive force from spring is bigger 15 where the weight factor inthe the Janus integration metric measure eq. originates ( from the conformal factor of than string tension. for some weights share the same operatorproduct L these boundary conditions and,for from partially that, massless representations intuitively in explain AdS the origin ofSturm-Liouville non-unitarity problems with HD BCs that we will consider have the following form: We now apply ourtwo understanding field of in the AdSsituation HD waveguide BC that studied in in HD the section conditions, BCs previous start subsection to tothe the appear component spin- and, fields. among In three this possible subsection, Dirichlet we boundary construct the extended inner product for instability or non-unitarily at thecontrast, level the of boundary equation action of clearlyit motion displays is and the simply origin boundary of a conditions. instability consequence In or of non-unitarity negative and mass7.2 of the endpoint Spin-two particles. waveguide with higher-derivative boundary conditions and this equation hasare a indications solution of for instability ofof the this system. instability. In However, in termsthe terms of origin of HD of boundary BC, instability it degrees is is of the hard freedom, negative to it spring see is constant. the immediateunderstanding origin that of underlying physics. For from 0 and monotonicallysinh increasing. Also the HD BC of generic odd function with negative eigenvalue is and this equation always has solutions because for some modes with negative eigenvalue. Generic even (with respect to JHEP11(2016)024 are  1 and (7.31) (7.32) (7.27) (7.28) (7.29) (7.30) N B.C. 1 − . . functional = α α 2 b + − . L  6= 0 with arbitrary n  3, c α α ) , , ) Θ + − θ −  1 1 m σα n d ( Θ , n  ) c )Θ ) are positive-definite, = ) )Θ m α a σα m ( Θ , HD BCs are imposed on . in table in table are ) 7.29 a n b Θ ) a + ( L ( σα a )Θ L ( ) c + (sec σα −  ( α ) n σα n ( L (  ( n B.C. 3 ). in eq. ( n B.C. 3 is not self-adjoint on − m )Θ Θ )Θ )  type IV type II a a (sec m Θ m )Θ L 7.25 α N σ σα Θ ( L ( Θ b m a N a )( m θ Θ − cot σ i n c a Θ ( L ( i 1   1 and the normalization constants L Θ n sec b − ( L ( b a = − ) − c X + Θ 2 σ L ) b ) a , ( L ( c θ ) n ) N = + . For any conformal factor (sec − – 40 – α − m ( L ( Θ b i c = c a n + n m Θ (sec 2 a a | θ 2, Θ Θ ) ( L ( Θ 1 1 = ( L , d a θ  m − b Θ L = (sec m α σ − α b Θ d − i Θ ( L ( N L h  Z n h σ σ m = −N θ , Θ −   , Θ b ≡h − i = ) i = =  i i X X σ σ m tan σ ) n 2, + ) = . In the last subsection, whether a given HD BC lead to non- θ b b is indeed self-adjoint with respect to the extended inner product: n Θ n θ Θ N − N 2 + + a a , Θ self-adjoint. It is σ a a Θ a L L ) ) = m 5.2 a b a  b sec sec L α α Θ L = a b h 1 3 ( L ( X b ( L ( σ h m N N ( L ( + , Θ = = c i−h m = (sec = (sec ) ) 2 2 θ | | n Θ h 2 1 2 0 in section h Θ , the HD BCs are imposed on spin-zero mode with Θ Θ a (sec L ( θ b , we integrate by part 2. We see that the normalization constants d c ( L ( α − α , − B.C. 3 B.C. 1 We apply the extended inner product to the ground modes for the HD BCs, d m Z Θ = h which correspond to the PMcondition spin-two of and massless spin-one spin-two mode fields, function respectively. is Boundary one-derivative boundary condition and its norm is For c so the norm-squared is positive-definite.spin-two mode In contrast, with for negative-definite. More explicitly, the ground modes of The last expression vanishes by the HD BCs inand eq. ( unitarity or not depends onnorm-squared parameters specifying is the positive boundary conditions. definite The if extended unitary, while it is negative definite if non-unitary. We can confirmL that where space, By inspection, however,Liouville we operatorL find an extended inner product which renders the Sturm- weight Using this, one finds that the differential operatorL We thus take the weight factor as (sec JHEP11(2016)024 2 L MN ¯ h type ¯ h . N (7.36) (7.35) (7.33) (7.34) i ¯ ∇ µν h MN ρ ¯ h ∇ ,  , M α ) is decomposed  ¯ µν ∇ α  h  2 sin − ρ ¯ h α 7.36 tan α . This negative norm 1 2 h ∇ 1 2 π ML sin α − − g ¯ h 2 d α − − N 1 d 2. When one of the Kaluza-Klein √ sec ¯ − MN ∇ d π/ . ¯ x h 1 sec ∼ NL , which can be expressed as +1 sec  2 1 − θ < α < d ¯ h φ α MN α d 1 d 2 , there is no massless spin-two field, − ¯ h ≤ M α 2 d α −  − Z ¯ ∇ d and d θ 4. For example, − = + µ 1) tan θ − for 0 − + 1) sin A 2 µν θ d − 2 sec
d α , h θ MN 2 ( d ρ d ¯ h = ( tan µν + 2 − sin sin h L s ∇ θ − sec 2 – 41 – ¯ h ¯ 0 and positive for 2 θ ) which corresponds to the PM mode, is always ∇ , d µν θ L − d − ∼ h d d dθ θ < ¯ ∇ d ρ α MN MN 7.33 sec α ¯ h g sec sec ¯ h 0 ∇ = ; ¯ L Z 4 L b dθ ¯ − dθ dθ ∇ α ¯ ∇ d 1 α + MN ) 1 2 α α 0 α α 1 2 ¯ h θ 2 a − − − Z appears instead. + Z Z 2 2 sin d − 2 2   (sec   L  1 2 2 2 2 θ x x 3 1 ) is negative for 1 1 N d ) with N N +2 +2 N − α type II α d d 7.34 − d d = = = < N = Z 7.26 i i i ¯ ¯ g g i i i g 2 2 2 | | √ √ | 2 0 2 1 − 2 1 Θ Θ Z Z √ Θ , , , x 2 2 2 background is = = | | | by the following estimate: 2 0 2 1 2 1 +1 Θ Θ d Θ +2 h h 16 h two d d h h h − Z With the extended inner product, we can construct the boundary action which reveals The other norm eq. ( spin , in the spectrum and I 16 of which obeys differenthave boundary three condition. kinds of From the boundary spectrum conditions: generating complex, we mass hits zero mass, thisIII norm vanishes. In this specific value of As in the open string case,by we appropriate require that inner each term product offunctions. which the quadratic ensures action We orthogonality to now be andof expressed completeness know these of that, terms the depending need modeboundary to on action. be the The the nature situation extended is of more inner boundary involved product as conditions, which there some contain are the three component contribution fields of each After the compactification oninto the quadratic AdS terms waveguide, of eachinner term component product of fields, eq. eq. ( ( physical properties of the imposedon HD BCs. AdS The action of free massless spin-two field Here, the inequality holdsimplies because that sin the kinetic term of PM mode has the wrong sign. It can be shownnegative that the norm eq. ( positive-definite. In contrast, the norms of spin-two modes are JHEP11(2016)024 , , . i  2 i | 1  σα . φ σ Θ 4 = φ , (7.40) (7.37) (7.38) (7.39) (7.41) θ φ φ φ − 2 | 3 d µ 3 1 − L −  ∇ Θ d d h φ L h, µ σα 2 2 . We will show = i − i ∇ − θ 2 d ρ µ

| L 0 . µ h A A Θ . ρ 1 1 A , 1 − 2  d , ∇ − | = 0 ,L ,L i , belongs to − 0 L µ d µ ) µ φ α µν L i Θ component of modes: A  µ h A g A  φ  1  2 3 2 i− s = ∇ + − − − C θ = φ , d d

− 4 µ  L 2 + σ | 2 − µ ( A 0 d  φ, φ 1 A 2 ). Using the extended inner 3 L Θ ν ) − − 3 σ d L ,L 2, we obtain the corresponding 2 φ, ∇ − ,L + 1 , L h φ d 4 , 7.37 h ( µν d − µν L + 1 − µν h d 2 h h d d ν σα 2 + 2 L − 2 − +1 2 = − h A d − d θ d φ = L µ 2

µ 1 + φ c ∇ + 1) − ∇ σ , 2 +1) / φ φ −  d 1 µν i µ ( µ d ( d h  ∇ ∇ :L 2 4 : :L σ 1 and 1 2 + andL − φ, φ − 2 – 42 – i | 2 φ 2 h d | − L |  0 n α µ ) h 2 n h 1 n 2 α − ∇ 17 = . Note that the sign of kinetic term for boundary − + 2 tan + 1 2 2 1 2  L | b i / 0 d 1 µ   (sec
h, + 1) Θ 3, = 2 − 4 2 + X σ λ ,A  − − i 1) − , d µ d 4 g − obtained by integration by parts belongs to L φ ) d A − d h − i µν − µ h d α = α 1 2 g φ ) d √ 3 d ; ∇ = 2 ( α | . In this case, the spin-zero component field obeys the HD BC: 2 − + x 0 d µν i tan (sec a  φ, L 0, respectively. h Θ +1 α 1 µ µν , ( 3 d (sec i − = − 1 h 2 d − 2 L , σ 2 d µν h∇ cot L φ − Z is standard in our convention. This matches precisely with the result of L 1) φ, B.C. 1 4 2 1 2 h 1 d L ,F ) with  − − = 2  1) , −  d d = d µν φ − s ) L X 1 ( σ d µν θ ( F 7.29 h d g h = − 2 for − 1 2 d − (sec i = L √ 2 BC1 θ | − − , h x s d 2 1 C   Θ spin-one Dirichlet, expanded by Θ spin-zero Dirichlet, expanded by Θ spin-two Dirichlet, expanded by Θ +1 , Z d + + − Consider first . They are the terms in the third bracket of eq. ( 2 | The classification appears somewhat arbitrary. For instance, • • • d 2 s | = 17 0 boundary Θ Z I where spin-zero fields but its another form that the total action is nevertheless the same provided we keep track of boundary terms. we get the boundary action as I Redefining the boundary values of spin-zero field as product eq. ( boundary action from theproduct: difference between extended inner product and original inner So, we need toΘ adopt the extended inner product for terms involving the mode function The first line, theh second bracket and the third bracket are spin- By a straightforward computation, we find that the action is decomposed as JHEP11(2016)024 2, to − N ), the (7.43) (7.44) (7.42) d R = ⊕ . This is 7.43 i b 2 yields non- φ L µ 2, ∇ − to 3 . − 2 = , equivalently, the d
L  L a . = 0 B.C. 3 ) , φ µ α + 1   A . Alternatively, we can h = = , d θ 

) with σ φ µν 2 as ( | g 2 ; i only yields unitary spectrum. . We will revisit this issue at Θ φ 7.29 σ µν 2 µ 17 h − , ∇ L , 2 1 σα µ − L = B.C. 1 A θ | g 1 2) representation theory.

− − µν L d, √ ( h h x 2 so / many boundary degrees of freedom. 1 +1 andL d  d – 43 – N 4 − Z d α ) that contains the kinetic and mass-like terms of )  2 | α is sourced by the boundary field = 2 X tan σ µ 7.37 Θ turns on the boundary field λ − (sec A , the boundary action of boundary spin-two fields have , the boundary action of boundary spin-zero fields have µ − 1 = A 1 has kinetic terms of wrong sign. Again, this fits nicely with − = 5.2 5.2 . In this case, the spin-two component field is subject to the d 2 | BC3  µν , 0  that the waveguide compactification with h Θ 2 = − B.C. 3 5.2 L σ µν 2 in section h in section boundary − . Using the extended inner product eq. ( I d µν L h that the waveguide compactification with B.C. 3 B.C. 1 5.2 1, we get the boundary action as kinetic term of wrongis sign. non-unitary This in explained AdS why space. the partiallyFor massless spin-two field kinetic term of conventionalunitary. sign. This explain why the massive spin-zero field is From the extended innerHD product, BC. we The constructed boundaryof the action waveguide boundary enabled spectrum. to action directly for trace a the given For origin of (non)unitarity For the HD BC,ensure the we Sturm-Liouville need operator to self-adjoint.be extend physically We understood showed the that as functional this adding extension space can from Summarizing, Consider finally . It is the first term in eq. ( − • • • • 2 | 2 = massless spin-two field without ever invoking boundary spin-two field the result ofunitary section spectrum for the partiallyanticipated that massless this spin-two fields. is an Asproduct, obvious stressed we result already, now just we have hardly from a the firm HD understanding BCs. for the With origin the of extended the inner non-unitarity of partially where we renamed the boundary value of the bulk spin-two field by Most significantly, with extra minus sign in front of the boundary action eq. ( Θ spin-two field c the end of this section. HD BC: We thus need to adopt the extended inner product for terms involving the mode function For the second termcan of interpret boundary that the action, bulkboundary we field value may of interpret bulk iteliminate field two this alternative term ways. byrelated writing to We the the freedom cross which term is explained in the footnote section JHEP11(2016)024 s ≤ ,M (8.1) ··· , (7.45) 2 for 0 ) , ,M k ) ( 1 k φ M µ 1) , ¯ φ ··· − α 3 k  ( µ ξ = θ . 2 field with surface term σα  µ 1 s 3) = φ i µ θ 3 3) ( φ −  − g 3 d − φ 2 − k . Such term was classi- 3 L spin- d i k − 2 ) − L φ d k − 1 + + 2 3 higher-spin field +2 d L − − d − s s d 1 L d ( L − +1 L φ − + d L d 5) ( , so needs to be refined. Though φ,  d i L φ, σ φ 2 3 − | 2 −h  N 0 − a 1) ( σ k d  Θ L N + , − = h 2 X σ + 2 k |  k are the same, up to boundary conditions. 0 µ ( = d + X i σ ( ··· Θ +1) k 1 i h k φ s ( µ 3 + φ ξ 3 − i = ], as we can determine the equation of motions d − 1) field φ – 44 – d 2 L 3 − 15 a L s k − φ, d + 3 φ, s L ( 3 − 1 , so appears not to be refined. On the other hand, and the term d − − i field and relations among mode functions. We shall − d L 2 L | L L s 1 1 and a −h φ, Θ h −h , + 2 | ) = = 1 B.C. 1 and k i i i µ Θ k i h ··· ) φ s φ − 2 k 3 3 . This term belongs to ( µ s − α α ξ − d d + − 1 L

= L µ 1  ( 1 1 − φ space, which is a linear combination of AdS − a ∇ 3 L L ), this term can also be rewritten as − s k +1 d φ, d φ, h L h = h h . Consider the 7.27 φ k 4 µ 17 − ··· ) d 1 k ) in AdS ( µ α s For spins greater than three, however, yet another issue crops out. In the spin-three Before concluding this subsection, let us revisit the ambiguity mentioned in the δφ ≤ (sec where the coefficients are 8.1 Stueckelberg gaugeIn transformations this subsection, we show thatk the gauge variation of AdS after the AdS waveguide compactification, is given by parameters. Fortuitously, analogousconstruct to the the Stueckelbergcompare lower spin- our spin results cases, with previousand we work the [ are mass able spectrumhigher-spin for to fields. all explicitly possible boundary conditions, including partially massless Fronsdal formulation of higher-spinalso fields, higher-spin fields not are onlydecomposition doubly does gauge not traceless. parameters yield(doubly) are the As Stueckelberg traceless traceless in fields conditions but the after compactificationwill for spin-three because case, explicitly gauge of naive fields construct the component or the gauge correct parameters. linear combination In of this higher-spin section, fields we and gauge From lower-spin field cases, wecompactification found are that determined the linearizedsection, uniquely we field shall by equations take the of the the same linearized route waveguide gauge and extend symmetries. thecase, results we to In arbitrary had higher-spin this to fields. sort out the issue that the gauge parameter is traceless. Now, in the One can start with anyno bulk ambiguity. action, but the extended action is always8 the same. There is Waveguide spectrum of spin- we see that footnote fied as originating from using eq. ( this appears to pose an ambiguity, it actually is not. From JHEP11(2016)024 ’s ) k ( (8.6) (8.4) (8.5) (8.2) (8.3) φ . ) k µ higher-spin , s. ··· s. . Taking the +1 +2 d ··· ≤ m 2 , m,k ) k µ ) exactly matches c ’s. So, our ansatz m ) +1 2 ` ≤ = 1 ( is totally symmetric 8.1 m − 2 ψ k s ( µ M ψ is totally symmetric and ··· m 2 ) , 2 s , k s 1 µ ( M space. These fields were first 1 ¯ for 4 − ξ 1) M k − ! − ··· ) expanded by the same mode s µ ) of irreducible decompositions, , +1 m | 1 ) s and their differential relations. 2 θ d 1 ··· s s s µ ( M | 1 m,k | ··· − 8.1 θ 8.1 ( k n g ¯ µ k n θ φ k n ··· ζ 1 ) 2) θ Θ Θ − ··· m m k gauge transformations are reduced to 1 m | − 2 − m,k ) are not. To get the double traceless 1 − − s µ − c field s 2 k − ψ -direction. A complication is that, while µ 1 | +2 µ / ( k n 2 k n θ µ d s ··· ( c ··· m ) c ) 1 1 s g ζ s 2 − ( µ ). With these higher-spin fields and higher- ( µ k ¯ ¯ φ ξ 2 + 1)

= − / spin- 8.3 d k k ≡ ≡ ) = – 45 – µ 1 ··· m 2] +2 + ( ! − 1 + 1) m,k =0 − d s ( µ | ) s m k/ k X [ d m 1 µ − ψ m s ) s − | ··· − ≡ µ k n k n m 1 ). s + ( ( µ ··· 1 − Θ Θ 1 k s − ψ ( µ 8.1 k ζ + 1) Γ ( µ in the gauge transformation. Requiring the double traceless ··· where ) ! m 1 k ··· 2 ( µ 2 ξ − k + 1) Γ ( µν ) ) − µ ξ k k µ ··· k − 0 1 s m,k ( Γ( ∇ ( µ Γ( -fields is not included in this linear combination as it would contain − ψ ! ψ d = 1, the solution is field leads to recursion relations to the coefficients , we thus need to consider linear combination of ) m ]. We now show that the gauge transformations eq. ( ) k ,k 1 ( m,k k is defined similarly to eq. ( 0 m ( c 2) c 15 φ 4 ]. φ − 1) 2] k =0 15 − = + 0L k/ X [ s m ( indices are taken along the waveguide m,k fields = − ( m,k k L ζ m c µ +1

) obey the (double) traceless conditions, We first construct the correct linear combinations that render the corresponding higher- d ··· ) 1 ¯ k φ ( µ ( φ ¯ we now derive the relationsRequiring between expansion each modes term Θ infunctions, the we gauge get transformations the complexes eq. ( spin gauge transformation parameters,the the gauge AdS transformations eq. ( 8.2 Kaluza-Klein modesHaving and identified the ground correct modes gauge transformations eq. ( By a similarparameters: method, we get the traceless linear combination of gauge transformation where condition to normalization for double traceless higher-spin field is The trace part of the divergence term where ξ AdS spin field doubly traceless.doubly traceless, The while AdS itsand traceless. gauge Upon transformation the parameter AdSfields waveguide compactification, and we gauge define transformation the AdS parameters in terms of the combinations, are the candidates of higher-spinconsidered Stueckelberg field in in [ AdS with the higher-spin Stueckelbergdirectly gauge in [ transformations, which were previously derived Extending the pattern of lower-spin fields in the previous sections, we expect that JHEP11(2016)024 s ) as (8.7) (8.8) (8.9) (8.12) (8.10) (8.11) 8.10 ’s for all
s s |
| ` n 3) k n 4) − Θ k
s − | s , + k | 3) ) k s n,k k a + − µ + 1) M s s ··· d − 3 , . k ) are equivalent, as the ( = + µ + s s | | ξ 1)( s d d k n k n | 2 8.8 ]: 1 − µ Θ Θ )( ) as the Stueckelberg gauge 1 − k 15 µ k k k 1) ( k n | | ( 1) − . 1 . g 8.1 Θ − s − field. They in turn determine 3 − +1 k 2 − 3) k k s s β s − − c c +( ) 1. This spectrum-generating com- m − 1 2 k k n + , + ( + ( 3) − +1 ) and eq. ( d − − k k k 2 s 2 M n n | | ( µ − + 2 k k + n 2 ··· n 8.7 − ··· c c M +1) M k , d d 1 k − k 2 M ( µ = = = ( + ξ + − − = − − s s s k s | | | s ...... k n k n | +1 = = α = are put together to , s + spectrum generating complex: m +1 Θ Θ s s s 1 + 1) (2 +1 | -th characteristic value of the Sturm-Liouville | | andL + d n L – 46 – 1 1 s ( ) precisely gives rise to the Stueckelberg spin- n 2) k ) 2 n,k 1 − − 2 2 2 2 n,k n,k n, n,s k k − ) s − µ s -th mode of spin- − k k | M 8.1 n M M M M 1 ··· n + − ) s L 2 s s − = k ( − − − − = ( ( ( µ k n ξ s − − k − | Θ = 2) d 1 L space previously derived in [ n s L − µ s | 2 2 | n,k ( L 2 k − 1) +1 2) ) ∇ a n,k M +1) m d − k − 2) k L s k k 1) k − M − s + s − +( + s ( s s s s | = ( ( (2 ( ( − k ). One can show that eq. ( − d − − − − k a µ L L L L L L − 8.8 ··· ) s s s 1 | | | k = ( µ s n 0 n k n ...... ), (     s Θ Θ | Θ δφ k n is used to represent the 8.7 1 2 2 2 Θ − − − − s is the mass-squared of | ) s d 1) 2) k : s − L | 2 − − − d n,k s k k , s L ( 2 n,s − + M − s s ( ( d − M ··· L − − , d L L = 0 Then, the gauge transformationgauge transformation eq. in ( AdS where factors independent of mode index Here, mass-squared of lower spinplex fields, enables us to interprettransformations. the Let gauge us transformations choose, eq. for convenience, ( the relative normalizations in eq. ( All relations are summarized by the spin- Here, problems eqs. ( identity relates the Sturm-Liouville operators each other, and the eigenvalues each other by ` These relations determine the Sturm-Liouville differential equations of Θ JHEP11(2016)024 . . 2 n n M field. (8.14) (8.13) (8.15) (8.16) (8.17) k := 2 = 0 as well. M , s ) decomposes |
, s +1 3) (∆ n,k . − V ) k M and for each mode + , k + 1) field whose mass- 1 s , s . k . 2 − 1). Their mass spectra are + and forms the Stueckelberg and forms the Stueckelberg ··· /` s 2) . d
− , 2
3) 1 + − k , /` +1) 1) ( d s 1, however, (0) − k − ( ( φ ) represents the partially massless 4) k − + s − φ = 0 , s k − d 1) and the Stueckelberg spin- k + k 1 ··· t ⊕ D -th mode is simply the Stueckelberg = ( s is tuned to special negative values, it − ··· − , + n ) . In turn, these gauge transformations − t 2) ( , − s 2 n k 1) + n d k s 1) , s − − d M − 1 k M + − field decomposes into two subsystems: the + ( ( s s ]: s d ( ( 2 s + 2 − ( , φ 28 Together, they constitute the ground modes: d − k ) D M , φ = ( + 1)( – ( k ) ) – 47 – ( t s t + k d 18 ( 26 φ − enters only through the mass-squared d 2) φ − − ( fields have the same structure of gauge transforma- n = s − D 2 k k = 0. = ( s ) represents the massive spin-( field is also part of the spectrum, since it is nothing but M + µ ) = · 2 1) field with mass s 2 s . , k = ∆ (∆ µ 1 , s M 1 k − 1 + 2 − 1 µ ` k k − d 2)-modules. At generic conformal weight ∆, the Verma module φ 2 − α
s gauge transformation of )( d, 2 s field of depth-0. system of depth k m + 2 ( κ k 5) + k s s d + so + d ( − − 1) ( 2 d 2 s = 0. These special values are the values at which ( k s ∇ − D k V k α + 2 ( d + 1) ( ( k field. Their mass spectra are given by ( − ), the irreducible representation k 2) irreducible representations [ = = d, 2 k k ) is irreducible. At special values of ∆ = 8.16 field, while ( β α The lower subsystem consistsspin- of The upper subsystem consists of system of partially massless fieldgiven with by depth s , s so Group theoretically, the decomposition pattern of the ground modes can be understood As in the lower-spin counterparts, were if Note that our conventions of the mass-squared of higher-spin field is such that it is zero when the • • (∆ 18 squared is set by the conformal weight ∆: higher-spin fields have gaugePauli-Fierz equation, symmetries. So, it differs form the mass-squared that appears in the AdS In eq. ( spin- in terms of the Verma V into partially massless spin- can happen that In this case, thepartially Stueckelberg massless system spin- of spin- Importantly, the massless spin- Here, the dependence onApart mode from label this, alltions. modes of Therefore, spin- spin- gauge transformation of spin- completely fix the equationsThey of constitute motion the for Kaluza-Klein each modes. spin where JHEP11(2016)024 . . 2 := /` = 0, ) can 2 3) + 1 (8.20) (8.21) (8.18) (8.19) n,` = 0 for 2 n,` k − . First, M ` 8.10 a − M − ` , 7 s | − G . ` . k Kaluza-Klein + } . a < k a < ) ) +1 ` k ` + s
, s , s ··· − s , form the massive 6= 0. If 4) U 2 + , k ··· ··· := Θ 2 − d n,` , , ` − =1 ` ` , − a M + 1) possible Dirichlet + + 1 + 1 s +1)( ,G ··· k 6= 0 for all for ` = 0 and a ` − i 1 ` ` + a, k, k k, k . After inductively applying + − a − form the Stueckelberg system l s K G 6= 0 for all − n ` k = = G { ` ` a + , a ` ` = − D = G +1 d s ` ` ` } , ··· − )( s , there are ( and Θ ` a D | ` = ( U s ` n ` G k, is positive (negative). This additional − 2 +1). See our earlier expositions on this − ··· k ≤ − ` ··· a k s M Θ k . Then, the sub-complex of eq. ( − for negative +1 ’s are the ground modes which satisfy the { have the same eigenvalues. Their eigenval- k +1 − k = ` a k ≤ k i 1 − s ` ` D G − U − = 0= 0 ( ( l , − s k k ` a α α ) when Θ + ( D U – 48 – +2 1 l G   s 2 n − U = = and − ` 0 θ θ . For 0 M ,

l k . s s (Θ | | ` 7 =L ) then fix the boundary conditions for all other fields − `k ` 0 Θ ` , Θ 2 = } D 2 ` − 8.10 ` − 2 ··· n,` + , 1) s = 0 with 2 and − M , L ` ` field with depth-( k − − a − =1 s + s = 0 for some l a ` ( G − α with the same subscript 2+ = 0 with d = 0 for positive  ,G +1 ` i − L ` ` ` a i : = 0, one can show that the structure of mode function is given by s = l 0 G − θ G ` − ` | K Θ d U ········· ··· s ` { | l D k 2 D 1) D 1 = ··· − − ` n ) − :=L a } k ` n k  ··· s Θ − ` | + − ) and thereafter. Fields corresponding to Θ consists of three parts. The first part is the set of massive spin- ` ) Stueckelberg field with mass-squared s s U U ( ( +1 a + . Finally, fields corresponding to ` ’s are the Kaluza-Klein modes, and s , a | k − − ` ` i s U k d − +1 | 8.13 L D Θ ` a K L k Mass spectrum for theΘ boundary condition characterizedtower, by whose mass-squared Dirichlet is condition given by at the eigenvalue of the Sturm-Liouville problem, { a + − Summarizing, • D U 2 n, k ( These spectra are depicted in figure ues can be obtained by thepositive first-order differential equation of partially massless spin- in eq. ( spin-( The ground modes this relation from Here, equations By this complex, there isthere one-to-one exists map one between additional Θ modemode Θ satisfies to counter cluttering indices,M we define simplifyingbe written notations in as the Θ form Below, we show that the pattern of mass spectrum takes the form of figure We next classify all possiblespectra. AdS As waveguide boundary before, conditions weconditions shall and on only determine Θ the consider mass boundaryconditions. conditions derived The from relations theoriginating eq. Dirichlet from ( the same mode functions: 8.3 Waveguide boundary conditions JHEP11(2016)024 + s field 1, as + (8.22) s − d k ··· , + 1)( field that are 1 ` , s − s = 0 1 ). ` − = ( 1 k 2 8.17 − − M , k , s ··· ··· , , 1 1 , , = 0 = 0 t ` ) for 1 with mass-squared, – 49 – , s 2 ) for − 1). The third part consists of the set of massive − , ` − t 1 , k k − − ··· s s − , 1 + + , , s d d ( ( . The second part is the set of partially massless spin- D D ··· = 0 s , ` 2 ( = , 1 k , ). 8.22 2) representations of ground modes are . 2 d, ) with ( /` 8.7 so 3) . Mass spectra for all possible boundary conditions characterized by Dirichlet conditions − . Each point represents a mode function. Points in the same rectangle form the Stueckelberg s whose masses are set in terms of conformal weights by eq. ( Stueckelberg spin ` The eq. ( with depth-(0 | k • + 1) possible boundary conditions led to the mass spectra encompassing all of massive, s So far, we studiedcharacterized a by class Dirichlet of condition on AdS( a waveguide mode boundary function. conditions Quite forpartially satisfactorily, spin- massless the and resulting massless higher-spin fieldsfurther upon on the compactification. variations of Here, this we dwell idea. 8.4 More on boundary conditions massless field, whiledescribed the in lower eq. ( triangle consists of the Stueckelberg spin Figure 7 on Θ system with the highest spin. The the upper triangle consists of the Stueckelberg system of partially JHEP11(2016)024 . ). 7 0 We and , = 0 1 8.15 (8.25) (8.23) (8.24) 19 α −  , type II = 0 and = k > s θ k | ··· s , k . | , k 2). ··· d, + 2 ) and eq. ( , ( | Two remarks are . o k s
8.14 + 2 −| , + 3 , s d + 1 | + 1 − . , the ground modes consist k s s 5 = 0 to the regime, 2 −| = α l −  = k > s = 2 l n θ | s M | k ) for − ) for , s = 2 , s s | 2 − l +1 − l – 50 – 2 + n,s and additional towers d 2)-modules. For + s M ( ) that a new class of boundary conditions are possi- d d, More generally, we can also consider extended form − D ( = ( D k so 8.10 +1 , s 2 2 , − 2 /` L /` . 4) s | s s | 4) − s n +1  l s n − Θ l Θ + 2 0 case gives rise to non-unitary representations of s + − 2, the spectrum contains partially massless higher-spin fields. Their 1, one can also show that all kinetic terms in the boundary action have s d + L − d − + k < is associated with the first spectrum in figure s d s )( , we will discuss other class of extended boundary conditions. s 2 ≥ )( ≤ 10 s − k l k − 0, the ground modes also consist of two parts: the ground modes of l = ( Explicit analysis for spin-threeboundary and degree of spin-four freedom fields isFor suggests proportional instance, to that in the the masses spin-three in masses system,in eq. of table we ( the verified that the boundary action of that the boundary actionindefinite has norm some in of the theis kinetic extended precisely terms Hilbert with associated space. wrong withwith sign, These the leading boundary partially to degrees massless ofpositive fields. sign. freedom Correspondingly, In they contrast, are for all unitary. the cases For the boundary conditionsfor derived from thenon-unitarity Dirichlet can be conditions traced Θ to the non-unitarity of boundary action. One can show 2 = ( 0. Though there is no corresponding physical excitations, formal analytic extension is k < In section • • 2 M 19 M additional towers, We see that the ground modes and relateof them two to parts: the ground modes of For possible; By analysis identical to the ones in the previous cases, one can derive the spectrum of the of the boundary conditionscan beyond see those from characterized the byspectrum the pattern generating Dirichlet of complex conditions. raising eq. and ( ble. lowering To operators this between end, let adjacentk us < spins extend in the the mode functions Θ Extended Dirichlet conditions. analysis simplified. Basedpatterns on for the arbitrary intuition spin- gained so far, we here mention the expected in order regarding the boundarythat, conditions for and spin-two boundarywith action. system, nontrivial We HD showed dynamics BC in section describedcomplexity, imply extension of by the the their analysis presence to boundary higher-spinHowever, of is the action. not boundary performed factorization yet degrees property with Mainly full of of due generality. Sturm-Liouville freedom to differential algebraic equations renders the Higher-derivative boundary conditions and boundary action. JHEP11(2016)024 ) α + 3) 2)- d − ( 8.26 d, (8.30) (8.29) (8.27) (8.28) (8.26) ( s space. . The D so 2 +1 + 2 d . ). In order d space arises ( 2) , . This seems 1 2 (d +1 . 8.22 +2 d so d = ) ]. If the wedge 2)-module: µ = 0 , l , ]: 1 : 8 α d s . In other words, the | d −  µ ν α, α s ), we would then need + 1 s but different conformal = , Q +2 − θ d d | + s µ ( ν s )) d | 8.26 ( s θ field in AdS so = 0) in eq. ( D AP s for even for odd α 1 (sin  − =0 µ ν l s M = µ θ 2 cos Q precisely matches with the above 2)-modules in the right hand side |
⊕ 2 s  | 2 − 1 c d, s 2) ( ,  2 where Θ 2 (Θ (d so ) ) + boundary. As such, one might anticipate )) µ so θ µ ν s
µ ) field are determined by the Sturm-Liouville field form the  2 Q +2 s ( = d , s s , (sin 1 ) Γ( 2 2 s k 4 . In the decompactification limit, the boundary µ | ν  π − spectrum from the viewpoint of AdS s n ) – 51 –  P s µ π µ Θ 2 s − 1 − + 1 2 c n 2 4) ( n ) Γ(1 M µ to match with the the first set of modules. Below, we ν (cos( µ − + )  − P d 1. We see that the second set of modules in eq. ( s s θ A ( ( A  = − = D + 2 sin s A s | k limit of flat space waveguide we studied in section sin s n d π =0 ( ∞ 2) module in the left side of eq. ( 1 waveguide asymptotes to the spectra of AdS Θ 2 n 2 π M sin , = (cos massless spin- − s 1) − − 2 | = π +1 1 s n − d + 1 s → ∞  −    +2 2) Θ , d d 2( 1 4 ( = 0 take the form L ' − o − L α s 0 = (d+1 2 n  2 are the associated Legendre functions with arguments, so ) and the Dirichlet condition for mode functions Θ case, the mass spectra of spin- = − ) M µ ν θ d . Representation theoretically, we can decompose this module into | s 2, the waveguide decompactifies to the entire AdS L Q s , s 8.7 2) | , 1 s = π/ ): the first set of modules have the same spin, spin- − k and s (d+1 2 hyperplanes correspond to the AdS + 1) = µ + ν so 8.26 ν ) d P π/ ( ( , s  ν In the Consider the AdS D 1 square-integrable modes of massless spin- = 2 − where and conditions Θ equation eq. ( The solution is given by ferent spins ranging overcoincide 0 with to the setto of reconstruct ground the modesthe for Kaluza-Klein modes from demonstrate this affirmatively. We can relate these modules withfield states in that the arise from AdS thedimensional waveguide, compactification degression. as of higher-spin the There foliationof are of eq. two figure ( kindsdimensions, of while the second set of modules have the same conformal dimension but dif- s modules, a procedure referred to as the “dimensional degression” in [ only for special setdecompactification limit. of boundary conditions. Here, weThis discuss is subtleties just involved likeL in the the In our setup, theapproaches AdS waveguide rangesα over the Janusthat wedge the [ spectra ofto AdS be in tension with our result, as the mass spectra of spin- 8.5 Decompactification limit JHEP11(2016)024 - ]” d 8 are field (9.1) + 1)- ν space. s d + +1 µ d refers to the . ), it immedi- I 8 g 4. However, the 8.17 , 2 ) ). The mode functions fields in AdS φ s < d < z ∂ are physically meaningful. 8.26 ( B x g d d I Z = 2. The holographic dual would in eq. ( I s 2 2) z, 1 , g =1 ∞ (d+1 0. Therefore, it must be that z X so ’s relative to ) space. As the bulk is strongly interacting, + I g 2 , s ) vector model in 2 ) α > 2). 1 +2 ) Heisenberg magnet. This system exhibits con- d φ N – 52 – ( − -dimensional derivatives, respectively. This CFT π/ N sin ∇ d ( s ( O − . This however is no longer true. The point is that → ) contain the modules which are not in the massless O x + s ) goes to the spectrum of “dimensional degression [ α d +1 s ( 6= d 8.22 = 1 d D + 1)-dimensional Heisenberg model in the bulk and = k  B d k Z  2 B 1 g ). Conversely, this explains transparently why the dimensional space, to unity, such that space does not generate “massless” spin- ). = + 1)-dimensional bulk is strongly interacting, so the dynamics is B +2 , s d g 5.36 d refers to the coupling parameter in the bulk, while +2 1 +1)-dimensional d ) and 1 CFT B d − I ν g + 1)-dimensional and s + d : the ( + space belongs to one of these singular modes. For spin-two case, this was µ ( n k,I ] of AdS g π +2 are ( 8 + d d =  ( , ∂ 2 B A D modules of AdS ∇ served (even) higher-spin currents startingbe from higher-spin gauge theorythe on energy-momentum of AdS the interfacedimensional is energy-momentum not separately tensor conserved. istensor Only conserved. currents. the ( In The terms same of goes AdS for waveguide, this higher-spin is the decompactification limit. g described by ( s The behavior of the system is extremely rich, controlled by the relative strength of the In the action, All are well so far, so one might anticipate that the spectral match with the dimensional Spectrum for the cases of ( =0 • ∞ n However, for the sake of classification, we will keepcoupling the parameters. normalizations as We can above. classify them as follows. system is adimensional juxtaposition isotropic of Lifshitz Heisenberg ( model in the interface.coupling See parameters figure in the interface.pling By parameters, normalization, say, one can always set one of the cou- dimensions. The CFT action is schematically structured as where In this section, we discuss anexhibit interface the conformal AdS field theory waveguide whose andshall holographic higher-spin dual study field would this theory in onsetup it. is the completely For context general, concrete of and discussion, can free we easily be extended to other critical models and higher in the AdS already shown in eq.degression ( [ 9 Holographic dual: isotropic Lifshitz interface degression continues to holdsome for of the ground modesspin- in eq. ( that would potentially matchfactor with goes are to actually zero) singular in (equivalently, the the normalization decompactification limit. In particular, massless spin- integer-valued in the decompactificationately limit. followsL that From the the modules relation that eq. correspond ( to the Kaluza-Kleinin modes the are decompactification precisely limit (i.e. where JHEP11(2016)024 I ∼ z, +2 d ) , g + 1) (9.2) (9.3) y B d g measures space. In α +1 ) Heisenberg d ) singlet. The . N 2 N ( ) = 1, this critical ( = 0 as Φ( φ O z O z y ∂ -dimensional Lifshitz d ))( and the dilaton field as y y Φ( 1 − z y ∂ -dimensional ( d y d Z x . d I 2 B d 2 z, g g ) Heisenberg magnet as Z N ' 2 ( ≥ ∞ α O z X – 53 – -dimensional and cannot interpolate to ( d + tan 2 ) ) Heisenberg magnet controlled by the dilaton field is φ N ∇ + 1) dimensions in which the bulk Heisenberg magnet ( ( 1. The dynamics is described by a d O y d -dimensional interface is strongly interacting with Lifshitz d -dimensional interface is strongly interacting with the leading k > Z d x = d z d : the : the 2 B Z 2 B , g = = 1. The dynamics is described by I , g , I k, z 6=1 2 6= z CFT 2 z g Heisenberg magnet. As the bulk has the Lifshitz exponent I g ), the second term gives rise precisely to the type of interactions of the form z ) y   ( N I δ I , ( = 2. The holographic dual would be higher-spin gauge theory on AdS ). This argument then provides the existence proof for the CFT dual of the higher- 1 2 2 k, 1 − behavior is achieved only ifinterface the maintains interface is nontrivial fine-tuned. interactions (even Thisthe though is interface possible weak) is only an with when open thein the system bulk. general interacting with Thus, non-unitary. the reservoir bulk system) and hence dimensions. In termslimit is of singular. holographic dual AdSg waveguide, the decompactification Lifshitz exponent O exponent magnet. This systems exhibits conserved (even)terms of higher-spin AdS current waveguide, starting thisspin corresponds from field. to the The Dirichlet condition interface to CFT each is higher- g k 9.1 The critical behavior as above is quite rich and intricate. Nevertheless, it can be easily The critical behavior classified as above fits perfectly with the AdS waveguide spectra ). The anisotropic, Lifshitz y • • I y 2 k, of higher-spin fields can be performed. We showed that, upon compactification of AdS In this paper, weall developed known massive a and newapproach is partially approach geometric massless for and utilizes higher-spin realizingthe the fields, Janus (inverse) Kaluza-Klein wedge compactification. at Higgs provides the We the mechanism AdS pointed waveguide linearized out of with that which level. the Kaluza-Klein compactification Our If the dilatong field is fine-tunedeq. ( to behavespin across system the compactified on interface the Janus at waveguide. 10 Discussions and outlooks Φ( described by the action achieved within the Lifshitz-Janus system. Thecan idea be is arranged that by the a couplinginterface nontrivial parameters kink is profile of a the hypersurfacelive dilaton field, in. of which is ( Denote the local coordinate normal to the interface as we analyzed in the previousthe sections. extent In the the extra gravity dimension dual,parameters extends. the of AdS the Thus, wedge we above angle expect interface Lifshitz that it is related to the coupling JHEP11(2016)024 α +1 d from is an space. s M +2 d = 0. We can ). where α  α 8.22 =  , θ = | radius. Again, their θ s | | k s | = 0 = 0 (10.1) k +1 α α d Θ   = = θ θ M | | s s | | k k ≥ ≥ l l -dimensional isotropic Lischitz system Θ Θ d , and become infinitely heavy as 2 2 interface α − − g l 1) ) Heisenberg magnet. The bulk of the CFT Stueckelberg couplings coming from the + s − l N L ( − +1 2 2 s d O ( ··· bulk bulk − g d g (∇Φ) (∇Φ) 1) – 54 – L − 2 space. The mass spectrum of ground modes fits 2)-modules. The ground modes can be massless, k Φ ··· + d, k s +2 ( ( 2 d ∂ and are just set by the AdS − − so ) L k α − s M ( − d L M for the set of Dirichlet boundary conditions Θ +1 , two classes of higher-spin fields appear. The first-class, which we referred d 1. +1 d z > . The interface configuration of Lifshitz to AdS We could consider morearbitrary general differential boundary operator. conditions, This would lead to a set of new boundary conditions +2 In this paper, we mainly concentrated on the compactification of massless spin- d + 1)-dimensional system. The boundary of the CFT is • d to the breaking patternpartially of massless Verma and massive with specific masses as describedAdS in eq. ( extend the analysis along the following directions. The second-class, which weall referred possible to types: as groundindependent massive, modes, of partially consists the massless of and wedgeequations higher-spin angle of massless. fields motion of are Their organizedhigher-spin masses, by gauge however, the symmetry are AdS in AdS down to AdS to as the Kaluza-KleinTheir modes, masses depend comprises on ofgoes the infinite to waveguide zero. tower wedge ofStueckelberg size As couplings, massive in whose higher-spin origin flat fields. is space, the these higher-spin Kaluza-Klein gauge symmetry masses in are AdS generated by the AdS Figure 8 is ( of exponent JHEP11(2016)024 ] +2 42 d . We 1)-dimensional . It is the AdS − C ]. This generaliza- for the Dirichlet α < θ < α d 29 8 − 20 in appendix and impose different boundary d 2 ) Heisenberg magnet in the spin- N ( O to AdS ]. k + < θ < α 45 d – 1 α 43 – 55 – on codimension-two defects. +2 d ]. It would also be interesting to investigate the reduction of a 41 , 40 , ]. 36 27 , – 26 30 spondence should match and, especially,to one-loop 1/N free correction energy of of free AdS energywill of side provide CFT correspond clues side. to The the Casimir quantum free aspects energy calculation of [ dual CFTs. conformal field theories.in For partially [ massless fields,partially dual massless field CFTs on were AdS conjectured In the context of AdS/CFT correspondence, the free energy of both side of the corre- ideal candidates for the concreteterface and realization Lifshitz of behaviors dual are conformalconfiguration. realizable in field Type theory. IIB Both string in- theory from theFurther Janus interesting direction of researchin is the the infrared emergence of and partially subsequent massless renormalization fields group flow to ( We argue that isotropictronics Lifshitz and interface the of Gross-Neveu fermion system in graphene stacks at Dirac point are space. Partially massless fieldssymmetries. and their Stueckelberg At couplings the retainenough partial linearized to gauge level, fix these thethat partial action, retain gauge so massless symmetries itsymmetries. fields are should and sufficient be understand possible physical to mechanism classify of boundary the conditions new gauge We can also considerhigher-spin fields. the The fermionic latter isary higher-spin particularly conditions, interesting fields as from supersymmetry as the is viewpoint well generally of as bound- reducedAlthough far by super less the interesting, symmetric boundary we can conditions. also start from partially massless fields in AdS We can also compactifydimensional more reduction method than from onegeneralization AdS of directions. the non-abelian Kaluza-Klein We compactificationtion [ worked also provides out a details geometricin framework of [ for the the colored higher spin theory proposed necessarily include theboundary ground condition. modes we discussed inWe assumed section thatcan the generalize waveguide it wedge toconditions is a at parity-symmetric, general each boundary. domain The spectrum is more complicated but, by construction, the new ground modes We are currently pursuing these issues and will report our results in future publications. Further development of the Higgs mechanism we studied from the dual conformal field For similar calculation on HS/CFT duality, see [ • • • • • • • 20 theory viewpoint are within the reach and would be very interesting. JHEP11(2016)024 ) ). ). ). µ , s and , k dx ,Y (A.1) (A.2) (A.3) (A.4) 1 ν a (∆ ). For µ (∆ − , e D 1) is the , s µ s Y 2)-module − = + (∆ d d, a ( d ( V , e ( d ) so 2 D or ` νρ σ 2 g µ a . − e b µσ ρ g A A a − b ρ µν µ νσ Γ ω g . The Verma − − s µρ , ν a ) g 2 ( A A µ µ dz ∂ ∂ 1) 1) for dS. Λ = − − = = − 1 a 2 Λ ν d − 1, there is a submodule ( A A 2 d 2)-module. Consider a finite dimensional µ d µ − d~x s d, ∇ ∇ ( ). We use ∆ to denote conformal dimension − ≤ ) = d so , . Covariant derivatives are defined as follow. ( 2 , b k and (sometimes its inverse – 56 – νρ ρ µν ). For the analysis of symmetric higher spin, we dt so g A a ( ≤ b g A d µ . We will use most positive sign for metric and ( a 2 2 ⊕ µσ e d ` ν ρµ µ z dimension. Space-time coordinate indices g ) is irreducible and therefore coincides with = so ω − d (2) + ν , s + Γ + b = so e ν a νσ (∆ ν is (+1) for AdS and ( g A A ab V µ µ η σ dx µρ ∂ ∂ µ ) for the irreducible quotient of Verma module µ g a = = e − , s dx ( ρ a 2 µν (∆ σ A A ` g run from 0 to 1). We will call µ µ D , 1 with an integer 0 b = = ∇ ∇ − 2 , radius and ··· a k , ) of sub-algebra ds is spin connection. 1 µνρσ metric and Riemann tensor are given as following. + dS , µ ) 1 R ,Y d A +1 − dx d (∆ ab is ( Y µ to denote Young diagram of ) is the space generated by action of the raising operators to the module ` ω Y , s (A)dS = = diag( (∆ ab ab V We will also denote generic value, Verma module However, for specific values,stance, it ∆ = becomes reducible with a non-trivial submodule. For in- B Verma module andHere partially we massless(PM) recall field themodule definition of the Verma and limit ourself to the Young diagram of a single row of length where cosmological constant. η as vielbein ifω it satisfies A AdS space Conventions are demonstrated in tangent indices Theory” (September 21–25,(November 2015), 4–6, Singapore 2015), Munich Workshop IAPP “Higher2–27, Workshop 2016), Spin “Higher where Spin Gauge Theory various Theories” in and parts part Duality” of by (May the this National2012-K2A1A9055280. work Research were Foundation presented. Grants 2005-0093843, This 2010-220-C00003 work and was supported We are gratefulPolyakov, Augusto for Sagnotti, useful Kostasand discussions Skenderis, JWK Misha acknowledges to the Vasiliev Karapet(11)” and APCTP Focus for Yuri Mkrtchyian, Program Zinoviev. providing “Liouville, Tassotality stimulating SHG Integrability and Petkou, scientific and stimulating Branes environment. Sasha atmosphere of SJR Tessalloniki Workshop acknowledges “Aspects warm of hospi- Conformal Field Acknowledgments JHEP11(2016)024 . +2 . ) is s d ··· with ) ) µ (B.3) (B.4) (B.1) (B.2) , s ··· 1 + 21 2 , s , k ) µ 1 1 s 2) − 1 , µ µ − − k 2)-modules (d ··· 1. This can s k so . δ φ d, + +2 ) + + − t ( s d d µ d t , l ( ( ( ξ so 1 − V ) D D +1 − d t s µ ' 1, − ) + − d , s ( · · · ∇ . 1 . ··· D 1 k − µ 2) + ( 1 + k = ∆ (∆ d k < s ) − =0 − ∇ Gauge variation: l + 2 s s M ].) The action for PM field has ` s µ d ≤ ( 2 ⊕ ··· 28 + κ – 4) D +2 2) d , t 26 µ − (d ξ , t 2) ( so can be decomposed into ) ) +1 − . For 0 t − s , s µ s , k s 2 1 +1 ( 1 d m for brevity. − − + 2 − s · · · ∇ s ) d 2) – 57 – , ( d 1 + + t (d µ d ( − ( 2 n ) is unitary and its field theoretical realization is so σ ` ∇ +1 , s + − d 1 = ]: . Partially massless(PM) field. d ⊕ D 8 ( = 0 which is given as s ) − µ 2 s D µ = ∆ (∆ ··· · , s 1) − 2 ) for PM spin-two case. The properties of PM fields are given 2 1 2 µ =0 t ∞ ` µ 1 n M − − + 1 Table 4 2 µ − ) is not equal to Verma module but is to the quotient of Verma k µ 5.23 s ∆ = m s to (A)dS φ + δ φ , s
2) d k + 1 , + ( 2 ( + κ 2)-module for massless spin- d field propagating in AdS d − d , 1). (For more general cases, see [ ( + (d+1 s k is defined by the following convention. By the mass of a field, we refer to D − 'D 2 so +1 + ) ) k d m ∇ ( 1, d , , s ( − , s 1 so , we described the AdS waveguide by using Poincar´ecoordinate of AdS 4 1 − ]: s D − s − 3 . PM field s k 47 = ( , t 4 = + t + 46 d d Field type ( k Finally, In table ( Extrapolating our convention, the massless higher-spin field is the partially-massless higher-spin field V D 21 depth- However, it can beresults which generalized are to given inmore other general the coordinate AdS main waveguide system text which do and can not be one change. built can from In show (A)dS with this depth-zero. appendix, that we the analyze the In main text we open omit subscripts C From (A)dS In section AdS [ by the following branching rules [ this convention, the relation between mass-squared and conformal dimension is given by Note that this is different from the mass-squared which appears in Fierz-Pauli equation in See paragraph below eq. ( in table the mass in flat limit. Therefore, it is zero when the higher spin gauge symmetry exist. In module: the massless spin- non-unitary and their field theoreticaldepth realizations are partially massless(PM)the fields PM gauge symmetrybe which contains derived covariant by derivatives Stueckelberg up form to of order PM field. Therefore, For JHEP11(2016)024 : ) +1 = 1 d C.5 (C.4) (C.5) (C.6) (C.7) (C.8) (C.1) (C.2) (C.3) . The σ i dθ . constrain i σ , ∂f ∂θ A mn , = ) with ¯ E L =1 are indices for k i m M, are solutions of = d θ C.6 . b N d M 0 P ) is automatically θ f and , m 0 dimension. Finally, 1 mn a = M σ = 0 metric are, N ¯ Ω k . To see more explicit C.5 b AB = = − . The one-form tensor δf + . +1 +2 ¯ 2 µ E Ω 0 , d d = tan d θθ x m a ∧ θ ν d θ d N ¯ a Γ E N A . After imposing this ansatz . I.e. . The equations in eq. ( i k , m k θ , and eq. ( + covariant derivative in term of σ E + µ m d N ν d a and k λ + = 0 N ¯ δ Ω + = b θ d n c + ¯ 0 → E , σf Ω am f a i ¯ ∧ Ω m ∧ MN m must be tensor in the lower dimension. NM, θ ¯ E = sec m B N ac ¯ Γ is Christoffel symbol. The covariance 0 = = , ¯ internal space indices in lower dimensional E f a m are for internal space. “Bar” are used to 0 0 µν 0 i µ νθ ab i M σ σ and independent of θ θ + Ω λ ¯ n Ω d θ and = 2. d f ¯ Γ + ): i ¯ are the vielbein and spin-connection of AdS Γ + = ab – 58 – θ k = Ω ln − ab Ω M 0 . ¯ C.2 Ω and and nm a )), one can easily check the covariance conditions, f ab ) are defined in similar ways: 2 ν ¯ ∧ Ω ¯ → Ω l µν n B m dθ n i C.5 , d µ n θ 2 N ¯ and Ω a, m Ω ), Cristoffle symbols and (A)dS , ¯ Γ m ∇ a = 1 and indices and ), ( + M n dimension and = dimension are also tensors in AdS = 0 M E , = k C.4 b = ( k 1 for dS. The first equation is the torsionless condition and can be constructed by solving constraints ( + dθ m k mn . +1 a + C.4 E νλ n d − µ d ν ¯ Ω A g + +1 ¯ x +1 ∧ m B are vielbein and spin connection for the Euclidean space with 2 d d d N are functions of µ ab M 0 M , δ d s ¯ n ∇ f mn 2 0 = : m ¯ Ω σ , − f implies that + Ω A = 0 m M a a ¯ = = = E ¯ n ν E In this case, ¯ 2 ¯ , δN E and B dE νλ +2 are for (A)dS N θ µ 2 d m and , ∧ m ¯ a ¯ Γ ¯ E ∇ − ¯ m d s E E m mn 2 local space-time indices, µ, ν 0 0 N ¯ Ω f N is +1 for AdS and f represents the exterior derivatives of internal space language, , = 1. It can be checked that 0 + 0 +1 σ = = δ σ f 0 d m and vielbein +1 0 is also independent of a σ ¯ To achieve these covariance conditions, we shall take the ansatz for spin connection Here, The main consistency condition for the waveguide is the covariance condition: covariant d E ¯ = 1 case). E is introduced to represent the curvature sign of (A)dS δf δ AB mn 0 k ¯ ¯ AdS waveguide from AdS and ( satisfied. The constraints eq. ( σ imply that constant curvature which is equal(or to solving the constraints curvature eqs. ofas AdS ( in the explicit examples for unknowns quantities in the ansatz eq. ( where L where second is locally (A)dS conditions. The similar conditions for Ω Here, space-time point of view. (A)dS represent quantities which are tensorsΩ or covariant is derivatives in spincondition connection of in derivatives of tensors in AdS form of covariant conditions,AdS let us write down a AdS JHEP11(2016)024 ) θ ] 2 ρ . θ dθ sinh ) can 2 (C.9) θ θ / ` θθ (C.10) are the 1 θ 2 C.5 + − tanh (1998) ¯ Γ tan tan = sinh ¯ Γ = − 2 (2002) 058 +1 05 d M and , 04 θ θ 2 AdS 1 θ ,N ). For the other ¯ Γ θ ds cot arXiv:1407.5597 JHEP ( 2 ) and eq. ( [ tanh , θ ¯ θ Γ C.6 = 0 JHEP / tanh 2 . tan 2 1 1 2 , C.4 − cos = 1 sec , 0 a Causality Constraints on = ρ du f E 2 + 3)-dimension are, θ θ i or θ ρ , N +2 (2016) 020 d d . However, we cannot obtain θ N 1 ¯ Γ tan tan +1 02 = 2 AdS d tanh − − + sinh ai ds 2 = sinh = cosh ¯ and related factors. Ω θ 0 N θ d ρ θ JHEP +1 , , . d N i ν sec tan ) + and AdS , f tanh µ dθ δ j +1 i ! d 2 = 1 and +1 ¯ d θ Γ 0 M du M θ θ 2 AdS to (A)dS – 59 – = , = θ M ds sech i du νθ sec sec 0 ( +2 µ ¯ d -D gravity in anti-de Sitter space − E − ρ ¯ Γ ), we can construct various type of (A)dS waveguide. 4 2 ,

=cosh , θ ρ 0 ]. θ θ C.6 a µν f because of the first equation in eq. ( 0 g ), which permits any use, distribution and reproduction in E f 1 0 sec tan ]. ¯ Chiral perturbation theory for tensor mesons Γ sech f +1 = cosh d SPIRE = = = cosh +3 IN d a waveguide from dS µν ij ][ SPIRE In this case metric is given by ¯ θ E ¯ d Ω ¯ Γ IN ) 2 AdS (1/1) 22 : σ CC-BY 4.0 (1/-1) ][ ds / ). , (-1/-1) 0 νθ +1 µ This article is distributed under the terms of the Creative Commons σ d +1 + 3)-dimension can be written as ! . ¯ Ansatz for vielbein and spin connection in ( Γ d +1 C.6 is an arbitrary locally AdS metric as we advertised. Higgs phenomenon for d d . But they are just coordinate change of first one. 5 θ represent two compactifying indices. Constraints eq. ( +1 ]. and ρ du j d to dS dρ to AdS sinh to dS µν hep-ph/9708355 / θ 2 AdS +2 sinh and ¯ . Various waveguide from (A)dS [ Γ +2 d SPIRE +2 d ds i Waveguide( = 1 d

IN hep-th/0112166 [ [ 010 Corrections to the Graviton Three-Point Coupling By solving conditions in Eq ( N M. Porrati, C.-K. Chow and S.-J. Rey, X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, There are other cases, for examples = dS = 2 case). AdS AdS i 22 [3] [1] [2] k ¯ E and References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. The metric in ( One can easily generalize this for an arbitrary k. where be solved as parts of constraints eq. ( One can obtain thethe dS AdS waveguide fromcases, dS see table ( Table 5 where JHEP11(2016)024 , D ]. ] , (2003) (1921) SPIRE ] ]. IN (1926) 895. ][ A 18 Nucl. Phys. Phys. Rev. , , 37 SPIRE ]. Super Yang-Mills IN [ = 4 arXiv:0804.2907 and its field theory ]. SPIRE . [ ]. N 5 IN Z. Phys. [ (2016) 667 hep-th/0105181 , [ AdS SPIRE SPIRE ]. Mod. Phys. Lett. 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