Three dimensional higher spin holography

Transcription

1 Three dimensional higher spin holography I. Lovrekovic The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 AZ, United Kingdom arxiv: v [hep-th] Jan 08 January 3, 08 Abstract We consider linearised Vasiliev s equations around the background AdS field in three dimensions for the scalar-scalar-higher spin field correlation function. Relating this with the higher spin field determined in the metric formulation allows determination of the corresponding coupling coefficient. The result agrees with the analogous computation for the spin three field. Corfu Summer Institute 07 School and Workshops on Elementary Particle Physics and Gravity, 6 September - 3 September, 07 lovrekovic@hep.itp.tuwien.ac.at

2 Introduction and Motivation Higher spin (HS) theories have started receiving more attention in the recent years due to a fact that they offer an answer to some important questions in string theory. They have been introduced in the early years by Vasiliev [,, 3, 4]. We can describe them characterising free fields by spin and mass and considering the consistent interaction among them. Writing consistent interaction for fields of arbitrary spin and mass turned out to be difficult, therefore one usually starts by looking at the massless fields. Their particular property is that interaction terms due to the masslessness need to be restricted by gauge symmetry, making them interesting and simpler to study. For the fields with spin higher than, the spectrum of the theory is necessarily infinite and contains the fields of all spins. The theory is minimal, which means that each spin state appears only once unlike in the string theory where spectrum contains degenerate states. Another property of the theory is that it does not contain dimensional parameter like string theory therefore it cannot come from spontaneous symmetry breaking. In the early days of its development the theory encountered number of no-go theorems due to a fact that interacting higher-spins do not propagate in the Minkowski space. This issue was solved be consideration of the HS fields on AdS while first equations of motion and action for arbitrary HS fields were introduced by Fronsdal whose action described a tower of massless non-interacting HS fields [5, 6, 7]. The issue that remained is construction of the interacting action of the theory for the spin higher than three. The interacting theories can be constructed order by order using perturbative interaction procedure. One deforms a quadratic Lagrangean of a particle spectrum of certain spins and masses by cubic terms, while keeping the gauge invariance which he repeats at the following step for quartic terms, etc. The procedure also deforms the gauge transformations. The solution of the cubic deformations gives cubic interaction vertices [8]. HS theory can also be considered in the sense of AdS/CFT correspondence, where the HS theory of massless HS fields corresponds to a limiting case of the string theory for the string tension going to zero. In [8] they have been studied in relation with N=4 SYM theory. Large N superconformal field theories were studied as holographic duals for higher spin gauge theories in perturbative expansion around AdS spacetime. The initial proposal of the duality dates to Klebanov and Polyakov [9, 0] and correspondence between the large N limit of the 3d O(N) vector model and higher spins on AdS 4 which has through further studies provided evidence which imply that the theory is correct. The analog of AdS 4 conjecture by Klebanov and Polyakov appeared in AdS 3 conjecturing a duality between a complex scalar coupled to higher-spin fields in Vasiliev s gravity in 3 dimensions and W N minimal model CFT in t Hooft limit denoted by coset representation SU(N) k SU(N) SU(N) k+, () where we define the t Hooft limit with N,k for λ N k+n. This duality has been verified by the number of studies, correspondence of global symmetries in bulk and at the boundary [], correspondence of the -loop partition function in the bulk and at the large NCFT [], andpartitionfunctionofthehsblackholeathightemperatureinthebulkandattheboundary CFT, as well as for the 3-point functions when λ = and s =,3,4 for the scalar-scalar-hs field (00s) correlator in the t Hooft limit. The 00s correlator for the general λ has as well been shown to agree from the bulk and CFT side in [0]. In this work, we extract the coupling of the 00s three-point correlator by considering the linearised Vasiliev s equations of motion, and we verify it by choosing the spin to be three and

3 comparing with result in [0]. The result corresponds to coupling of the three-point correlation function up to selected normalisation. The structure of the work is as follows: In the section two we consider linearised equations of motion in the Vasiliev s theory, in the section three we consider the higher spin field in the metric formulation, and in section four we conclude. Linearized equations of motion Let us first consider the coefficient coming from the Vasiliev s linearised equations. Vasiliev s theory contains five equations for the master fields W which is spacetime -form, B and S α which are spacetime 0-forms. The generating functions are dependent on the coordinates of the spacetime, auxiliary bosonic twistor variables (referred to as oscillators ) and Clifford element pairs, where in definitions we follow conventions from [0]. The oscillators and various other ingredients are used to define the deformed oscillator star-commutation relations which give rise to hs[λ] higher spin algebra. Two of the above mentioned equations that will be of the interest here are We can rewrite W with projector operators dw = W W () db = W B B W (3) P ± = ±ψ (4) for ψ elements of the Clifford pairs such that W = P + A P A for P ± ψ = ψ P ± = ±P ± P ± ψ = ψ P (5) where A are Chern-Simons gauge fields which take value in the Lie algebra hs[λ]. In this formulation the equation dw = W W (6) gives da+a A = 0 (7) DA+A A = 0 (8) where A and A are positive polynomials of the positive degree in products of deformed oscillators. (7) and (8) are in that case equal to field equations hs[λ] hs[λ] Chern-Simons theory. The generators of hs[λ] are defined with spin index s and mode index m as Vm s for s while m < s and obey star product V s m V t n = s+t s t u=,,3 g st u (m,n;λ)v s+t u m n (9) where g st u (m,n;λ) = ( )u+ gu ts (m,n;λ) (0) 3

4 are specific coefficients dependent on λ and defined according to conventions [0]. The equations describe interaction of arbitrary higher spin background with lienarized scalars. The coupling that we are interested in can be extracted from expanding the master field C in the deformed oscillators ỹ α in the equation dc +A C C A = 0. () That allows us determining the generalised Klein-Gordon (KG) equation in the background of HS fields. While the expansion of the master field C in formalism of bosonic Vasiliev theory is given by C = C 0 +C αβ ỹ α ỹ β +C αβσλ ỹ α ỹ β ỹ σ ỹ λ +... () with implied star product and symmetric C components. Master field components are now separated in physical scalar field C0 and higher ones, related to it on-shell by derivatives. The expansion of C is given by C = CmV s m s (3) s= m <s for Cm s C α α..α s where m and number of oscillators ỹ versus ỹ are related with m = N N and Cm s are functions of spacetime coordinates. Auxiliary tensors are absorbed within a field. The fields A and Ā are expanded analogously A = s= m <s A s m V s m A = s= m <s A s m V s m. (4) The standard procedure of finding the generalised KG equation consists of inserting the expressions for A, Ā and C in () and determining the smallest possible set of equations needed to find the scalar equation in arbitrary background. Standard procedure can be described considering equation () in AdS background since it is a foundation for the following computations. The vacuum C0 equation without AdS fields is ordinary KG equation while one can determine the higher components in the terms of C0. The AdS connection consists of the spin- generators that form SL(), subalgebra of hs[λ] with AdS metric A = e ρ V dz +V 0 dρ (5) Ā = e ρ V d z V0 dρ (6) ds = dρ +e ρ dzd z. (7) The higher spins fields vanish, and we are working in Euclidean metric and Fefferman-Graham gauge. The general form of the C equation () in the AdS background is ρ Cm s +Cm s+ +Cm s+ g (s+) 3 (m,0) = 0 (8) ( Cm s +e ρ Cm s + gs (,m )Cm s + ) g(s+) 3 (,m )Cm s+ = 0 (9) ( Cm s eρ Cm+ s gs (,m+)cs m+ + ) g(s+) 3 (,m+)cm+ s+ = 0 (0) for m < s, = z, = z and the λ-dependence in the structure constants suppressed. 4

5 In the simplest case choosing s =,s = one can solve for the higher components in C and obtain the Klein-Gordon KG equation [ ρ + ρ +4e ρ (λ ) ] C0 = 0. () Consistency condition on equations is that all the components of C have smooth solution when expressed using C0. The strategy for determining the minimal set of equations for C 0 is to select components of C that are of the form C±m m+ and therefore the smallest spin for fixed m (e.g. C0,C ±,..). That are minimal components. One needs V m,ρ s equations for fixed m, solve for non-minimal components in terms of minimal ones and ρ derivatives, for A ρ = A ρ = V0. After solving for minimal ones, one needs to solve Vm,z s and Vm,z s equations in terms of C 0 and its derivatives. Once that we have expressed the higher components of C in terms of C0 we can determine the part that defines the KG equation and the generalised part that appears due to the HS background. To obtain the equation of motion for thescalar field upto linear order weconsider the variation of the gauge field and apply the KG equation on it. This and the standard procedure for obtaining the linearised equation of motion for the scalar field described above should be equal once the gauge parameter is chosen conveniently. That approach can be written in the following way. First we express the higher components of the C field in terms of the combination of the derivatives on C0 in the background AdS. Focusing on the master field C, the equation () is invariant under the hs[λ] hs[λ] gauge invariance when for Λ(ρ,z, z) = s n= C C +C Λ Λ C () (n )! ( )n λ (s) (z, z)e (s n)ρ V s s n. (3) Where we take Λ to be chiral, so Λ = 0. The field in the higher spin background is obtained by transformation C s m = Cs m (Λ C)s m. (4) The field Cm s we express in terms of the C0. To do that we focus on the set of equations (8), (9),(0). The product of the C field with Λ gives combination of higher components of C in AdS background which can, as we will show, be expressed in terms of C0. On the field C 0 we can use the transformation () and obtain C 0 = C 0 (Λ C) 0. (5) Since we are at the linear order, once we have C0 we can rewrite it as C 0 which is defined on the higher spin background. From the expression for the gauge field Λ (3) and the relation for the star product (9) we can determine the variation of the scalar field C0 s (δc) 0 = (n )! ( )n Λ (s) gss s (s n,n s)c (s n) s e(s n)ρ, (6) n= 5

6 for C (s n) s an arbitrary component of the master field C. Taking m m in the set of equations (8), (9), (0) and s = m+ in (9), we can iteratively determine the dependence of the Cm m+ on the C0. From the equation (9) we obtain z C m+ m + eρ g(m+) 3 (, m )C m+ m = 0 (7) takingintoconsiderationthatforcertaincomponentscn s itisrequired n s thisiteratively leads torelation of Cm m+ andc 0, andfromthe (0) analogously forcm+ m and C 0. Thegeneral form of the C± s is then given in terms of Cm+ ±m and coefficients gts u (m,n). Knowing Cs ± and allows to obtain [0] C m+ ±m and fs,n ± (δc) 0 = s n= f s,n ± (λ) n z Λ (s) z s n φ (8) (λ) expressed in terms of coefficients gst u (m,n) Using the replacement (λ), allows to determine its for φ C0 ρ ( ± λ) and writing explicitly first few n values for f s,n ± general expression f s,n ± (λ) = ( )s Γ(s+λ) Γ(s n+±λ) n (( n ))!!( n n )! Substituting (9) in (8) one obtains the variation of the scalar field s (δc) 0 = ( ) s Γ(s±λ) Γ(s n+±λ) n= ( n ) j= n ( ( n j= ) ) (!! n s+ n s j. (9) )! (30) s+j n s j n z Λ (s) z s n C0. (3) To consider the coefficient in front, we focus on the term with the lowest number of z derivatives on the gauge field Λ (s), obtained for n=. Then, (3) becomes (δc) 0 n= = ( ) s Λ (s) s C 0. (3) To obtain the linearised equation of motion for the scalar field we act on (3) with KG operator (). This can be written as KG C 0 = KG C 0 + KGδC 0. (33) Taking ρ (±λ) in f s,n ± (λ) we have taken and considering the term with highest number of derivatives on C0 leads to KG highest number of derivatives (δc) 0 = (34) = ( ) s 4e ρ Λ (s) (s ) C0 (35) = ( ) s 4e ρ[ Λ (s) (s ) C0 + Λ (s) s C0 + Λ (s) (s ) C0 +Λ (s) ] s C0. (36) The term in (36) that is of further interest is the one multiplying 4e ρ acting on δc 0 which is convenient to compute in the metric formulation. 6

7 3 Metric formulation In the metric formulation we can express the higher spin field of arbitrary spin s with ( ) φ µ...µ s = tr ẽ (µ...ẽ µs Ẽ µs) (37) where Ẽµs = õ Ā µ and õ and Ā µ we define below. The dreibein is determined from the background AdS metric (7) e z = eρ (L +L ) = eρ (V +V ) (38) e z = eρ (L L ) = eρ (V V ) (39) e ρ = L 0 = V 0. (40) The invariance of the equation () under the gauge transformation for hs[λ] hs[λ] for the fields A means A A+dΛ+[A,Λ] à (4) Ā Ā+d Λ+ [ Ā, Λ ] Ā. (4) Since Λ parameter is chiral it means Λ = 0 and the field Ā is essentially unchanged. The field à µ is then à = A AdS +dλ+[a AdS,Λ]. (43) dλ reads dλ = s n= (n )! V s s n e(s n)ρ [( ) n Λ (s) (z, z)dz (44) +( ) n Λ (s) (z, z)d z +( ) n Λ (s) (z, z)(s n)dρ] (45) and [A AdS,Λ] = [ e ρ V dz +V 0 dρ, s n= ] (n )! ( )n Λ (s) (z, z)e (s n)ρ Vs n s (46) To read out the coupling we focus on z... z component of the field C0 with lowest number of derivatives on gauge field Λ (s). The multiplication of the dreibeins in (37) in that case contributes only with first gu st (m,n;λ) coefficient with the each following dreibein that is being multiplied. More explicitly e z e z = eρ( V ) V (V V ) (47) From (3) we notice that the lowest number of derivatives on Λ will appear for lowest n, i.e. for n = in summation (3). Knowing the relation for the trace of higher spin generators, the required generator Vs n s will than be of the form Vs s, as we see below, which means that multiplication of HS generators we have to consider is V V... V. (48) 7

8 Then where the g conclude while V V = ( g (, )V 3 +g (, )V +g 3 (, )V ) (49) (, ) = g 3 (, ) = 0. Multiplying with following V, etc. on e can V V... V = }{{} s g(s ) (, (s ))V (s ) s (50) s g (s ) (, (s )) = g (s ) 3 (, (s )) = 0. (5) That means we have found the contribution to the z... z component multiplied with lowest derivative on Λ (s) due to definition of trace for generators V s n [] for tr ( VmV s n t ) ( ) s m = Ns Γ(s+m)Γ(s m)δ st δ m, n. (5) (s )! N s 3 4s 3 πq s 4 Γ(s) (λ )Γ(s+ ) ( λ) s (+λ) s (53) and (a) n = Γ(a+n) Γ(a) ascending Pochhammer symbol. The overall constant is set to tr(v V ) =. (54) Let us go back to φ z... z component. The star product e z... e z will contribute with e (s )ρ V s s (s ) if we consider as explained above the lowest derivative on Λ(s). We can denote this as e z... e z (V... V ) = s e(s )ρ V s (s ). (55) The Ẽ z s = Ã z s Ā zs needs to be able to satisfy the conditions of the trace (5) in star multiplication with e z... e z, the only HS generator that contributes is Vs s generator. When we gauge the field A µs, d z component appears in dλ while A AdS and [A AdS,Λ] do not have d z component. The Ā zs has d z component that comes from ĀAdS part and it is e ρ V d z. This however will not appear with the right number of derivatives on Λ. Since we have chosen Λ to be chiral and Λ = 0, that was the only contribution from Ā z. Altogether, we can write φ z... z component for the Λ (s) derivative as [ φ z... z Λ (s) = tr s e(s ) V (s ) s Vs s e (s )ρ Λ ] (s) (z, z) (56) = s e(s )ρ Λ (s) N s. (57) Inserting the normalisation N s we obtain φ z... z Λ (s) = s e(s )ρ Λ (s) 3 4 π4 4 s Γ(s)Γ(s+λ)Γ(s λ) (λ )Γ(s+ (58) )Γ( λ)γ(+λ). The expression φ z... z we want to compare with expression (36) for highest derivative on C 0 and Λ (s). In the computation of the vertex this would be a term φ z...z φ z... z φ (59) 8

9 for φ z...z higher spin field with s indices and φ scalar field. Raising indices contributes with a factor s e sρ, so that the field φ z...z becomes φ z...z = e ρ Λ (s) s Γ(s)Γ(s+λ)Γ(s λ) (λ )Γ(s+ (60) )Γ( λ)γ(+λ). When we take the ratio with KG highest number of derivatives(δc 0 ) Λ = ( ) s 4e sρ Λ (s) s C 0 we get (schematically written) φ z...z Λ (s) ( ) s KG highest number of derivatives(δc) s Γ(s)Γ(s+λ)Γ(s λ) π (λ )Γ(s+ Λ (s) )Γ( λ)γ(+λ). (6) which taking into account the normalisation gives the coupling for the 00s three point function. 4 Conclusion and Outline We have considered the three-point coupling using metric-like formation to express the higher spin field and using the linearised Vasiliev s equations of motion. The obtained result can also be verified using the alternative methods, for example following the procedure by [3]. The generalisation of the result to higher point functions would be non-trivial since in order to compute higher order vertices, one would have to consider perturbations around the background AdS field with higher spin fields up to that required higher order. 5 Acknowledgements I would like to thank Stefan Fredenhagen for guidance and discussions, Arkady Tseytlin, and Jan Rosseel for discussions. The work was made possible by the hospitality of Imperial College London, University of Vienna, and the organisers of the School and Workshops on Elementary Particle Physics and Gravity at Corfu Summer Institute 07, and by the FWF project J-49 in the framework of Schrödinger Programme. References [] E. S. Fradkin and Mikhail A. Vasiliev. Cubic Interaction in Extended Theories of Massless Higher Spin Fields. Nucl. Phys., B9:4 7, 987. [] Mikhail A. Vasiliev. Consistent equation for interacting gauge fields of all spins in (3+)- dimensions. Phys. Lett., B43:378 38, 990. [3] E. S. Fradkin and Mikhail A. Vasiliev. On the Gravitational Interaction of Massless Higher Spin Fields. Phys. Lett., B89:89 95, 987. [4] Mikhail A. Vasiliev. Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions. Int. J. Mod. Phys., D5: , 996. [5] Christian Fronsdal. Massless Fields with Integer Spin. Phys. Rev., D8:364, 978. [6] Christian Fronsdal. Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7. Phys. Rev., D0: ,

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