Massive totally symmetric fields in AdS(d)

Free totally symmetric arbitrary spin massive bosonic and fermionic fields propagating in AdS(d) are investigated. Using the light cone formulation of relativistic dynamics we study bosonic and fermionic fields on an equal footing. Light-cone gauge actions for such fields are constructed. Interrelation between the lowest eigenvalue of the energy operator and standard mass parameter for arbitrary type of symmetry massive field is derived.


Introduction
A study of higher spin theories in AdS space-time has two main motivations (see e.g. [1,2]): Firstly to overcome the well-known barrier of N ≤ 8 in d = 4 supergravity models and, secondly, to investigate if there is a most symmetric phase of superstring theory that leads to the usual string theory as a result of a certain spontaneous breakdown of higher spin symmetries. Another motivation came from conjectured duality of a conformal N = 4 SYM theory and a theory of type IIB superstring in AdS 5 × S 5 background [3]. Recent discussion of this theme in the context of various limits in AdS superstring may be found in [4]. As is well known, quantization of GS superstring propagating in flat space is straightforward only in the light-cone gauge. It is the light-cone gauge that removes unphysical degrees of freedom explicitly and reduces the action to quadratical form in string coordinates. The light-cone gauge in string theory implies the corresponding lightcone formulation for target space fields. In the case of strings in AdS background this suggests that we should study a light-cone form dynamics of target space fields propagating in AdS space-time. It is expected that AdS massive fields form spectrum of states of AdS strings. Therefore understanding a light-cone description of AdS massive target space fields might be helpful in discussion of various aspects of AdS string dynamics. This is what we are doing in this paper.
Let us first formulate the main problem we solve in this letter. Fields propagating in AdS d space are associated with positive-energy unitary lowest weight representations of SO(d − 1, 2) group. A positive-energy lowest weight irreducible representation of the SO(d − 1, 2) group denoted as D(E 0 , h), is defined by E 0 , the lowest eigenvalue of the energy operator, and by h = (h 1 , . . . h ν ), ν = [ d− 1 2 ], which is the highest weight of the unitary representation of the SO(d − 1) group. The highest weights h i are integers and half-integers for bosonic and fermionic fields respectively. In this paper we investigate the fields whose E 0 and h are given by h = (h 1 , 0, . . . , 0) (1.1) The fields in (1.1) are massive bosonic fields while the ones in (1.2) are fermionic fields. The massive fields in (1.1),(1.2) with h 1 ≥ 1, are referred to as totally symmetric fields 1 . In manifestly Lorentz covariant formulation the bosonic(fermionic) totally symmetric massive representation is described by a set of the tensor(tensor-spinor) fields whose SO(d − 1, 1) space-time tensor indices have the structure of the respective Young tableauxes with one row. Covariant actions for the bosonic totally symmetric massive fields in AdS d space were found in [10]. 2 . Fermionic massive totally symmetry AdS d fields with arbitrary E 0 and h have not been described at the field theoretical level so far 3 . In this paper we develop a light-cone gauge formulation for such fields at the action level.
Using a new version of the old light-cone gauge formalism in AdS space developed in [7], we describe both the bosonic and fermionic fields on an equal footing. Since, by analogy with flat space, we expect that a quantization of the Green-Schwarz AdS superstring with Ramond-Ramond charge will be straightforward only in the light-cone gauge [14] it seems that from the stringy perspective of AdS/CF T correspondence the light-cone approach is the fruitful direction to go.
2 Light-cone gauge action and its global symmetries In this section we present new version of the old light cone formalism developed in [7]. Let φ(x) be a bosonic arbitrary spin field propagating in AdS d space. If we collect spin degrees of freedom in a ket-vector |φ then a light-cone gauge action for φ can be cast into the following 'covariant form' An operator A does not depend on space-time coordinates and their derivatives. This operator acts only on spin indices collected in ket-vector |φ . We call the operator A the AdS mass operator.
We turn now to discussion of global so(d − 1, 2) symmetries of the light-cone gauge action. The choice of the light-cone gauge spoils the manifest global symmetries, and in order to demonstrate that these global invariances are still present one needs to find the Noether charges which generate them 5 . Noether charges (or generators) can be split into kinematical and dynamical generators. For x + = 0 the kinematical generators are quadratic in the physical field |φ , while the dynamical generators receive corrections in interaction theory. In this paper we deal with free fields. At a quadratic level both kinematical and dynamical generators have the following standard representation in terms of the physical light cone field [7] Representation for the kinematical generators in terms of differential operators G acting on the physical field |φ is given by of E 0 were discussed in [12]. Light cone actions for AdS 5 self-dual massive fields with arbitrary values of E 0 were found in [13]. 4 We use parametrization of AdS d space in which ds 2 = (−dx 2 0 + dx 2 i + dx 2 d−1 + dz 2 )/z 2 . Light-cone coordinates in ± directions are defined as x ± = (x d−1 ± x 0 )/ √ 2 and x + is taken to be a light-cone time. We adopt the conventions: Vectors of so(d − 2) algebra are decomposed as X I = (X i , X z ). 5 These charges play a crucial role in formulating interaction vertices in field theory. Application of Noether charges in formulating superstring field theories may be found in [15] (2.9) while a representation for the dynamical generators takes the form while the operators A and B admit the following representation Q AdS is eigenvalue of the second order Casimir operator of the so(d − 1, 2) algebra for the representation labelled by D(E 0 , h): Here we use sign ≈ to write instead of equations X|φ = 0 simplified equations X ≈ 0. As was noted the operator B I transforms in vector representation of the so(d − 2) algebra Making use of the formulas above-given one can check that the light-cone gauge action (2.3) is invariant with respect to the global symmetries generated by so(d − 1, 2) algebra taken to be in the form δĜ|φ = G|φ . To summarize a procedure of finding light cone description consists of the following steps: i) choose form of realization of spin degrees of freedom of the field |φ ii) fix appropriate representation for the spin operator M IJ ; iii) using formula (2.19) evaluate an eigenvalue of the Casimir operator; iv) find solution to the defining equations for the operator B I (2.20). Now following this procedure we discuss bosonic and fermionic fields in turn.

Bosonic fields
To discuss field theoretical description of massive AdS field we could use so(d − 1) totally symmetric traceless tensor field ΦÎ 1 ...Îs ,Î = 1 ′ , 1, . . . , d − 2. Instead of this we prefer to decompose such field into traceless totally symmetric tensors of so(d − 2) algebra φ I 1 ...I s ′ , I = 1, . . . , d − 2; s ′ = 0, 1, . . . , s: As usual to avoid cumbersome tensor expressions we introduce creation and annihilation and make use of ket-vectors |φ s ′ defined by The |φ s ′ satisfies the following algebraic constraints Eq.(3.25) tells us that |φ s ′ is a polynomial of degree s ′ in oscillator α I . Tracelessness of the tensor fields φ I 1 ...I s ′ is reflected in (3.26). The spin operator M IJ of the so(d − 2) algebra for the above defined fields |φ s ′ takes then the form We are going to connect our spin s field with the unitary representation labelled by Eigenvalue of the second order Casimir operator for totally symmetric representation under consideration is given then by (see (1.1), (2.19),(3.28)) An action of the operator B I on the physical fields |φ s ′ is found to be Operator A I s ′ which appears in definition of the operator B I (3.30) is given by Let us outline procedure of derivation above-mentioned results. Most difficult problem is to find solution to the defining equation (2.20). Fortunately, for the case of totally symmetric fields this equation simplifies due to the following relation for M IJ : This formula can be checked directly by using representation for M IJ given in (3.27). Plugging (3.34) in (2.20) we get the following simplified form of defining equation Applying this equation to |φ s ′ and using (3.30) we get a relationship for the coefficients Now we exploit a requirement the operator B I be hermitian 8 with respect to scalar product defined by

Fermionic fields
Light cone action for fermionic fields takes the following form This action is invariant with respect to transformations δ|ψ = G f erm |ψ , where differential operators G f erm are obtainable from the ones for bosonic fields ((2.5)-(2.14)) by making there the following substitution x − → x − + 1 2∂ + . In addition to this in expressions for generators in ((2.5)-(2.14)) we should use the spin operator M IJ suitable for fermionic fields. Defining equation (2.20) for the operator B I does not change. Before to discuss concrete form of the spin operators M IJ and B I we should fix a field theoretical realization of spin degrees of freedom collected in |ψ .
To discuss field theoretical description of massive AdS field we could used totally symmetric traceless and γ-transversal tensor-spinor field ΨÎ 1 ...Îsα ,Î = 1 ′ , 1, . . . , d − 2 which corresponds to irreducible spin s + 1 2 representation of so(d − 1) algebra. Instead of this we prefer to decompose such field into traceless totally symmetric tensor-spinor fields of so(d − 2) algebra ψ I 1 ...I s ′ α , I = 1, . . . , d − 2; s ′ = 0, 1, . . . , s : 9 (4.43) As before to avoid cumbersome expressions we exploit creation and annihilation oscillators α I andᾱ I (3.23) and make use of ket-vectors |ψ s ′ defined by Eq. (4.45) tells us that |ψ s ′ is a polynomial of degree s ′ in oscillator α I . Tracelessness of the tensor-spinor field ψ I 1 ...I s ′ α is reflected in (4.47). Realization of spin operator M IJ on the space of the ket-vectors |ψ s ′ is given by We are going to connect our massive tensor-spinor field with unitary representation labelled by D(E 0 , h) (1.1) whose h 1 is related with integer s: Eigenvalue of the second order Casimir operator for totally symmetric fermionic representation is given then by (see (1.2),(2.19),(4.49)) An action of the operator B I on the physical fermionic fields |ψ s ′ is found to be As before the coefficients q s ′ , a s ′ , b s ′ turn out to be functions of E 0 , s, s ′ and are given by where we use the notation (4.55) Operators A I s ′ and Y I s ′ which enter definition of basic operator B I (4.51) are given by Now let us outline procedure derivation of these results. We start with general representation for B I given in (4.51) and the problem is to find coefficients q s ′ , a s ′ , b s ′ which satisfy the defining equation (2.20). To this end we evaluate expression for ( where a sign ≈ indicates that (4.57) is valid by module of terms which are equal to zero by applying to |ψ s ′ i.e. to derive (4.57) we use constraints (4.45)-(4.47). After this using Eqs.(2.20),(4.50),(4.57) we find solution for q s ′ and product a s ′ +1 b s ′ . Exploiting then the hermicity condition for the operator B I (3.37),(3.38) we arrive at the solution for q s ′ , a s ′ , b s ′ given above. Some helpful formulas to evaluate commutator [B I , B J ] are given by (4.61) 5 Interrelation between lowest energy value E 0 and mass parameter m In previous sections we have expressed our results in terms of lowest eigenvalue of energy operator E 0 . Because sometimes formulation in terms of the standard mass parameter m is preferable we would like to derive interrelation between E 0 and m. Before to going into details let of first present our results. Given massive fields with massive parameter m corresponding to unitary representation labelled by D(E 0 , h) we find the following relationship between E 0 and m (for even AdS space-time dimension d; ν = (d − 2)/2) for bosonic fields; (5.62) where a number k is defined from the relation We remind that for bosonic fields the labels h σ are integers while for fermionic fields the h σ are half-integers. We note that relations (5.62),(5.63) are valid also for those massive fields in odd dimensional AdS d whose h (d−1)/2 = 0. Now let us outline procedure of derivation these results. Let φ µ 1 ... m=0 be massless field in AdS d . As was demonstrated in [5], [6] the massless fields associated with unitary representation labelled by D(E m=0 0 , h) should satisfy the equation of motion where D 2 is a covariant D'Alembertian operator in AdS d and E m=0 0 is given by Eq. (5.65) reflects the well-known fact that equations of motion for AdS massless field involve mass-like term which is expressible in terms of E m=0 0 . As is well known to discuss gauge invariant description of massive fields one introduces the set of fields including some fields φ µ 1 ... m which we shall refer to as leading field plus Goldstone fields (sometimes referred to as Stueckelberg fields). By definition, the structure of Lorentz indices of the leading massive field φ µ 1 ... m is the same as the one for massless field φ µ 1 ... m=0 . If we impose on the leading field φ µ 1 ... m and the Goldstone fields an appropriate covariant Lorentz gauge and tracelessnes conditions then for the leading field one gets the equation One other hand an analysis of Ref. [5], [6] implies that the leading field satisfies the following equations of motion The same arguments can be applied to the fermionic fields. In this case equation for massless field in AdS d is given by (see the second Refs. in [5]) where D µL is a covariant derivative with respect to local Lorentz rotation, e µ a is an inverse of the vielbein e a µ and E m=0 0 is given in (5.66). Because on the one hand the leading massive tensor-spinor field ψ α µ 1 ... m taken to be in appropriate covariant Lorentz gauge satisfies an equation and on the other hand this equation should be representable as (see [5]) In the AdS/CFT correspondence the E 0 is connected with dimension of conformal operator as E 0 = ∆. The ∆ for bosonic massive totally antisymmetric and bosonic massive symmetric spin two fields were evaluated in Refs. [16], [17]. Our results coincides with the ones obtained in these references. For instance for the case of bosonic massive totally symmetric fields we have h 1 = s, k = 1 and this leads to For the case of s = 2 this is result of Ref. [17]. For fermionic massive totally symmetric fields we have h 1 = s + 1 2 , k = 1 and formula (5.63) leads to For particular value of s = 1 (Rarita-Schwinger field) appropriate ∆ was evaluated in Refs. [18], [19]. Note that our result taken to be for s = 1 differs from the one obtained in these references. The reason for this is that in this paper we use normalization of mass parameter such that the point m = 0 corresponds to massless fields (see Eqs. (5.70),(5.71)). In Refs. [18], [19] another normalization was used.
Because while derivation of our results for E 0 we used arguments based on gauge invariant formulation for massive fields our relations (5.62),(5.63) are not applicable to the scalar field and spin one-half field. Conformal dimension for these fields are well known (see [20], [21]).
Conclusions. The results presented here should have a number of interesting applications and generalizations, some of which are: i) In this paper we develop light cone formulation for massive totally symmetric fields. It would be interesting to extend such formulation to the study of massive mixed symmetry fields (see Refs. [22], [23], [24]) and then to apply such formulation to the study of AdS/CFT correspondence along the line of Ref. [25], [26]. ii) As is well known massive and massless fields can be connected via procedure of dimensional reduction. Procedure of dimensional reduction AdS d → AdS d−1 was developed in [8] (for discussion of alternative reductions see [27], [28]). It would be interesting to find mass spectra of AdS massive modes upon dimensional reduction of massless AdS fields.
While deriving of these formulas we use commutation relations of so(d − 2) algebra spin operators M IJ = M ij , M zi given in (2.16) and exploit the relation (A.10) Our basic observation is that if we introduce the quantities These commutation relations together with the ones following from so(d − 3) covariance imply that the operator B I is indeed transformed in vector representation of so(d − 2) algebra (cf. (2.21)). All that remains is to analyse Eq.(A.6). Making use of (A.11) we can rewrite (A.6) as Now taking into account that a state obtainable by acting the spin operator M zj on wave function |φ should also belong to |φ we conclude that constraint (A.14) should commute with M zj . This gives a new constraint Constraints (A.14),(A.15) are collected into the ones given in (2.20). Thus we proved that the old light cone formalism of Ref. [7] is equivalent to the one used in this paper. Remarkable features of new formalism are i) appearance of so(d − 2) vector B I which was hidden in old formalism; ii) manifest so(d − 2) invariance of defining equations for B I .

Appendix B Decompositions of tensor-spinor fields
Here we wish to describe an decomposition of totally symmetric tensor-spinor field ΨÎ 1 ...Îsα transforming in irreps of the so(d − 1) algebra in terms of tensor-spinor fields |ψ s ′ which are irreps of the so(d − 2) algebra. Consider a generating function We start with analysis of the second constraint in (B.2) which can be considered as the second order differential equation with respect to oscillator variable α 1 ′ (ᾱ 1 ′ᾱ 1 ′ + ω 2 )|Ψ(α 1 ′ , α I ) = 0 , ω 2 =ᾱ IᾱI . The traceless tensor-spinor fields of so(d − 2) algebra |ψ s ′ satisfy by definition the constraintsᾱ IᾱI |ψ s ′ = 0 , (αᾱ − s ′ )|ψ s ′ = 0 . (B.9) Because the tensor-spinor |ψ s ′ does not satisfy γ-transversality constraint this field is a reducible representation of the so(d−2) algebra. It can be decomposed into two irreducible representations of the so(d − 2) algebra by formulas It is the set of traceless and γ-transversal tensor-spinor fields |ψ s ′ that we used to formulate light cone action for massive fermionic representations of so(d − 1, 2) algebra.