Test of the local form of higher-spin equations via AdS/CFT
aa r X i v : . [ h e p - t h ] M a y FIAN/TD/10-17
Test of the local form of higher-spin equations via
AdS/C F T
V.E. Didenko and M.A. Vasiliev
I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,Leninsky prospect 53, 119991, Moscow, Russia [email protected], [email protected]
Abstract
The local form of higher-spin equations found recently to the second order [1] is shownto properly reproduce the anticipated
AdS/CF T correlators for appropriate boundaryconditions. It is argued that consistent
AdS/CF T holography for the parity-brokenboundary models needs a nontrivial modification of the bosonic truncation of the orig-inal higher-spin theory with the doubled number of fields, as well as a nonlinear defor-mation of the boundary conditions in the higher orders.
Higher-spin (HS) theories (see e.g. [2] for a review) have attracted much of interest providinga relatively simple playground for
AdS/CF T correspondence [3]-[5]. Studying these modelsmay shed light on the nature of holography itself. Particularly, some dualities relate compli-cated theory of gravity and infinitely many HS fields in the bulk with simplest CFT dualsbeing just free theories. The HS
AdS/CF T story dates back to stringy tensionless limitargument by Sundborg [6] (see also [7]-[10]) asserting free boundary theory as a HS dual. Aconcrete proposal of Klebanov and Polyakov [11], was that what is known as HS A -modelshould be dual to either free or critical O ( N )-model. The conjecture was later generalizedto supersymmetric theories [12] and to HS B -model in [13]. However, due to the lack ofconventional action principle for HS theory it was not clear how to test those conjecturesat the level of correlation functions until an impressive calculation by Giombi and Yin [14]based on a certain setup for extracting tree-level correlators from equations of motion. In[14] and [15] a substantial piece of evidence in favor of the proposed dualities at the level ofthree-point correlation functions was given. Later on Maldacena and Zhiboedov showed [16]that the presence of infinitely many exactly conserved HS currents in d = 3 constrains CFTtheory drastically leaving one with either a theory of free bosons or free fermions. Yet, even1f one allows for a slight HS symmetry deformation, the CFT is still highly constrained [17].Though in this paper we focus on the AdS /CF T HS holography, it should be noted thatthe important proposal on the
AdS /CF T HS duality was put forward in [18].Despite noticeable success of the
AdS /CF T HS holography tests some loose ends stillremain even at the level of three-point analysis especially in the sector of holographic dualityof parity-noninvariant 3 d conformal theories proposed in [19, 20] exhibiting difficulties inextracting correlation functions from the parity-noninvariant bulk HS theories [21] (wheresome were nevertheless obtained). The main origin of those problems and inconsistenciescan be traced back to the nonlocal setup in HS equations used in the original papers. Indeed,as was noticed in [14] the natural procedure of extracting HS interaction vertices from HSequations results in a nonlocal interaction even at the lowest nontrivial level leading toinfinities in the boundary limit. The origin of these nonlocalities is due to natural ambiguityin field redefinitions in HS equations. Particularly, the procedure of extracting HS verticesamounts to solving some differential equations in the auxiliary spinorial space which resultsin unavoidable problem of fixing a representative. More generally, this is the problem ofthe choice of proper (minimally nonlocal) class of functions respecting physical properties ofnonlocal theories such as HS theory.Partly, the class of functions that respects nonlinear structure of HS equations was pro-posed in [22] and later further narrowed in [1, 23] for the special case of quadratic correctionsin the 0-form sector. In [1] it was shown that the proper field redefinition that brings HSequations into a manifestly local form does exist, fixing relative coefficients of the secondorder HS interaction vertices. In this paper we show that the structure of second-order localHS interactions in four dimensions is in perfect agreement with the CFT expectations.The paper is organized as follows. In section 2 we briefly review HS equations in fourdimensions presenting perturbative expansion up to the second order. Then, in section 3 wediscuss boundary conditions and truncations respecting the AdS/CF T duality. In section 4we extract three-point correlation functions from the 0-form sector of HS equations and insection 5 we leave our conclusion.
HS equations in four dimensions have the form [24]d W + W ∗ W = 0 , (2.1)d S + [ W, S ] ∗ = 0 , (2.2)d B + [ W, B ] ∗ = 0 , (2.3) S ∗ S = − iθ α ∧ θ α (1 + F ∗ ( B ) ∗ k ∗ κ ) − i ¯ θ ˙ α ∧ ¯ θ ˙ α (1 + ¯ F ∗ ( B ) ∗ ¯ k ∗ ¯ κ ) , (2.4)[ S, B ] ∗ = 0 . (2.5)Here master fields W ( Z ; Y ; K | x ), B ( Z ; Y ; K | x ) and S ( Z ; Y ; K | x ) depend on spinorial vari-ables Z A = ( z α , ¯ z ˙ α ) and Y A = ( y α , ¯ y ˙ α ) ( α, ˙ α = 1 , = ( k, ¯ k ). W is a space-time 1-form, B is a 0-form and S is a 1-form in the exterior Z A -directions with anticommuting differentials θ A . Functions of spinor variables Z A = ( z α , ¯ z ˙ α )and Y A = ( y α , ¯ y ˙ α ), α, ˙ α = 1 , f ∗ g )( Z, Y ) = 1(2 π ) Z dU dV f ( Z + U, Y + U ) g ( Z − V, Y + V ) e iU A V A (2.6)( V A = ( ǫ αβ V β , ǫ ˙ α ˙ β V ˙ β ). Inner Klein operators κ and ¯ κ are κ = e iz α y α , ¯ κ = e i ¯ z ˙ α ¯ y ˙ α . (2.7)Outer Klein operator k (¯ k ) is defined to anticommute with all (anti)holomorphic variables { k, V α } ∗ = 0 , k ∗ k = 1 , (2.8)where V α = ( y α , z α , θ α ). This formula extends the star product to k, ¯ k -dependent elements.In this paper we focus on the purely HS sector of the theory where B is linear in k and¯ k while W and S contain the k, ¯ k -independent part as well as bilinear k ∗ ¯ k . Function F ∗ ( B )is set to be linear with an arbitrary constant complex parameter F = ηB , ¯ F = ¯ ηB . (2.9)Now, since fields depend on outer Klein operators k and ¯ k , we assume these to enter on themost right, for example, B ( Z ; Y ; k, ¯ k ) := B ( Z ; Y ) k + ¯ B ( Z ; Y )¯ k . (2.10)System (2.1)-(2.5) can be analyzed perturbatively. One starts with the vacuum solutionthat corresponds to pure AdS space-time B = 0 , (2.11) S = Z A θ A , (2.12) W = i ω αα y α y α + ¯ ω ˙ α ˙ α ¯ y ˙ α ¯ y ˙ α + 2 e α ˙ α y α ¯ y ˙ α ) . (2.13)We take Poincare coordinates as well adopted for the boundary analysis ω αα = − i z d x αα , ¯ ω ˙ α ˙ α = i z d x ˙ α ˙ α , e α ˙ α = 12 z ( d x α ˙ α − iǫ α ˙ α dz ) , (2.14)where x αβ = x βα denote the three boundary coordinates (independently of whether theycarry dotted or undotted indices) while z is the Poincar´e coordinate.First-order equations reduce to twisted-adjoint flatness condition for the 0-form B = C ( Y ; k, ¯ k ) DC = D L C + ie α ˙ α ( y α ¯ y ˙ α − ∂ α ∂ ˙ α ) C = 0 , (2.15)3nd First on-shell theorem for HS potentials ω ( Y ) Dω = i (cid:0) η ¯ H ˙ α ˙ α ∂ α ¯ C (0 , ¯ y ; k, ¯ k )¯ k + ¯ ηH αα ∂ α C ( y, k, ¯ k ) k (cid:1) . (2.16)At second order the local form of HS equations was extracted from (2.1)-(2.5) in [1] for0-form C ( Y ) and in [26] for 1-form ω ( Y ). The equation for 0-form reads DC = i ηe α ˙ α Z e i ¯ u ˙ α ¯ v ˙ α y α ( t ¯ u ˙ α + (1 − t )¯ v ˙ α ) J ( ty, − (1 − t ) y, ¯ y + ¯ u, ¯ y + ¯ v ) k + c.c , (2.17)where J ( y , y , ¯ y , ¯ y ; k, ¯ k ) := C ( y , ¯ y ; k, ¯ k ) C ( y , ¯ y ; k, ¯ k ) , (2.18)and we use the short-hand notation for integrals Z F ( t , . . . , t n ; ¯ u, ¯ v ) := Z [0 , n dt . . . dt n Z R π ) d ¯ ud ¯ vF ( t , . . . , t n ; ¯ u, ¯ v ) . (2.19)Similarly for integrals that contain both holomorhic u, v and antiholomorhic ¯ u, ¯ v integrationvariables. HS equations (2.1)-(2.5) admit various truncations. Due to dependence on Klein operators k and ¯ k there are two copies of fields of every spin. In the bosonic case, the spectrum canbe reduced down to a single copy by setting B ( Z, Y ; k, ¯ k ) → B ( Z, Y )( k + ¯ k ) , W ( Z, Y ; k, ¯ k ) → W ( Z, Y )(1 + k ¯ k ) . (3.1)While bosonic truncation (3.1) can be imposed to all orders reducing the spectrum of thetheory, it is not a priori guaranteed that it has any CFT dual at all in the HS theorieswith broken parity [25] . Within the perturbation theory however one can impose conditionrelating fields of the full theory with the doubled spectrum in such a way that the theorybecomes bosonic yet different from the one resulting from (3.1). To explain the origin of themodified conditions driven by the AdS/CF T requirement let us analyze the boundary limitof the full fledged HS system in perturbation theory.
Free-level analysis has been carried out in [25]. According to it the field-current correspon-dence is reached via the following identification C ( y, ¯ y ; k, ¯ k ) = ze y α ¯ y α T ( w, ¯ w ; k, ¯ k ) , (3.2)4here w = √ zy , ¯ w = √ z ¯ y . (3.3)Eq. (3.2) says that if C is on-shell, that is satisfies (2.15), then T enjoys the unfolded formof the 3 d conformal current conservation equationd x T − i αα ∂ α ¯ ∂ α T = 0 . (3.4)HS potentials are sourced by the field C in the bulk in accordance with (2.16). Its boundarypushforward reads D x ω x = 14 H αβ xx ∂ ∂w + α ∂w + β (cid:0) ¯ ηT ( w + , k − ηT (0 , iw + )¯ k (cid:1) . (3.5)One concludes that, in general, boundary HS fields, which are gauge fields of the boundaryconformal HS theory, are sourced by currents. To make AdS/CF T work in the standardsense, i.e., for the usual boundary CFT with the well-defined stress tensor, one has to imposesuch boundary conditions that make the right hand side of (3.5) vanish allowing to get rid ofboundary HS gauge fields which can make the boundary stress tensor gauge non-invariant.For η = 1 or η = i proper conditions read T ( w, ¯ w ) k = ± T ( − i ¯ w, iw )¯ k . (3.6)It is important that one can exclude scalar and spinor fields from (3.6) since they do not affect(3.5) (at higher orders this will not be the case) opening the way to alternative boundaryconditions in this sector, corresponding to the critical boundary models in accordance withthe original proposal of [11]-[13]. Let us also note that there is no way to include generalparameter η into (3.6) demanding η = 1 or η = i .However, for general η conditions (3.6) can be modified as follows. Having two fields inthe decomposition C ( Y ; k, ¯ k ) = C ( Y ) k + ¯ C ( Y )¯ k (3.7)one can identify positive helicity component of a bosonic Weyl module with C ( Y ), whilenegative helicity part of the same field with ¯ C ( Y ), i.e. C ( Y ) := C + ( Y ) , ¯ C ( Y ) := C − ( Y ) , (3.8)where by the doubled helicity of a spin s field we mean the difference between the numberof y and ¯ y variables, in other words, the eigenvalue of the following operator n = y α ∂∂y α − ¯ y ˙ α ∂∂ ¯ y ˙ α . (3.9)Particularly, C + carries more y variables than ¯ y and C − other way around. Let us stressthat this way one truncates the spectrum to the bosonic system in a way different from (3.1)allowing to get rid of the sources in (3.5) in the parity broken case by setting¯ ηT + ( w, ¯ w ) = ηT − ( − i ¯ w, iw ) . (3.10)5o make contact of the introduced boundary conditions with those usually imposed in theHS literature consider HS boundary to bulk propagators. In the 0-form sector the positiveand negative helicity parts have the following form [14] C + = ηKe if α ˙ α y α ¯ y ˙ α + iξ α y α , C − = ¯ ηKe if α ˙ α y α ¯ y ˙ α + i ¯ ξ ˙ α ¯ y ˙ α (3.11)(no η -factors for a scalar), where K = z (x − x ) + z , (3.12) f α ˙ α = − z (x − x ) + z (x − x ) α ˙ α − i (x − x ) − z (x − x ) + z ǫ α ˙ α , (3.13) ξ α = Π αβ µ β , Π αβ = K (cid:18) √ z (x − x ) αβ − i √ zǫ αβ (cid:19) , (3.14)and the reality conditions for polarization spinors are µ α = i ¯ µ α . (3.15)While we will not use it in this paper, let us give for completeness the explicit formula for1-form ω propagator ω = − i Ke α ˙ α ξ α ¯ ξ ˙ α Z dte itξ α y α + i (1 − t )¯ ξ ˙ α ¯ y ˙ α . (3.16)Scalar part of the propagator (3.11) corresponds to the ∆ = 1 solution. Another scalarbranch that stands for ∆ = 2 reads C ∆=2 = K (1 + if α ˙ α y α ¯ y ˙ α ) × e if α ˙ α y α ¯ y ˙ α . (3.17)Let us show how these propagators match different reality conditions just spelled out.Using boundary prescription (3.2) one finds for (3.11) T + = η | x − x | e − i (x − x ) − αα w α ¯ w α + i (x − x ) αβ µ β w α , (3.18) T − = ¯ η | x − x | e − i (x − x ) − αα w α ¯ w α + i (x − x ) αβ ¯ µ β ¯ w α (3.19)and T ∆=2 = w α ¯ w α | x − x | × e − i (x − x ) − αα w α ¯ w α . (3.20)One can see now that condition (3.6) for η = 1 is fulfilled for (3.18), (3.19) while for η = i one has to use (3.20) in accordance with parity-odd scalar condition for HS B -model. For This form of the propagator was found by one of us (V.D.) with Zhenya Skvortsov in 2014 but was neverpublished. η (3.18) and (3.19) as well as (3.20) for alternative scalar boundary condition satisfy(3.10).Having HS equations to the second order one may wish to examine them in the boundarylimit. Particularly, the expectation for (3.6) boundary condition for the A and B HS theoriesis that in these cases HS symmetry remains undeformed leading to conservation of boundarycurrents yet leaving no HS gauge fields at the boundary. We will show that this is indeedthe case. For boundary conditions like (3.10) or for alternative scalar like in the criticalcase on the contrary it turns out that HS potentials get sourced on the boundary and oneshould introduce certain nonlinear completion for (3.10) at higher orders to make themvanish. This implies among other things that without such a nonlinear completion the tree-level correlation functions extracted from the bulk are anticipated to differ from boundaryexpectation starting from the 4-point functions.
Let us carry out boundary limit for (2.17). This will give us the deformed version of currentequation (3.4). The limit is quite straightforward using (3.2). The final result isd x T − i αα ∂ α ¯ ∂ α T = − η αα w α Z ( t ¯ ∂ α − (1 − t ) ¯ ∂ α ) I (cid:0) tw, − (1 − t ) w, ¯ w + i (1 − t ) w, ¯ w − itw (cid:1) k + c.c. , (3.21)where I ( w , w , ¯ w , ¯ w ) = T ( w , ¯ w ; k, ¯ k ) T ( w , ¯ w , k, ¯ k ) . (3.22)Note, that while field-current correspondence (3.2) contain potentially dangerous projector e iy α ¯ y α which may cause infinities at the boundary it turns out that no divergencies appeardue to specific dependence on the homotopy parameter t in (2.17). One observes thatcurrents receive contributions originated from current-current interaction that may lead tononconservation. Indeed, from (3.21) it follows that ∂∂w α ∂∂w β ∂∂ x αβ T == − η Z dt (cid:18) w α ∂∂w α (cid:19) (cid:16) t ¯ ∂ β − (1 − t ) ¯ ∂ β (cid:17) ∂∂w β h T ( tw, ¯ w + i (1 − t ) w ) T ( − (1 − t ) w, ¯ w − itw ) i k + c.c. (3.23)which is nonzero in general resulting in ∂ · J s = 0 . (3.24)Let us analyze this issue starting from the parity preserving models. In this case with theboundary conditions (3.6) one finds that despite the deformation is nonlinear the boundarycurrents remain conserved ∂∂w α ∂∂w β ∂∂ x αβ T = 0 ⇒ ∂ · J s = 0 . (3.25)7his can be most easily seen from noting that under (3.6) the right-hand side of (3.21) getsrewritten asd x T − i αα ∂ α ¯ ∂ α T = − η αα w α ∂∂w α Z dtI (cid:0) tw, − (1 − t ) w, ¯ w + i (1 − t ) w, ¯ w − itw (cid:1) , (3.26)from where (3.25) immediately follows. The fact that for free theories (3.26) results incurrent conservation means that there is a local field redefinition that brings (3.26) to thecanonical conserved current form (3.4). For parity broken boundary condition (3.10) theHS currents no longer conserve. In obtaining (3.26) the structure of (3.21) was important.Particularly one uses the symmetry with respect to the exchange t → − t . The checkcarried out for parity preserving boundary conditions (3.25) alone is sufficient to justify theagreement between bulk vertices given in (2.17) and boundary free theory 3pt correlationfunctions. Indeed, according to Maldacena-Zhiboedov theorem [16] the conservation of HScurrents inevitably implies free boundary theory.A soft spot in this argument is the following. As a matter of principal it may happenthat while HS currents do conserve the theory still contains sources for the boundary HSconnections, in which case the standard AdS/CF T correspondence can be lost. So let uscheck out the conditions at which sources for HS connections do vanish. To do so we shouldanalyze the 1-form sector found in [26] in the boundary limit.It is easy to perform boundary limit for current interaction equation in the 1-form sectorof [26] following the logic of [25], arriving at the equation D x ω x ( w + , v − ) = i η ¯ η Z d tδ ′ (1 − t − t ) H αα xx (cid:18) ∂∂u α (cid:19) × (3.27) × n I ( t ( w + + u ) , − t ( w + + u ) , it w + , − it w + ) − I ( t w + , − t w + , it ( w + + u ) , − it ( w + + u )) o(cid:12)(cid:12)(cid:12) u =0 , where the following variables have been introduced w = w + + izv − , ¯ w = iw + + zv − . (3.28)Just as well as at the linearized level, one observes that sources for HS connections do notvanish in general (although they almost do since the two terms on the right-hand side of(3.27) are equal to each other at u = 0). However, imposing free theory boundary conditions(3.6) one finds exact cancellation and the theory becomes free of boundary HS connectionsin accordance with the AdS/CF T expectation.An important comment is now in order. One may expect that for parity breaking bound-ary conditions (3.10) or for those of critical theories one has vanishing sources for boundaryconnections too. This is not the case as (3.6) is likely to be the only linear relation thatcancels out sources. This implies that on the way of proposed dualities with critical modelsand vectorial models with Chern-Simons matter one has to modify boundary conditions tocompensate the nonlinear corrections.Also it should be noted that in the CFT-based HS literature terms on the r.h.s. of (3.21)are usually interpreted as “slight breaking” of HS symmetry [17]. From the perspective of8he original HS equations, however, they are naturally interpreted as a deformation ratherthan breaking of HS symmetry. Indeed, consistency of nonlinear terms on the r.h.s. ofHS field equations implies that the HS gauge symmetry transformations receive nonlinearcorrections as well. The tricky point is that, at the boundary, the resulting deformationmay go beyond the standard class of CFTs with well defined (gauge invariant) stress tensorbecause the deformed HS gauge transformation in most cases mixes HS 0-forms, that haveclear meaning from the boundary CFT perspective, with the HS 1-forms at the boundary,which are conformal HS gauge fields on the boundary not allowed in the standard CFTs.
In this section we venture to extract correlation functions of a dual theory from the 0-formsector of the bulk field equations. We restrict ourselves to the case with two sources on theboundary s and s that generate spin s such that s ≥ s + s . (4.1)This constraint comes from the fact that so far we have taken into account only currentinteractions given in (2.17), which is only consistent when restriction (4.1) is imposed sinceotherwise the contribution of HS 1-forms also has to be taken into account. For the oppositecase of three spins obeying the triangle inequalities, the original current interaction is sup-ported by the HS 1-forms and is local in the original setup of Giombi and Yin [14], givingthe proper answer.According to the proposal of [14] a solution to the second order equation for Weyl 0-form(2.17) generated by two boundary sources can be associated with a properly normalized 3ptfunction via h J J J i ∼ lim z → z − G ( wz − , ¯ wz − ) (cid:12)(cid:12)(cid:12) ¯ w =0 , (4.2)where G ( y, ¯ y ) is a Green’s function for equation (2.17). The remaining w -variable is tobe associated with a polarization spinor for the outgoing leg of spin s . Though such aprescription for the computation of correlation functions may need some further justification,for a time being we take it as a working tool. Before going into technical details of thecomputation we give general arguments on the dependence of the boundary correlators onthe phase parameter in the HS theory. In this section we reconsider the analysis of HS holography of [1] in a more conventionalsetup leading to the same conclusions. To this end, consider HS equations of [26] Dω ( y, ¯ y ) = i (cid:16) η ¯ H ˙ α ˙ β ∂ ∂y ˙ α ∂y ˙ β C − (0 , y | x ) + ¯ ηH αβ ∂ ∂y α ∂y β C + ( y, | x ) (cid:17) + η ¯ η Γ loc ( J ) , (4.3)9here Γ loc is the second-order current interaction, C ± denote positive and negative helicityparts of C ( y, ¯ y ) and the dependence on the Klein operators is discarded. Though as shown in[1, 26] the quadratic J -dependent deformation is independent of the phase of η = | η | exp iϕ ,the linear part is phase-dependent. Introducing the new fields C ′− = ηC − , C ′ + = ¯ ηC + , (4.4)Eq. (4.3) takes the form Dω ( y, ¯ y ) = i (cid:16) ¯ H ˙ α ˙ β ∂ ∂y ˙ α ∂y ˙ β C ′− (0 , y | x ) + H αβ ∂ ∂y α ∂y β C ′ + ( y, | x ) (cid:17) + Γ loc ( J ( C ( C ′ ))) , (4.5)where, setting for simplicity | η | = 1, C ( C ′ ) = exp iϕ C ′ + + exp − iϕ C ′− . (4.6)Clearly, redefinition (4.6) is an U (1) electromagnetic duality transformation with the phase ϕ . The linear term in Eq. (4.5) tells us that it is the 0-form C ′ that has to be identifiedwith the generalized Weyl (Faraday for s = 1) tensor associated with the curvatures of theFronsdal HS fields contained in ω ( y, ¯ y ). In these terms, the vertex which was ϕ -independentin terms of C acquires the nontrivial ϕ -dependence in terms of C ′ Γ loc ( J ) = Γ loc (exp 2 iϕJ ++ ( C ′ ) + exp − iϕJ −− ( C ′ ) + 2 J + − ( C ′ )) . (4.7)Since the A -model with ϕ = 0 and B -model with ϕ = π are known to correspond to bosonicand fermionic parity-invariant boundary vertices, we set J b := J ++ ( C ′ ) + J −− ( C ′ ) + 2 J + − ( C ′ ) , J f := − J ++ ( C ′ ) − J −− ( C ′ ) + 2 J + − ( C ′ ) . (4.8)The remaining parity-odd boundary vertex is associated with J o = i ( J ++ ( C ′ ) − J −− ( C ′ )) . (4.9)In terms of these currents, Γ loc ( J ) acquires the form Γ loc ( J ) = cos ( ϕ ) J b + sin ( ϕ ) J f + 12 sin(2 ϕ ) J o (4.10)coinciding with the expression obtained in [1] by slightly different arguments. This expressionprecisely matches the dependence on the phase ϕ anticipated from the HS holography [17,19, 20, 21].To summarize, the proper phase dependence of the current interactions in the phase-independent vertex results from that in the terms linear in the 0-forms upon the identificationof the genuine HS Weyl tensors. 10his simple analysis is useful in many respects. In particular it shows that, to find thephase dependence of the boundary correlators it suffices to know it for any three different datain ϕ . For instance it is enough to find the boundary correlators in the A -model with ϕ = 0, B -model with ϕ = π/ ϕ -derivative at its B -model value ϕ = π/ d conformal structure proposed in [27]as a massive deformation of 3 d fermionic currents. Indeed, from the boundary perspective theparameter η is closely related to the 3 d massive boundary deformation though at a nonzeroVEV of the 0-form B, which, though making sense in the model including topological fieldsnot considered in this paper, has to be set to zero in the end of the computation. The righthand side of equation (2.17) contains two pieces proportional to η and ¯ η , respec-tively. Therefore the Green’s function can be found as a sum of two G = G η + G ¯ η , (4.11)where G η (similarly G ¯ η ) obey the equation DG η = i ηe α ˙ α Z e i ¯ u ˙ α ¯ v ˙ α y α ( t ¯ u ˙ α + (1 − t )¯ v ˙ α ) J ( ty, − (1 − t ) y, ¯ y + ¯ u, ¯ y + ¯ v ) k . (4.12)In terms of power series, the Green’s function was analyzed in [23]. Here we would like tohave its representation suitable for practical calculations. So, let us use the following Ansatzfor G η , G η = η Z f ( t , t , t ) e iu A v A J ( u + t y, t v − t y, ¯ y + ¯ u, ¯ y + ¯ v ) k , (4.13)which is most general in the holomorphic sector of spinor variables. Since the measure in(4.47) is compact and assuming that a function (distribution) f ( t , t , t ) is well defined wewill be freely integrating by parts. Substituting (4.13) into (2.17) one finds (for more detailsee [23]) f ( t , t , t ) = 12 δ ′ (1 − t − t − t ) , (4.14)and, therefore, G η = η Z δ ′ (1 − t − t − t ) e iu A v A J ( u + t y, t v − t y, ¯ y + ¯ u, ¯ y + ¯ v ) k . (4.15)So defined Green’s function does not satisfy (2.17) in general in the first place because (2.17)is not everywhere consistent in particular because the contribution of 1-forms should be takeninto account if the constraint (4.1) is not respected. But even for s ≥ s + s when (4.1)is fulfilled the Green’s function, (4.15) is only valid for those sources in which constituentfields C ( y, ¯ y ) and C ( y, ¯ y ) have opposite chiralities (3.9), i.e., n ( C ) n ( C ) < . (4.16)11ince the extension of the Green’s function to the general case with arbitrary signs of chi-ralities yet remains to be constructed, our strategy will be as follows. Assuming that thecoefficients in correlation function depend solely on spins in the vertex, i.e., modules of he-licities, we will take sources ( i.e., constituent fields) of opposite chiralities in the calculation.There are two sets of primary currents stored in the boundary limit of the Green’sfunction: those depending only on w or only on ¯ w . We focus on the w -dependent oneswhich makes it possible extracting correlation functions from the holomorphic part G η . Inaccordance with the general analysis of section 4.1, there are three different structures thatarise upon substituting propagators (3.11) that satisfy boundary conditions (3.10) h J J J i boson ∼ G ++ + G −− + G + − + G − + , (4.17) h J J J i fermion ∼ G ++ + G −− − G + − − G − + , (4.18) h J J J i odd ∼ G ++ − G −− , (4.19)where pluses and minuses denote chirality signs. The dependence on the phase parameter η isfixed according to (4.10). Particularly, it follows that G ++ + G −− and G + − + G − + correspondto free theories correlators. Substituting propagators (3.11) into (4.15) and performing simpleGaussian integration leads to the following result in the leading order in zG + − = Z d t K K ∆ δ ′ (1 − t − t − t ) e t t Q + t ((1 − t ) P + zt ˜ S ) − t ((1 − t ) P + zt ˜ S ) , (4.20) G − +12 = Z d t K K ∆ δ ′ (1 − t − t − t ) e t t Q + t ((1 − t ) P − zt ˜ S ) − t ((1 − t ) P − zt ˜ S ) . (4.21)Here, indices 1 and 2 label points at the boundary and∆ = (1 − t ) + z ǫ t + O ( z ) , ǫ = 2 x x x , (4.22)where the outgoing leg x is denoted by x x := x . (4.23)The parity-preserving conformal structures are denoted by P and Q , while ˜ S denote parity-odd ones. We specify these later upon taking the boundary limit.An important comment is that, naively, it looks like in the boundary limit z → S vanish, because they are accompanied by a factor of z . This is notthe case due to the pole at z = 0 resulting from ∆ upon integration over t . Indeed, carefulanalysis shows that the terms (1 − t ) and zt in exponentials (4.20) and (4.21) give the samecontribution in z as z →
0. Carrying out the boundary limit z → t t one arrives at the following result G ++12 = z K s s s Q s − s − s | x || x || x | Z ∞ dτ τ s ( τ P + S ) s ( − τ P + S ) s (1 + τ ) s + s + s +1 , (4.24) G −− = z K s s s Q s − s − s | x || x || x | Z ∞ dτ τ s ( τ P − S ) s ( − τ P − S ) s (1 + τ ) s + s + s +1 , (4.25) G + − = z K s s s Q s − s − s | x || x || x | Z ∞ dτ τ s ( τ P + S ) s ( − τ P − S ) s (1 + τ ) s + s + s +1 , (4.26) G − +12 = z K s s s Q s − s − s | x || x || x | Z ∞ dτ τ s ( τ P − S ) s ( − τ P + S ) s (1 + τ ) s + s + s +1 , (4.27)(4.28)where K s s s = 2 s − s − s ( s + s + s )!(2 s )!(2 s )!(2 s )! . (4.29)Note that for equal chirality signs, i.e. for G ++ and G −− , the coefficient K s s s would bedifferent should we still used (4.15) in this case as a Green’s function.The conformal structures appear in the following combinations P = i (x ) αα w α µ α | x | , P = i (x ) αα w α µ α | x | ; Q = (cid:18) x | x | − x | x | (cid:19) αα w α w α , (4.30) S = (x ) βα (x ) αγ µ γ w β | x || x || x | , S = (x ) βα (x ) αγ µ γ w β | x || x || x | . (4.31)(4.32)To identify three-point correlation functions from (4.24)-(4.27) one uses prescription (4.17)-(4.19) and symmetrization over the sources at x and x .As noted above, G ++ + G −− and G + − + G −− correspond to the parity-preserving three-point functions. To verify these against free theory correlators let us start with G + − + G − +12 = z K s s s Q s − s − s | x || x || x | Z ∞−∞ dτ τ s ( τ P + S ) s ( τ P + S ) s (1 + τ ) s + s + s +1 . (4.33)Naively it may seem that (4.33) has nothing to do with correlators of currents of free bosonand free fermion as it depends on the parity-odd structure S . However, since the integrationin (4.33) is carried out along the real axis, the parity-odd structures will appear in bilinearcombinations leading to a parity-even result. Since conformal structures (4.30)-(4.31) arenot algebraically independent (see e.g., [29] for a list of identities on these structures), itis hard to identify in this expression the product of cosines and sines found in [15]. (Theform of the final result is sensitive to a particular representation choice.) For a simple check13howing that the result matches free theory correlators it is convenient to fix boundary pointsas follows x = 0 , x = x , x = x − δ , | δ | ≪ | x | (4.34)and take equal polarization spinors w α = iµ α = iµ α = λ α . (4.35)In addition it is convenient to require −→ x · −→ λ = 0 . (4.36)In this limit, which was also used in [14] and is similar to the light cone limit of [16], the 3ptcorrelators calculated in a free theory amount to [14] h J s (x , λ ) J s (x − δ, λ ) J s (0 , λ ) i λ · x=0 ∼ Γ( s + s + )Γ( s + ) πs ! s ! s ! ( λ · δ ) s + s + s | x | s +2 | δ | s +2 s +1 . (4.37)Let us see what (4.33) gives in this limit. From (4.30)-(4.31) one finds P = 0 , P = δ · λ | x | , Q = − δ · λ | x | , (4.38) S = S = i (x · δ ) αα λ α λ α | x | | δ | . (4.39)Since δ ≪ x we can neglect P in (4.33) and therefore G + − + G − +12 = z K s s s Q s − s − s S s +2 s | x | | δ | Z ∞−∞ dτ τ s (1 + τ ) s + s + s +1 . (4.40)Using Fierz ( i.e., Schoutens) identities it is easy to see, that (note, that (4.24)-(4.27) do notapply for half-integer spins) S = − ( δ · λ ) | x | | δ | (4.41)leading to G + − + G − +12 ∼ K s s s ( λ · δ ) s + s + s | x | s +2 | δ | s +2 s +1 Z ∞−∞ dτ τ s (1 + τ ) s + s + s +1 . (4.42)Integrating by residues, Z ∞−∞ dτ τ s (1 + τ ) s + s + s +1 = Γ( s + )Γ( s + s + )( s + s + s )! (4.43)and substituting K s s s one finds G + − + G − +12 ∼ Γ( s + )Γ( s + s + )(2 s )!(2 s )!(2 s )! ( λ · δ ) s + s + s | x | s +2 | δ | s +2 s +1 , (4.44)14hich is consistent with the free theory prediction (4.37) upon an appropriate 2pt-normalization.Same is true for G ++12 + G −− .The parity-odd contribution resides in G ++12 − G −− and the corresponding three-pointfunction can be obtained from that expression by symmetrizing sources at x and x . Up tothe two-point function normalization h J s J s i the final result reads h J s J s J s i odd ∼ K s s s | x || x || x | Z ∞ dτ τ s (1 + τ ) s + s + s +1 Q s − s − s × (4.45) (cid:0) ( τ P + S ) s ( − τ P + S ) s − ( τ P − S ) s ( − τ P − S ) s (cid:1) + (x , µ , s ) ↔ (x , µ , s ) . Recall that spins are restricted by (4.1). To see that the result is nonzero it is enough toconsider the case of s = 1, s = 0 which gives h O ∆=1 (x ) J (x ) J s (x ) i ∼ s !(2 s )! 2 s − | x || x || x | ( Q s − + ( − Q ) s − ) P S . (4.46)Similarly, using (3.20) propagator one can calculate correlation functions corresponding tocritical models. We do not perform this calculation in our paper. Note that from theboundary side nonconservation of HS currents in the parity-broken case was recently studiedin [28], where some correlators were explicitly found. It will be interesting to compare themwith (4.45).As stressed earlier, the form of the final result (4.45) heavily depends on the freedom inusing relations on conformal structures (4.30)-(4.31). We expect (4.45) to admit a simplerrepresentation. In this respect it is interesting to note that typical integrals that show up inthe boundary limit of a Green’s function G ( C s , C s ) ∼ Z ∞ dτ τ s (1 + τ ) s + s + s +1 ( τ a + b ) s ( τ c + d ) s , (4.47)where a, b, c, d are some conformal structures among list (4.30)-(4.31), can be rewritten uponthe change of integration variable τ = tan φ as R Z π/ dφ sin s φ sin s ( φ + φ ) sin s ( φ + φ ) , (4.48)where R = ( a + b ) s ( c + d ) s , tan φ = ba , tan φ = dc . (4.49)This representation may be useful for finding a simpler representation for the parity-oddthree-point functions. The main findings of our work are the following. We have examined local form of HS equa-tions to the second order at the level of equations of motion in its most sensitive part of the15urrent interaction sector with spins obeying s ≥ s + s , i.e., outside the triangle inequalityregion. We have checked whether the coefficients obtained in [1] and [26] are consistent withthe boundary theory expectations and found perfect agreement. Particularly, the boundarylimit that describes deformation to current conservation condition is consistent with therequirement for free theories to have exactly conserved HS currents. For these boundaryconditions we have also checked that, in agreement with the conventional AdS/CF T pre-scription, no HS connections survive at the boundary. These facts, being crucially dependenton the structure of vertices obtained in [1], confirm that the prescription of [1] is the onlyproper one. Still we have carried out some calculation at the level of three point functionsextracted from the 0-form sector a la
Giombi and Yin [14]. Though details of the prescriptionof extracting correlators from the 0-form sector of HS equations is not entirely clear to us andperhaps needs some further analysis (particularly, this concerns the argument on the linearrelation between the Weyl module and HS connections) we found perfect agreement in caseof free theories. For parity broken case we have calculated correlation functions h J s J s J s i odd for s ≥ s + s using the same approach. The result is nonzero which seemingly contradictsto the analysis of [16] where parity-odd three-point functions were found within the triangleidentity s i ≤ s j + s k and it was claimed that for s ≥ s + s the result is zero. However, it isimportant to note, that in paper [16] all HS currents were supposed to be conserved, whilein our case we do not have current conservation for parity-odd case which is in agreementwith general analysis of [29].Another observation of our work highlighting the conjecture of [25] on the role of theboundary conditions is that, apart from the case of free boundary theories, no boundaryconditions linear in the HS 0-forms make the sources to the boundary HS connection van-ish. Particularly, for critical models and parity-broken models a nonlinear correction to thesource for boundary connections always springs out. This implies that boundary conditionsconsistent with the standard AdS/CF T prescription may need a nonlinear deformation an-ticipated to become important starting from the 4pt correlation functions. This deformationis similar to the one observed recently in N = 8 supergravity theory which requires boundarysupersymmetry modification in order to match superconformal correlation functions [30].Finally, the analysis carried out in this paper suggests that even in the purely bosoniccase there exist perturbatively different reductions of the full nonlinear HS equations withthe doubled set of fields compared to the naive reduction with a single set of bosonic fieldsof any spin. Note added
After completion of our work we learned that closely related problem was considered in [31]by E. Sezgin, E.D. Skvortsov and Y. Zhu 16 cknowledgements
We acknowledge a partial support from the Russian Basic Research Foundation Grant No17-02-00546. MV is grateful to the Galileo Galilei Institute for Theoretical Physics (GGI)for the hospitality and INFN for partial support during the completion of this work withinthe program New Developments in AdS3/CFT2 Holography. The work of MV is partiallysupported by a grant from the Simons Foundation. V.D. is grateful to professor HermannNicolai for kind hospitality at AEI where a part of this work has been done. We are gratefulto Olga Gelfond for important comments, Simone Giombi for the correspondence and ZhenyaSkvortsov for pointing out a misleading notation in Section 4.2 of the original version of thepaper. V.D. thanks Sasha Zhiboedov and Yi Pang for fruitful discussions.
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