aa r X i v : . [ h e p - t h ] A p r The Involutive System of Higher-Spin Equations
Rakibur Rahman a,b a Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)Am M¨uhlenberg 1, D-14476 Potsdam-Golm, Germany b Department of Physics, University of Dhaka, Dhaka 1000, Bangladesh
Abstract
We revisit the problem of consistent free propagation of higher-spin fields innontrivial backgrounds, focusing on symmetric tensor(-spinor)s. The Fierz-Pauliequations for massive fields in flat space form an involutive system, whose algebraicconsistency owes to certain gauge identities. The zero mass limit of the former leadsdirectly to massless higher-spin equations in the transverse-traceless gauge, whereboth the field and the gauge parameter have their respective involutive systems andgauge identities. In nontrivial backgrounds, it is the preservation of these gaugeidentities and symmetries that ensures the correct number of propagating degreesof freedom. With this approach we find consistent sets of equations for massive andmassless higher-spin bosons and fermions in certain gravitational/electromagneticbackgrounds. We also present the involutive system of partially massless fields,and give an explicit form of their gauge transformations. We consider the Liesuperalgebra of the operators on symmetric tensor(-spinor)s in flat space, and showthat in AdS space the algebra closes nonlinearly and requires a central extension. ontents
A.1 Involution Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 DoF Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.2.1 No Gauge Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.2.2 Irreducible Gauge Symmetries with Constrained Parameters . . . . . . . . . . . . . 50
B Involutive Deformations 51C Technical Details 53
C.1 Gravitational Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53C.2 Electromagnetic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Introduction
The construction of consistent interacting theories of higher-spin fields is a difficult task.Generic interactions of massless fields are incompatible with gauge invariance, and this factgives rise to various no-go theorems [1–5]. For massive fields, when interactions are turnedon, the dynamical equations and constraints may either lose algebraic consistency [6] orstart propagating unphysical/superluminal modes [7–10]. These pathologies show up evenfor a much simpler setup that we would like to consider in this article: the free propagationof higher-spin fields in nontrivial backgrounds (see [11] for a recent review).In this article, we employ the metric-like formulation, where the degrees of freedom(DoF) of higher-spin particles are encoded in symmetric tensors and tensor-spinors. Theflat-space free Lagrangians and the equations of motion (EoM) are well known for mas-sive and massless metric-like fields [11]. In nontrivial backgrounds, however, consistentpropagation is not at all automatic; one must ensure among other things that only thephysical modes propagate and that their propagation remains causal. This is the weakestlink of the Lagrangian formulation, for both massive [7–10] and massless fields [12], sincethe problems become manifest only at the EoM level. Moreover, the EoM’s often turn outto be surprisingly simple, but this simplicity is obscured at the Lagrangian level [13–16].It is therefore desirable to study the propagation of higher-spin fields solely at theEoM level, without recourse to the Lagrangian formulation. This is where the involutiveproperties of higher-spin equations come into play (see Appendix A for an expositionof involutive systems). Devoid of a parent Lagrangian, the mutual compatibility of thedynamical equations and constraints/gauge-fixing conditions in a nontrivial backgroundis no longer guaranteed. The good news is that this can be duly taken care of by the“gauge identities” of the involutive system. In fact, in the involutive approach, all theconsistency issues are under full control, so that one may systematically deform the flat-space system of higher-spin equations. This “involutive deformation method” has alreadybeen employed for the free propagation of massive bosons in various backgrounds [17–19].In this article, we would like to extend this approach to fermions as well as to gauge fields.The organization of this article is as follows. The remaining of this section gives a briefaccount of the operator formalism − a handy computational tool to be used throughoutthe article. Section 2 deals with the Fierz-Paui system for massive bosons, and rederivesits involutive deformations in gravitational and electromagnetic backgrounds using theelegant operator formalism. The extension of this construction to massive fermions ispresented in Section 3. Sections 4 and 5 respectively consider gauge bosons and fermions,where we first present the flat-space involutive systems in the transverse-traceless gauge,obtained in the zero mass limits of their massive counterparts. Then we construct theirrespective deformations in gravitational and electromagnetic backgrounds − a task made2hallenging by “unfree” gauge symmetries [20], whose parameters themselves are governedby involutive systems. Section 6 analyzes the involutive systems of partially masslessbosons and fermions along with their gauge transformations. In Section 7, we show howthe various operators acting on symmetric tensor(-spinor)s in AdS space form a nonlinearLie superalgebra with a central charge. Some concluding remarks are made in Section 8,in particular about the possible rˆole of mixed-symmetry fields. Three appendices providebrief accounts of involutive systems and deformations, and some technical details. The Operator Formalism
The operator formalism introduces auxiliary tangent-space variables u a and their deriva-tives: d a ≡ ∂∂u a , where fiber (world) indices are denoted by lower case Roman (Greek)letters. The vielbein e µa ( x ) and its inverse e aµ ( x ) give the contracted auxiliary variables: u µ ≡ e µa ( x ) u a , d µ ≡ e aµ ( x ) d a , (1.1)which comprise a set of oscillators that satisfies the Heisenberg algebra:[ u µ , u ν ] = 0 , [ d µ , d ν ] = 0 , [ d µ , u ν ] = δ νµ . (1.2)A symmetric rank- s tensor Φ µ ··· µ s ( x ) denotes a spin- s bosonic field, while a symmetricrank- n tensor-spinor Ψ µ ··· µ n ( x ), with the spinor index kept implicit, denotes a fermionicfield of spin s = n + . They are represented respectively by the generating functions:Φ( x, u ) = s ! Φ µ ··· µ s ( x ) u µ · · · u µ s , Ψ( x, u ) = n ! Ψ µ ··· µ n ( x ) u µ · · · u µ n , (1.3)The commutator of covariant derivatives acts on them in the following way:[ ∇ µ , ∇ ν ] Φ = R µνρσ ( x ) u ρ d σ Φ , (1.4)[ ∇ µ , ∇ ν ] Ψ = R µνρσ ( x ) u ρ d σ Ψ + R µνρσ ( x ) γ ρ γ σ Ψ , (1.5)with γ µ ≡ e µa ( x ) γ a , where γ a are the tangent-space gamma matrices. It is important tonote that the vielbein postulate results in the following vanishing commutators:[ ∇ µ , u ν ] = 0 , [ ∇ µ , d ν ] = 0 , [ ∇ µ , γ ν ] = 0 . (1.6)The index operator is: N ≡ u · d = u µ d µ , where a “dot” stands for the contraction ofa pair of indices. For any operator ˆ O , there is a corresponding weight w of N , given by:[ N, ˆ O ] = w ˆ O. (1.7)The weight w is an intrinsic property, which counts the tensor rank of the operator.The case of flat space is special, where the vielbein ˆ e aµ satisfies: ˆ e aµ ˆ e νa = η µν . Then, itsuffices to consider only world indices that can be lowered and raised by the Minkowskimetric and its inverse. In the absence of any gauge connections, one is left only withpartial derivatives ∂ µ that are of commuting nature: [ ∂ µ , ∂ ν ] = 0.3 Massive Bosonic Fields
In this section, we study the Fierz-Paui system for totally-symmetric massive bosons in theoperator formalism. We start with the free propagation in Minkowski background, wherewe properly identify all the gauge identities of the involutive system. Then, the involutivedeformations in gravitational/electromagnetic backgrounds [17–19] are rederived, rathermore elegantly, using the operator formalism. Despite having no new results, this sectionwill be immensely useful for the sake of familiarity with the concepts and methodology.
The Fierz-Pauli conditions for a symmetric bosonic field of mass m in flat space involvethe Klein-Gordon, divergence and trace operators [11], comprising the set: G = { g , g , g } , (2.1)where a subscript gives the negative weight ( − w ) corresponding to an operator. Table 1summarizes the various properties of these operators.Table 1: Operators in Bosonic Fierz-Pauli SystemOperator Symbol Definition Weight ( w ) Derivative Order ( k )Klein-Gordon g ∂ − m g d · ∂ − g d − different operators: c ≡ [ g , g ] , c ≡ [ g , g ] , c ≡ [ g , g ] , (2.2)all of which vanish on account of the commutativity of partial derivatives. Moreover,these linear operators have associative property, so that the Jacobi identity holds:[ g , c ] + [ g , c ] + [ g , c ] = 0 . (2.3)The Fierz-Pauli equations constitute an involutive system of differential equations [21]: g Φ = 0 , g Φ = 0 , g Φ = 0 . (2.4)From the point of view of an involutive system, the algebraic consistency of the sys-tem (2.4) is taken care of by the gauge identities [22] (see also Appendix A): c Φ = 0 , c Φ = 0 , c Φ = 0 , (2.5)4hich hold good because c i ’s themselves vanish. For the involutive system (2.4), however,the gauge identities (2.5) are not irreducible . To see this, let us define the operator: j ≡ g c + g c + g c . (2.6)Then, the Jacobi identity (2.3) implies the following on-shell identity: j Φ = 0 . (2.7)In other words, given the system of equations (2.4), we have a gauge identity at reducibilityorder 1. This exhausts the list of all possible gauge identities for our system.The system (2.4)–(2.7) of involutive equations plus gauge identities is of the kindconsidered in Appendix A.2.1. To check its absolute compatibility and find the DoFcount, let us first give the number of equations at order k in space-time derivatives . Fora symmetric boson of rank/spin s , in D is space-time dimensions, it is given by: t k = δ k (cid:18) D + s − s (cid:19) + δ k (cid:18) D + s − s − (cid:19) + δ k (cid:18) D + s − s − (cid:19) , (2.8)where a weight- w operator acting on a rank- s tensor gives (cid:0) D + s + w − s + w (cid:1) number of equations.On the other hand, the number of O ( k ) gauge identities at reducibility order j is: l k, j = δ k δ j (cid:18) D + s − s − (cid:19) + δ k δ j (cid:18) D + s − s − (cid:19) + (cid:0) δ k δ j + δ k δ j (cid:1) (cid:18) D + s − s − (cid:19) . (2.9)With the total number of field variables f = (cid:0) D + s − s (cid:1) , one finds from Eq. (A.23) that c = 0, i.e., the bosonic Fierz-Pauli system is absolutely compatible. The physical DoFcount per space-time point, computed from Eq. (A.24), turns out to be: D b ( s ) = 2 (cid:18) D − ss − (cid:19) + (cid:18) D − ss (cid:19) , (2.10)which is indeed the correct number of propagating DoF’s of a massive spin- s boson [11]. In order to describe the free propagation of a massive boson in a gravitational background,we would like to apply the involutive deformation method to the flat-space system ofthe previous section. As outlined in Appendix B, the zeroth-order deformations areobtained by replacing ordinary derivatives by covariant ones: ∂ µ → ∇ µ , while the first-order ones should be linear in the background curvature tensor, and so on. The mostgeneric deformations of the operators (2.1) take the following form:Klein-Gordon : ˆ g = ∇ − M + α R µνρσ u µ u ρ d ν d σ + α R µν u µ d ν + α R + O (cid:0) (cid:1) , Divergence : ˆ g = d ·∇ + O (cid:0) (cid:1) , Trace : ˆ g = d + O (cid:0) (cid:1) , (2.11)5here the deformed mass M and the dimensionless operators α , α , α have weight w = 0, and the mass scale Λ is larger than other scales in the theory. Note that the book-keeping parameter (see Appendix B) indicating the deformation order is implicit here.The deformations (2.11), of course, preserve the respective weights w of the operators.Because the deformations are smooth, M → m in the limit of zero curvature.Now, we would like to calculate the commutators between two different operators. Thetechnical steps of the explicit computations of the desired commutators: [ˆ g , ˆ g ], [ˆ g , ˆ g ]and [ˆ g , ˆ g ] are relegated to Appendix C.1. In order for having some deformed gaugeidentities in the first place, we should ensure that these commutators close within thegiven set of operators. Among other things, we have the following expression:[ˆ g , ˆ g ] = 2( α − W µνρσ ∇ µ u ρ d ν d σ + (cid:2) ( α + 1) − (cid:0) ND − (cid:1) ( α − (cid:3) S µν ∇ µ d ν + · · · , (2.12)where W µνρσ and S µν ≡ R µν − D g µν R are respectively the Weyl tensor and the tracelesspart of the Ricci tensor, and the ellipses stand for other kinds of terms whose explicit formsdo not matter at his point. In particular, some of the latter terms involve the gradient ofthe Riemann tensor, which can be decomposed into irreducible Lorentz tensors: ⊗ | {z } Gradient of Riemann = | {z } X ⊕ | {z } Y ⊕ | {z } Z ⊕ |{z} U , (2.13)where, with the convention that (anti)symmetrization of indices has unit normalization, X µνραβ ≡ ∇ ( µ W ναρ ) β − (cid:0) D +2 (cid:1) g ( µν ∇ σ W ρ )( ασβ ) ,Y µνρ ≡ ∇ ( µ R νρ ) − (cid:0) D +2 (cid:1) g ( µν ∇ ρ ) R,Z µνρ ≡ ∇ [ ρ R ν ] µ + (cid:0) D − (cid:1) g µ [ ρ ∇ ν ] R + ( µ ↔ ν ) ,U µ ≡ ∇ µ R. (2.14)For an arbitrary-spin field in D ≥
4, it is clear from Eq. (2.12) that the two terms on theright hand side must vanish for a gauge identity to hold good; this demands: α = 1 , α = − . (2.15)Then, the explicit form of Eq. (2.12) reduces to:[ˆ g , ˆ g ] = [ M , d ·∇ ] − R [ α , d ·∇ ] + h α − N − N + D − D − D +2) i d · U + ( u d · U + u · U ) ( D − D +2) d + X µνραβ u α u β d µ d ν d ρ + Z µνρ h u ρ d µ d ν + D − u ρ N d µ d ν + D − D − u µ u ν d ρ d i − Y µνρ (cid:2) u µ d ν d ρ − D − (cid:0) u µ u ν d ρ d + u d µ d ν d ρ − u µ N d ν d ρ (cid:1)(cid:3) + O (cid:0) (cid:1) . The last two lines in the above equation impose the following constraints: X µνραβ = 0 , Y µνρ = 0 , Z µνρ = 0 . (2.16)6ithout constraining the gravitational background any further, we can also choose: α = N − N + D − D − D +2) . (2.17)Finally, in order to deal with the commutator [ M , d ·∇ ], let us assume that the deformedmass M is a quadratic polynomial in the index operator N : M = m + µ (cid:0) N + βN + γ (cid:1) , (2.18)where β and γ are some numerical constants, and µ is some constant mass parameterthat vanishes in the limit of zero curvature. The justification of such an assumption canonly be given a posteriori, when we consider the massless case. Then, we have:[ M , d ·∇ ] = P ( N ) d ·∇ , P ( N ) ≡ − µ (2 N + β + 1) . (2.19)With the choices and constraints (2.15)–(2.19), the commutator (2.12) reduces to:[ˆ g , ˆ g ] = D − D +2) (cid:2) R (2 N + D −
2) ˆ g + (cid:0) u d · U + u · U (cid:1) ˆ g (cid:3) + P ( N )ˆ g + O (cid:0) (cid:1) . (2.20)Similarly, in view of the choices (2.15) and (2.17)–(2.18), we have the following result:[ˆ g , ˆ g ] = D − D +2) R (2 N + D −
1) ˆ g + Q ( N )ˆ g + O (cid:0) (cid:1) , (2.21)where Q ( N ) ≡ − µ (2 N + β + 2). The third and last commutator takes the simple form:[ˆ g , ˆ g ] = O (cid:0) (cid:1) . (2.22)Given the relations (2.20)–(2.22), we now identify the deformed counterparts of thecommutators appearing in Eq. (2.2). They are:ˆ c ≡ [ˆ g , ˆ g ] + D − D +2) [ R (2 N + D −
2) ˆ g + ( u d · U + u · U ) ˆ g ] + P ( N )ˆ g , ˆ c ≡ [ˆ g , ˆ g ] − D − D +2) R (2 N + D −
1) ˆ g − Q ( N )ˆ g , (2.23)ˆ c ≡ [ˆ g , ˆ g ] . Finally, we identify the deformed version of the operator j of Eq. (2.6) with:ˆ j ≡ ˆ g ˆ c + ˆ g ˆ c + ˆ g ˆ c . (2.24)On account of the Jacobi identity among the deformed operators { ˆ g , ˆ g , ˆ g } , we can usethe definitions (2.23) to express ˆ j in the following form:ˆ j = ˆ O ˆ g + ˆ O ˆ g + ˆ O ˆ g , (2.25)where ˆ O i is an operator of weight − i , whose explicit expression is given in Eq. (C.8).7ow we are ready to present our deformed involutive system with all the gauge iden-tities. Of course, the system of equations is given by:ˆ g Φ = 0 , ˆ g Φ = 0 , ˆ g Φ = 0 . (2.26)The gauge identities at reducibility order 0 can be written in the following form:ˆ c Φ = 0 , ˆ c Φ = 0 , ˆ c Φ = 0 , (2.27)provided that the ˆ c i ’s, given by Eqs. (2.23), vanish identically. This happens when the O (cid:0) (cid:1) terms in the commutators (2.20)–(2.22) are zero. Without explicit knowledge ofsimilar terms in the operators { ˆ g , ˆ g , ˆ g } , the latter condition can be ensured by taking :Λ → ∞ . (2.28)On account of the relation (2.25), we also have the following on-shell identity:ˆ j Φ = 0 , (2.29)which is the desired gauge identity at reducibility order 1. This completes our involutivedeformation analysis of a free massive boson in a gravitational background.To summarize, the consistent dynamical equation for a free massive boson reads: (cid:16) ∇ − M + R µνρσ u µ u ρ d ν d σ − R µν u µ d ν + N − N + D − D − D +2) R (cid:17) Φ = 0 , (2.30)where the deformed mass is of the type (2.18). The constraint equations are given by: d ·∇ Φ = 0 , d Φ = 0 . (2.31)The involutive nature of this system hinges upon the constraints (2.16) on the background.This result essentially captures those already found in [17, 18], and is valid for arbitraryspin in D ≥
4. Below we consider some important special cases.
Lower Spins
The constraints (2.16) on the gravitational background are necessary when the bosonicfield has spin s ≥
3. Because d µ d ν d ρ Φ = 0 for a spin-2 field, the quantity X µνραβ doesnot need to vanish in order for the commutator (2.12) to close. The constraints on thegravitational background therefore boil down to:For s = 2 : Y µνρ = 0 , Z µνρ = 0 . (2.32)Among others, these conditions admit manifolds with a covariantly constant Ricci tensor(Ricci symmetric spaces) reported in [17], and in particular Einstein manifolds [23, 24].No restriction on the gravitational background is imposed for s = 1 and s = 0. Alternatively, when O (cid:0) (cid:1) terms are judiciously included in the equations (2.26), similar contribu-tions should be absent in the commutators (2.20)–(2.22) modulo additional on-shell vanishing terms. Thismay pose additional constraints on the gravitational background. We would not consider this possibility. D Manifolds
The Weyl tensor vanishes identically in D = 3, and so does the tensor X µνραβ . Therefore,the necessary constraints on the gravitational background again take the form:For s ≥ Y µνρ = 0 , Z µνρ = 0 . (2.33)The constraint on Z µνρ is tantamount to the vanishing of the Cotton tensor. In otherwords, it is necessary that the 3D manifold be conformally flat. Let us assume that the massive boson possesses minimal coupling to the electromagnetic(EM) background field with an electric charge q . The zeroth-order deformations are ob-tained by the substitution: ∂ µ → D µ , where the covariant derivatives have commutators:[ D µ , D ν ] = iqF µν , with F µν being the background field strength. In this case, the mostgeneric parity-preserving deformations of the operators (2.1) can be written as:Klein-Gordon : ¯ g = D − ¯ M + iqαF µν u µ d ν + O (cid:0) (cid:1) , Divergence : ¯ g = d ·D + O (cid:0) (cid:1) , Trace : ¯ g = d + O (cid:0) (cid:1) , (2.34)where the deformed mass ¯ M and the dimensionless operator α have weight zero, andthe scale ¯Λ is larger than other mass scales in the theory. Here, the charge q plays therˆole of the parameter that keeps track of the deformation order (see Appendix B). Thedeformations (2.1), of course, preserve the respective weights w of the operators. Becausethe deformations are smooth, ¯ M → m in the limit of vanishing field strength.Let us calculate the commutators between two different operators in (2.34). Theyinvolve the gradient of the EM field strength, which can be decomposed as: ⊗ | {z } Gradient of F = | {z } A ⊕ |{z} V , (2.35)where the Young diagram does not contribute because of the Bianchi identity, and theother irreducible Lorentz tensors are defined as: A µνρ ≡ ∂ ( µ F ν ) ρ − (cid:0) D − (cid:1) h η µν V ρ − δ ρ ( µ V ν ) i , V ν ≡ ∂ µ F µν . (2.36)The commutators we are interested in ought to close within the given set of opera-tors (2.34), so that some deformed gauge identities to exist. We obtain (see Appendix C.2):[¯ g , ¯ g ] = iq ( α − F µν D µ d ν − iqαA µνρ u ρ d µ d ν + iq (cid:16) αN +( α − D − D − (cid:17) d · V − iqα (cid:0) D − (cid:1) u · V d − iq [ α, d ·D ] F µν u µ d ν + [ ¯ M , d ·D ] + O (cid:0) (cid:1) . (2.37)9n the right hand side of Eq. (2.37), the first term must vanish, which sets: α = 2 , (2.38)for a F µν = 0. On the other hand, the second and third terms require that for any bosonicfield of spin s >
1, the irreducible Lorentz tensors A and V vanish: A µνρ = 0 , V µ = 0 , (2.39)which is tantamount to the requirement of a constant EM background: F µν = constant.The remaining problematic term is the commutator [ ¯ M , d ·D ], which can be managed byassuming again that ¯ M is a polynomial function of the index operator N . This gives:[ ¯ M , d ·D ] = ¯ P ( N ) d ·D , [ ¯ M , d ] = ¯ Q ( N ) d , (2.40)where ¯ P ( N ) and ¯ Q ( N ) are polynomials in N of the same order. With the choices andconstraints (2.38)–(2.40), the commutator (2.37) and the other two can be written as:[¯ g , ¯ g ] = ¯ P ( N )¯ g + O (cid:0) (cid:1) , [¯ g , ¯ g ] = ¯ Q ( N )¯ g + O (cid:0) (cid:1) , [¯ g , ¯ g ] = O (cid:0) (cid:1) . (2.41)In view of Eqs. (2.41), we can identify the deformed counterparts of the commutatorsappearing in Eq. (2.2) as the following:¯ c ≡ [¯ g , ¯ g ] + ¯ P ( N )¯ g , ¯ c ≡ [¯ g , ¯ g ] − ¯ Q ( N )¯ g , ¯ c ≡ [¯ g , ¯ g ] . (2.42)Next, we identify the deformed counterpart of the operator j of Eq. (2.6); it is:¯ j ≡ ¯ g ¯ c + ¯ g ¯ c + ¯ g ¯ c . (2.43)Thanks to the Jacobi identity among the deformed operators { ¯ g , ¯ g , ¯ g } , we can use thedefinitions (2.42) to express ¯ j in the following form:¯ j = ¯ c ¯ g + (cid:2) ¯ c + ¯ g ¯ P ( N ) + ¯ Q ( N )¯ g (cid:3) ¯ g + (cid:2) ¯ c − ¯ P ( N )¯ g − ¯ g ¯ Q ( N ) (cid:3) ¯ g . (2.44)Let us now present the deformed involutive system of equations; it is:¯ g Φ = 0 , ¯ g Φ = 0 , ¯ g Φ = 0 . (2.45)Assuming that the ¯ c i ’s defined in Eqs. (2.42) vanish identically, we also have the followinggauge identities at reducibility order zero:¯ c Φ = 0 , ¯ c Φ = 0 , ¯ c Φ = 0 , (2.46)10hich holds if the O (cid:0) (cid:1) terms in Eqs. (2.41) vanish. Lacking the explicit knowledge ofsimilar terms in the deformed operators { ¯ g , ¯ g , ¯ g } , the latter condition is guaranteed if¯Λ → ∞ . (2.47)We also have a desired gauge identity at reducibility order 1; it reads:¯ j Φ = 0 , (2.48)and holds as an on-shell identity given the relation (2.44). This completes our analysis ofthe involutive deformation of a free massive boson in an EM background.The consistent of dynamical equations and constraints for a free massive boson read: (cid:0) D − ¯ M + 2 iqF µν u µ d ν (cid:1) Φ = 0 , d ·D Φ = 0 , d Φ = 0 , (2.49)where the deformed mass ¯ M is assumed to be a polynomial in the index operator N , suchthat in the limit of vanishing field strength: ¯ M → m . The consistency of this systemrelies on the constraints (2.39) on background field strength, which mean: F µν = constant.Already found in [17], this result holds for an arbitrary-spin boson. This section analyzes the Fierz-Paui system for totally-symmetric massive fermions in theoperator formalism. The starting point is the free propagation in Minkowski background,where we identify all the gauge identities of the involutive system. Then we derive theinvolutive deformations in gravitational and EM backgrounds.We use the metric convention ( − , + , · · · , +). The γ -matrices satisfy: { γ a , γ b } = +2 η ab ,and γ a † = η aa γ a . Totally antisymmetric products of γ -matrices, γ a ··· a p ≡ γ [ a γ a · · · γ a p ] ,have unit weight. A “slash” denotes a contraction with a γ -matrix, e.g., /∂ = γ a ∂ a . The Fierz-Pauli conditions describing a symmetric fermionic field of mass m involve theDirac, divergence and gamma-trace operators [11]. These operators form the set: F = { f , g , f } , (3.1)where again a subscript gives the negative weight ( − w ) corresponding to an operator.Table 2 summarizes the various properties of these operators. For s = 1, because d µ d ν Φ = 0, the constraint that necessarily follows from Eq. (2.37) is: V µ = 0, i.e.,the EM background satisfies the source-free Maxwell equations. For s = 0, on the other hand, there isno constraint on the background field strength. w ) Derivative Order ( k )Dirac f /∂ − m g d · ∂ − f /d − different operators: [ f , g ],[ g , f ] and { f , f } . The first two commutators vanish, while the last one is given by: { f , f } = 2 g − mf , (3.2)which closes within the given set F . Let us now define the following operators: h ≡ [ f , g ] , h ≡ [ g , f ] , h ′ ≡ { f , f } − g + 2 mf , (3.3) j ≡ f h − ( f + 2 m ) h + g h ′ . (3.4)Because the operators { f , g , f } are linear, we have the graded Jacobi identity: { f , [ f , g ] } − { f , [ g , f ] } + [ g , { f , f } ] = 0 , (3.5)which enables us to rewrite the operator j , defined in Eq. (3.4), as: j = h f + h ′ g − h f . (3.6)The Fierz-Pauli equations comprise an involutive system of differential equations [21]: f Ψ = 0 , g Ψ = 0 , f Ψ = 0 . (3.7)The mutual compatibility of the equations (3.7) is encoded in the gauge identities: h Ψ = 0 , h Ψ = 0 , h ′ Ψ = 0 , (3.8)which follow directly from the graded commutators of the operators in F . Moreover,because of the relation (3.6), we have the following on-shell identity: j Ψ = 0 , (3.9)which is a gauge identity at reducibility order 1, implying that the gauge identities (3.8)are not irreducible. This completes the list of all possible gauge identities of our system.Note that the system (3.7)–(3.9) of involutive equations and gauge identities is of thetype considered in Appendix A.2.1. In order to check its absolute compatibility and countthe DoF’s, we first give the number of equations at order k in space-time derivatives : t k = (cid:20) δ k (cid:18) D + n − n (cid:19) + δ k (cid:18) D + n − n − (cid:19) + δ k (cid:18) D + n − n − (cid:19)(cid:21) [ D ] / , (3.10)12here n is the rank of the symmetric fermion, and D is the space-time dimensionality.We also have the count of O ( k ) gauge identities at reducibility order j , given by: l k, j = (cid:20) δ k δ j (cid:18) D + n − n − (cid:19) + δ k δ j (cid:18) D + n − n − (cid:19) + (cid:0) δ k δ j + δ k δ j (cid:1) (cid:18) D + n − n − (cid:19)(cid:21) [ D ] / . (3.11)Given the total number of field variables f = (cid:0) D + n − n (cid:1) [ D ] / , we find from Eq. (A.23)that c = 0, i.e., the fermionic Fierz-Pauli system is absolutely compatible. The count ofphysical DoF’s per space-time point is given by Eq. (A.24): D f ( n ) = (cid:18) D + n − n (cid:19) [ D − / , (3.12)which is the number of propagating DoF’s of a massive spin- (cid:0) n + (cid:1) fermion [11]. The free propagation of a massive fermion in a gravitational background can be analyzedby applying the involutive deformation method to the flat-space system we just described.In accordance with Appendix B, the substitution of ordinary derivatives by covariantones, ∂ µ → ∇ µ , gives the zeroth-order deformations, while linear terms in the backgroundcurvature comprise the first-order ones, etc. The deformations of the operators (3.1) oughtto preserve the respective weights w ; they can be written as:Dirac : ˆ f = / ∇ − M + O (cid:0) (cid:1) , Divergence : ˆ g = d ·∇ + O (cid:0) (cid:1) , Gamma-Trace : ˆ f = /d + O (cid:0) (cid:1) , (3.13)where the deformed mass M has weight w = 0, and Λ is some mass scale larger thanother scales in the theory. The deformations (3.13) are assumed to be smooth, so that inthe limit of zero curvature: M → m . Here, the book-keeping parameter indicating thedeformation order (see Appendix B) is implicit.We will now compute the graded commutators between two different operators in (3.13).The details of the computations are given in Appendix C.1. We must ensure that thesecommutators close within the given set of operators, so that some deformed versions ofthe gauge identities exist at all. An explicit computation leads us to the following result:[ ˆ f , ˆ g ] = W µνρσ γ µ u ρ d ν d σ + (cid:0) D − (cid:1) (cid:2) /u S µν d µ d ν − (cid:0) N + D − (cid:1) S µν γ µ d ν (cid:3) − [ M, d ·∇ ]+ (cid:0) D − (cid:1) (cid:2) S µν γ µ u ν /d − S µν u µ d ν (cid:3) /d + D ( D − R (cid:2) /u /d − (cid:0) N + D − (cid:1)(cid:3) /d + O (cid:0) (cid:1) . (3.14)From the first line of Eq. (3.14) it is clear that, for an arbitrary-spin field, the gravitationalbackground is required to fulfill the following conditions: W µνρσ = 0 , S µν = 0 . (3.15)13n other words, the background manifold must be a conformally flat as well as an Einsteinone. This is tantamount to the requirement of a maximally symmetric space, for whichEqs. (C.2) apply. We also need to deal with the commutator [ M, d ·∇ ]. In order to do so,let us assume that the deformed mass M is a linear function of the index operator N : M = m + µ ( N + δ ) , (3.16)where δ is a numerical constant, and µ a constant mass parameter that vanishes in thezero curvature limit. Again, the justification of such an assumption is postponed untilwe consider the massless case. The constraints (3.15) and the choice (3.16) reduce thecommutator (3.14) to a desired form. In an AdS space of radius L , one obtains:[ ˆ f , ˆ g ] = µ ˆ g − L (cid:2) /u /d − (cid:0) N + D − (cid:1)(cid:3) ˆ f + O (cid:0) (cid:1) . (3.17)The other graded commutators, on the other hand, are simpler to compute. They read:[ˆ g , ˆ f ] = O (cid:0) (cid:1) , { ˆ f , ˆ f } = 2ˆ g − (2 M + µ ) ˆ f + O (cid:0) (cid:1) . (3.18)With the graded commutation relations (3.17)–(3.18), we can now identify the de-formed counterparts of the operators (3.3); they are given by:ˆ h ≡ [ ˆ f , ˆ g ] − µ ˆ g + L (cid:2) /u /d − (cid:0) N + D − (cid:1)(cid:3) ˆ f , ˆ h ′ ≡ { ˆ f , ˆ f } − g + (2 M + µ ) ˆ f , (3.19)ˆ h ≡ [ˆ g , ˆ f ] . We also identify the deformed counterpart of the operator j in Eq. (3.4) with:ˆ j ≡ ˆ f ˆ h − (cid:16) ˆ f + 2( M + µ ) (cid:17) ˆ h + ˆ g ˆ h ′ . (3.20)The graded Jacobi identity involving the operators n ˆ f , ˆ g , ˆ f o , however, gives:ˆ j = ˆ h ˆ f + ˆ h ′ ˆ g − h ˆ h − L (cid:16) { ˆ f , /u /d − N } − ( D −
1) ˆ f (cid:17)i ˆ f . (3.21)At this stage, we are ready to present the deformed involutive system along with allthe gauge identities. The dynamical equations and constraints read:ˆ f Ψ = 0 , ˆ g Ψ = 0 , ˆ f Ψ = 0 , (3.22)while the gauge identities at reducibility order 0 are:ˆ h Ψ = 0 , ˆ h Ψ = 0 , ˆ h ′ Ψ = 0 , (3.23)14hich follow from the graded commutators (3.17)–(3.18) provided that the O (cid:0) (cid:1) -termsappearing therein vanish. The latter conditions can be ensured by taking:Λ → ∞ . (3.24)Furthermore, the relation (3.21) gives rise to the following on-shell identity:ˆ j Ψ = 0 , (3.25)which is the desired gauge identity at reducibility order 1. This completes our analysis.To summarize, the involutive system of equations for a massive fermion reads: (cid:0) / ∇ − M (cid:1) Ψ = 0 , d ·∇ Ψ = 0 , /d
Ψ = 0 , (3.26)where the deformed mass M is assumed to be of the form (3.16). For a fermion of arbitraryspin, this system is consistent in D ≥ massive fermion can be consistently described in an Einsteinspace ( S µν = 0). This can easily be seen from Eq. (3.14) given that in this case d µ d ν Ψ = 0.No such constraints on the gravitational background appear for s = . We assume that the massive fermion has a nonzero electric charge q , which defines itsminimal coupling to the EM background. As usual, the zeroth-order deformations areobtained by the substitution: ∂ µ → D µ . So, the deformations of the operators (3.1) are:Dirac : ¯ f = / D − m + A , Divergence : ¯ g = d ·D + B , Gamma-Trace : ¯ f = /d + C , (3.27)where A , B and C contain all the higher-order deformations that are assumed to be smoothand parity preserving. Note that the deformation order is controlled by the charge q .Given the formal expressions (3.27), one can write down the graded commutatorsbetween two different operators. They read:[ ¯ f , ¯ g ] = iqF µν γ µ d ν + (cid:0) /∂ B − d · ∂ A (cid:1) + (cid:0) [ γ µ , B ] − [ d µ , A ] (cid:1) D µ + [ A , B ] , (3.28)[¯ g , ¯ f ] = [ B , /d ] − d · ∂ C − [ d µ , C ] D µ + [ B , C ] , (3.29) { ¯ f , ¯ f } = 2¯ g − m ¯ f + { /d, A} − B + 2 m C + {C , A} . (3.30)These commutators ought to close within the set of operators (3.27). The F µν γ µ d ν -termin Eq. (3.28) requires that the non-minimal couplings be present, i.e., the terms A , and B cannot both be zero because otherwise the commutator [ ¯ f , ¯ g ] does not close.15t is difficult to find the general solution for A , B and C for generic spin. In order toproceed, we will therefore make some simplifying assumptions. First, let us assume that C = 0 . (3.31)In other words, the γ -trace operator does not undergo any deformation at order one orhigher. This can be justified by noting that all the known consistent models of chargedmassive higher-spin fields enjoy this property [13–16]. Moreover, such deformations maynot show up even in a gravitational background, as we just saw. Next, we spell out thenon-minimal deformation of the Dirac operator (see Appendix C.2): A = iq (cid:0) a + F + µν + a − F − µν (cid:1) u µ d ν + iq ( a F ρσ γ ρσ + · · · ) + O (cid:0) q (cid:1) , (3.32)where F ± µν ≡ F µν ± γ µνρσ F ρσ , the a ± and a are operators of weight w = 0 and massdimension −
1, and the ellipses stand for terms containing derivatives of the field strength.Similarly, we can write down the non-minimal deformation of the divergence: B = iq ( b F µν γ µ d ν + · · · ) + O (cid:0) q (cid:1) , (3.33)with b being a weight-0 operator of dimension −
1, and the ellipses containing derivativesof the field strength. Given Eqs. (3.31)–(3.33), one can compute the graded commutatorsup to O ( q ), as in Appendix C.2. For spin s ≥ , the cancellation of the offending O ( q )terms obstructing the closure of the commutators (C.35) and (C.39) requires that:1 − m ( a + − a − + 2 b ) = 0 ,a − − b = 0 , (3.34)( D − a + − ( D − a − + 4 a + 2 b = 0 , which can be solved, with the introduction of a single free parameter ǫ , as: a ± = (1 ± ǫ ) m − , a = − (cid:0) D − (cid:1) ǫm − , b = (1 − ǫ ) m − . (3.35)Moreover, the irreducible Lorentz tensors A µν and V µ (see Eq. (2.36)) must vanish, i.e., F µν = constant . (3.36)With these choices and constraints, the graded commutators (3.28)–(3.30) reduce to:[ ¯ f , ¯ g ] = ( iq/m ) F µν (cid:2) − γ µ d ν ¯ f + ǫ (cid:0) γ µν ¯ g + 2 γ µ D ν ¯ f − γ µν / D ¯ f (cid:1)(cid:3) + O (cid:0) q (cid:1) , [¯ g , ¯ f ] = ( iq/m ) (1 − ǫ ) F µν γ µ d ν ¯ f + O (cid:0) q (cid:1) , { ¯ f , ¯ f } = 2¯ g − m ¯ f + ( iq/m ) F µν (cid:2) u µ d ν + ǫγ µν (cid:3) ¯ f + O (cid:0) q (cid:1) . (3.37)16e are now ready to identify the deformations of the operators { h , h , h ′ } given inEq. (3.3). Up to O ( q ) correction terms, they are:¯ h ≡ [ ¯ f , ¯ g ] + ( iq/m ) F µν (cid:2) γ µ d ν ¯ f − ǫ (cid:0) γ µν ¯ g + 2 γ µ D ν ¯ f − γ µν / D ¯ f (cid:1)(cid:3) , ¯ h ≡ [¯ g , ¯ f ] − ( iq/m ) (1 − ǫ ) F µν γ µ d ν ¯ f , (3.38)¯ h ′ ≡ { ¯ f , ¯ f } − g + 2 m ¯ f − ( iq/m ) F µν (cid:2) u µ d ν + ǫγ µν (cid:3) ¯ f . We also identify the deformed counterpart of the operator j in Eq. (3.4); it is:¯ j ≡ ¯ f ¯ h − (cid:0) ¯ f + 2 m (cid:1) ¯ h + ¯ g ¯ h ′ . (3.39)Thanks to the graded Jacobi identity involving the operators (cid:8) ¯ f , ¯ g , ¯ f (cid:9) , one can use thedefinitions (3.38) to rewrite ¯ j in the following form:¯ j = ¯ O ¯ f + ¯ O ′ ¯ g − ¯ O ¯ f + O (cid:0) q (cid:1) , (3.40)where the explicit expressions of the operators ¯ O , ¯ O ′ and ¯ O are given in Eqs. (C.40).Our deformed involutive system consists of the dynamical equations and constraints:¯ f Ψ = 0 , ¯ g Ψ = 0 , ¯ f Ψ = 0 . (3.41)The required gauge identities are valid up to O ( q ). At reducibility order 0, they read:¯ h Ψ = O (cid:0) q (cid:1) , ¯ h Ψ = O (cid:0) q (cid:1) , ¯ h ′ Ψ = O (cid:0) q (cid:1) , (3.42)thanks to the graded commutators (3.37). At reducibility order 1, the gauge identity is:¯ j Ψ = O (cid:0) q (cid:1) , (3.43)which is an on-shell identity that follows from the relation (3.40).Therefore, a free massive fermion of spin s ≥ in an EM background is described, upto O ( q ), by the following one-parameter family of an involutive system of equations: (cid:8) / D− m +( iq/m ) (cid:2)(cid:0) F µν + ǫγ µνρσ F ρσ (cid:1) u µ d ν − (cid:0) D − (cid:1) ǫF µν γ µν (cid:3) + O ( q ) (cid:9) Ψ = 0 , (3.44) (cid:8) d ·D + ( iq/m )(1 − ǫ ) F µν γ µ d ν + O ( q ) (cid:9) Ψ = 0 , /d
Ψ = 0 , (3.45)given that the background is a constant one: F µν = constant. In principle, one can proceedorder by order in the parameter q to find the higher-order deformations. However, it isnot clear at all whether a consistent deformation up to all order exists for arbitrary spin.The only known example of an all-order solution is for s = in D = 4 [16], to which our O ( q )-results (3.44)–(3.45) agree, with the parameter choice of ǫ = 1. It has the Dirac equation: (cid:2) / D − m + m (cid:0) B + µν − B µρ B ρν + η µν Tr B (cid:1) u µ d ν (cid:3) Ψ = 0, plus constraints: (cid:0) d ·D + mB µρ B ρν γ µ d ν (cid:1) Ψ = 0, and /d Ψ = 0, where B µν = ( iq/m ) F µν + Tr B B µν − Tr( B ˜ B ) ˜ B µν . Massless Bosonic Fields
In this section, we consider the zero mass limit of the involutive system of a higher-spinmassive boson. As we will see, in the massless limit the flat-space involutive system (2.4)acquires a gauge symmetry, whose gauge parameter itself is governed by the same kind ofinvolutive system. In other words, we obtain the description of a higher-spin gauge bosonin the transverse-traceless gauge. Given the discussion of Appendix A.2.2, we then confirmthat the involutive system of a gauge boson describes the correct number of physical DoF’s.Armed with this formulation, we then study the consistent free propagation of masslessbosons in nontrivial backgrounds.
For the massive spin- s boson Φ of Eqs. (2.4), let us consider the following transformation: δ Φ = g − λ, λ = s − λ µ ··· µ s − ( x ) u µ · · · u µ s − , (4.1)where we have introduced the symmetrized gradient operator g − , defined as:Symmetrized Gradient: g − ≡ u · ∂, with [ N, g − ] = g − . (4.2)We take note of the following commutation relations for the symmetrized gradient:[ g , g − ] = 0 , [ g , g − ] = g + m , [ g , g − ] = 2 g , (4.3)to find that the left-hand sides of the involutive equations (2.4) transform as: δ ( g Φ) = g − ( g λ ) ,δ ( g Φ) = g − ( g λ ) + ( g + m ) λ, (4.4) δ ( g Φ) = g − ( g λ ) + 2 g λ. We would like to see when, if at all, transformations of the type (4.1) may become asymmetry of the Fierz-Pauli involutive system (2.4). With this end in view, let us firstimpose that λ itself be governed by the following involutive set of equations: g λ = 0 , g λ = 0 , g λ = 0 . (4.5)Then, the right-hand sides of Eqs. (4.4) vanish if: m λ = 0 . (4.6)Therefore, a nontrivial gauge symmetry emerges in the massless limit: m → g ≡ ∂ = lim m → g , with [ N, g ] = 0 . (4.7)The operators relevant for the massless case are the massless cousins of (2.1) and thesymmetrized gradient, which we collect in the following set: G = { g , g , g , g − } . (4.8)Notice that the massless counterparts of the commutators (4.3) are:[ g , g − ] = 0 , [ g , g − ] = g , [ g , g − ] = 2 g , (4.9)which close completely within the set G\ { g − } . This fact plays a crucial rˆole in the exis-tence of transverse-traceless gauge symmetry. It is the closure of the commutators (4.9)that ensures gauge invariance, which in turn controls the DoF count, as we will now see.In order to make the DoF count, let us note that the gauge field Φ and the gaugeparameter λ are both governed by the same set of involutive equations, which is: g " Φ λ = 0 , g " Φ λ = 0 , g " Φ λ = 0 . (4.10)It is easy to see from Section 2.1 that the zero mass limit does not hurt the involutivestructure of the Fierz-Pauli system (2.4). Neither does it alter the DoF count (2.10).In this case, however, the aforementioned count is a naive one because of the emergenceof gauge symmetry. This is precisely the circumstances under which the analysis ofAppendix A.2.2 may apply. From formula (A.29), it is easy to write down the number ofphysical DoF for a spin- s gauge field; it is simply the difference between the DoF countof a massive spin- s boson and that of a massive spin-( s −
1) boson: D (0) b ( s ) = D b ( s ) − D b ( s − . (4.11)Then, it follows directly from the DoF count formula (2.10) that D (0) b ( s ) = 2 (cid:18) D − ss − (cid:19) + (cid:18) D − ss (cid:19) , (4.12)which is the correct number of propagating DoF’s for a massless spin- s boson [11].19 .2 Gravitational Background In a gravitational background, we would like to find the deformed counterparts of theoperators (4.8). The massless limits of the deformed operators in Eqs. (2.30)–(2.31),augmented by the deformed symmetrized gradient ˆ g − give following set:ˆ G = { ˆ g , ˆ g , ˆ g , ˆ g − } . (4.13)This includes the deformed d’Alembertian operator:ˆ g = ∇ − M + R µνρσ u µ u ρ d ν d σ − R µν u µ d ν + N − N + D − D − D +2) R, (4.14)where, we recall from the mass ansatz (2.18) that, M = µ (cid:0) N + βN + γ (cid:1) , (4.15)with µ being a constant mass parameter that vanishes in the zero curvature limit, and β and γ numerical constants. We also have the deformed divergence and trace operators:ˆ g = d ·∇ , ˆ g = d . (4.16)Last but not the least, we have the deformed symmetrized gradient. To write this, let usrecall from Eq. (2.28) that we choose to stay in a parametric regime where the suppressionscale Λ of higher-dimensional operators is taken to infinity. This allows us to drop all thepossible non-minimal terms to ˆ g − , and instead identify it as a zeroth order deformation:ˆ g − = u ·∇ . (4.17)The involutive system of a spin- s massless boson Φ is given simply by the masslesslimits of Eqs. (2.30)–(2.31), i.e., through the deformed operators (4.14)–(4.16), as:ˆ g Φ = 0 , ˆ g Φ = 0 , ˆ g Φ = 0 . (4.18)The spin-( s −
1) gauge parameter λ , on the other hand, is governed by a similar system:ˆ g ′ λ = 0 , ˆ g λ = 0 , ˆ g λ = 0 . (4.19)In order not to ruin the involutive structure of Eqs. (4.19), the deformed d’Alembertianˆ g ′ acting on the gauge parameter may differ from ˆ g only in the mass-like term:ˆ g ′ = ˆ g + M − M ′ , (4.20)where M ′ is some quadratic polynomial in N , in accordance with the ansatz (2.18).20e now consider gauge transformations of the form: δ Φ = ˆ g − λ , and find the variationsof the left-hand sides of Eqs. (4.18); they are given by: δ (ˆ g Φ) = [ˆ g , ˆ g − ] λ + ˆ g − (ˆ g λ ) = [ˆ g , ˆ g − ] λ + ˆ g − (cid:0) M ′ − M (cid:1) λ,δ (ˆ g Φ) = [ˆ g , ˆ g − ] λ + ˆ g − (ˆ g λ ) = [ˆ g , ˆ g − ] λ,δ (ˆ g Φ) = [ˆ g , ˆ g − ] λ + ˆ g − (ˆ g λ ) = [ˆ g , ˆ g − ] λ, (4.21)where the right-hand sides are obtained by making use of Eqs. (4.19)–(4.20). In orderto see how gauge invariance can be restored in a gravitational background, we thereforeneed the commutators of ˆ g − with the other three operators in (4.13). The commutatorswith ˆ g and ˆ g are rather easy to compute; they can be written as:[ˆ g , ˆ g − ] = ˆ g ′ + X , [ˆ g , ˆ g − ] = 2ˆ g , (4.22)where the weight-0 operator X is explicitly given in Eq. (C.9). In view of Eqs. (4.19),the necessary and sufficient conditions for the vanishing of δ (ˆ g Φ) and δ (ˆ g Φ), i.e., for thegauge invariance of the transverse-traceless conditions amount to: X λ = 0 . (4.23)Now, using the decomposition formula (C.1), it is possible to write: X = − W µνρσ u µ u ρ d ν d σ + (cid:0) D − (cid:1) S µν (cid:2) (2 N + D ) u µ d ν − u d µ d ν − u µ u ν d (cid:3) + · · · , (4.24)where the ellipses contain neither of the irreducible tensors W µνρσ and S µν . By inspection,it is clear that in order for Eq. (4.23) to hold, for arbitrary spin s >
2, the gravitationalbackground is required to be conformally flat as well as Einsteinian: W µνρσ = 0 , S µν = 0 . (4.25)In other words, the background must be a maximally symmetric space . Then, one canmake use of Eq. (C.2) to find the following simple expression: X L = − u ˆ g + M ′ L − (2 N + D )( N + D − / (1 + D ) . (4.26)In order for Eq. (4.23) to be fulfilled, the following identification must be made: M ′ L = (2 N + D )( N + D − / (1 + D ) , (4.27)which gives a justification to the mass ansatz (2.18). The constraints (4.25) and theparameter choice (4.27) ensure the gauge invariance of the transverse-traceless conditions. Fulfilled automatically by any maximally symmetric space, the constraints (2.16) are indeed weaker. g , ˆ g − ]. This can be computed easily by taking the hermitian conjugate of Eq. (2.20) in the limit Λ → ∞ and m →
0. Thus, we obtain:[ˆ g , ˆ g − ] = − ˆ g − (cid:2) µ (2 N + β + 1) + L D (2 N + D − / (1 + D ) (cid:3) , (4.28)given the constraint of maximally symmetric background. Now, let us take the firstequation of (4.21), and plug the expressions (4.15), (4.27) and (4.28) in it to write: δ (ˆ g Φ) = − L ˆ g − (cid:0) δ N + δ N + δ (cid:1) λ, (4.29)where the numerical coefficients δ , δ and δ are given by: δ = µ L − D +2 , δ = ( β + 2) µ L − D − D − , δ = ( β + γ + 1) µ L . (4.30)Each of these coefficients must be zero since otherwise the right-hand side of Eq. (4.29)does not vanish. This leads to a unique solution for the parameters µ , β and γ , whichcan be reexpressed through a solution for the mass-like term, as: M L = ( N − N + D − / (1 + D ) . (4.31)This again justifies the mass ansatz (2.18). For the massive case − as long as the involutivestructure of the system is concerned − any arbitrary polynomial in the index operator N would qualify as the deformed mass. Only in the massless limit does one see why this oughtto be a quadratic polynomial in N . Given the constraints (4.25), and the expressions (4.27)and (4.31), the deformed d’Alembertians (4.14) and (4.20) reduce to:ˆ g = ∇ − m , m L ≡ ( N − N + D − − N, ˆ g ′ = ∇ − m ′ , m ′ L ≡ N ( N + D − − N. (4.32)Now we are ready to present our gauge invariant involutive system. The transformationof the massless spin- s field Φ is given in terms of a spin-( s −
1) gauge parameter λ , as δ Φ = u ·∇ λ . They are governed by their respective involutive systems: (cid:0) ∇ − m (cid:1) Φ = 0 , d ·∇ Φ = 0 , d Φ = 0 , (cid:0) ∇ − m ′ (cid:1) λ = 0 , d ·∇ λ = 0 , d λ = 0 , (4.33)with the mass-like terms given by Eqs. (4.32). This system holds good in D ≥ s > In this regard, the hermitian conjugation is implemented by: u † µ = d µ and d † µ = u µ . Indeed one has:[ d µ , u ν ] = [ d µ , d † ν ] = g µν , which allows for interpretation in terms of creation and annihilation operators. ower Spins The constraints (4.25) on the gravitational background are necessary only for gauge bosonswith spin s ≥
3. The gauge parameter in the spin-2 case satisfies: d µ d ν λ = 0, and thereforethe Weyl tensor does not need to vanish in Eq. (4.24) for a field with s = 2. The necessaryconstraint in this case turns out to be:For s = 2 : S µν = 0 . (4.34)In other words, the gravitational background must be an Einstein manifold. Note thatthe conditions (2.32) in the massive case automatically holds for such a background. Thesystem is still described by Eqs. (4.33), with the substitution: L → D ( D − / | R | .The spin-2 result is quite expected in view of General Relativity. Einstein manifoldsare nothing but the vacuum solutions of Einstein equations. On such backgrounds, onecan always consider linearized graviton fluctuations, which of course will propagate consis-tently, thanks to General Relativity. Note that it is the absence of a stress-energy tensorthat enables one to take into account solely graviton fluctuations in the EoM’s.For s = 1, no restrictions on the gravitational background are imposed. In this case,it is easy to see that the gauge system will instead be described by: (cid:0) ∇ − R µν u µ d ν (cid:1) Φ = 0 , d ·∇ Φ = 0; ∇ λ = 0 . (4.35)In particular, the mass-like terms M and M ′ must be set to zero. In this section, we will consider the propagation of a charged bosonic field in an EMbackground, and will end up with a no-go for a higher-spin gauge boson, and a yes-go fora massless vector. The EM counterparts of the involutive systems (4.18)–(4.19), for thespin- s massless boson Φ and the accompanying spin-( s −
1) gauge parameter λ , read: " ¯ g
00 ¯ g ′ Φ λ = 0 , ¯ g " Φ λ = 0 , ¯ g " Φ λ = 0 , (4.36)with the deformed operators given directly from Eq. (2.49) as:d’Alembertian : ¯ g = D − ¯ M + 2 iqF µν u µ d ν , Divergence : ¯ g = d ·D , Trace : ¯ g = d , (4.37)along with ¯ g ′ = ¯ g + ¯ M − ¯ M ′ , where the mass-like terms ¯ M and ¯ M ′ are polynomialsin the index operator N that vanish in the limit of zero background EM field strength.23n the other hand, the deformed symmetrized gradient is identified as a zeroth-orderdeformation (for a reason analogous to that of the gravitational case), i.e.,Symmetrized gradient : ¯ g − = u ·D . (4.38)In order to consider gauge transformations: δ Φ = ¯ g − λ , one needs the commutatorsof ¯ g − with the other operators; they are easy to compute. Upon using the Eqs. (4.36),one ends up with the following variation of the involutive system: δ (¯ g Φ) = (cid:0) ¯ g − ¯ M ′ − ¯ M ¯ g − (cid:1) λ,δ (¯ g Φ) = (cid:0) ¯ M ′ − iqF µν u µ d ν (cid:1) λ,δ (¯ g Φ) = 0 , (4.39)where the constancy of background field strength has been taken into account. It is clearthat gauge invariance cannot be restored for a generic spin s >
1, irrespective of the massparameters. In particular, ¯ M ′ may only be a function of the index operator N , and soit cannot cancel the operation of the F µν u µ d ν -term in the variation δ (¯ g Φ).Thus, we come up with a no-go theorem: a charged gauge boson with spin s > U (1) gauge field [5]. Yes-Go for Massless Vector
For spin s = 1, the right-hand sides of Eqs. (4.39) may all vanish, i.e., we have a yes-goresult. To see this, let us note that d µ λ = 0 in this case, and so the variation δ (¯ g Φ)vanishes if ¯ M ′ is set to zero. Then, the variation δ (¯ g Φ) also vanishes with the choice¯ M = 0. This leaves us with the following involutive system for a massless vector Φ: (cid:0) D + 2 iqF µν u µ d ν (cid:1) Φ = 0 , d ·D Φ = 0 , (4.40)in an EM background: F µν = constant, along with the gauge symmetry: δ Φ = u ·D λ, D λ = 0 . (4.41)This yes-go result may not come as a surprise given the existence of Yang-Mills theoriesas consistent interacting theories of spin-1 gauge fields. Indeed, the system (4.40)–(4.41)can be obtained from a non-Abelian gauge theory linearized around some background.To see this, let us consider an SU (2) gauge field W aµ , whose field strength is given by: G aµν = ∂ µ W aν − ∂ ν W aµ + gǫ abc W bµ W cν , where g is the Yang-Mills coupling. The EoM’s are: ∂ µ G aµν + gǫ abc W µ,b G cµν = 0 , a = 1 , , , (4.42)24nd the infinitesimal gauge transformations read: δW aµ = ∂ µ λ a + gǫ abc W µ,b λ c . (4.43)It is easy to see that the EoM’s (4.42) admit the following solution: W µ = W µ = 0 , W µ = A µ = 0 , with F µν = 2 ∂ [ µ A ν ] = constant . (4.44)On this background, let us now consider small fluctuations w aµ . At the linearized level,the mode w µ behaves as if it were a U (1) gauge field: ∂ µ (cid:0) ∂ µ w ν − ∂ ν w µ (cid:1) = 0 , δw µ = ∂ µ λ . (4.45)The other two modes have the linearized field strengths: f iµν ≡ (cid:16) ∂ [ µ w iν ] + ( − ) i gA [ µ w j = iν ] (cid:17) , i, j = 1 , , (4.46)through which these modes are described by the coupled equations: ∂ µ f iµν + ( − ) i g (cid:0) A µ f j = iµν − F µν w µ,j = i (cid:1) = 0 , (4.47)that are invariant under the gauge transformations: δw iµ = ∂ µ λ i + ( − ) i gA µ λ j = i . (4.48)Now, we consider the following complex vector field and gauge parameter:Φ µ ≡ √ (cid:0) w µ + iw µ (cid:1) , λ ≡ √ (cid:0) λ + iλ (cid:1) . (4.49)At the linear level, the Yang-Mills coupling g can now be identified as the U (1) charge q of the vector Φ µ , on which acts the covariant derivative: D µ ≡ ∂ µ + igA µ . The EoM’sand the gauge symmetry of Φ µ read:2 D µ D [ µ Φ ν ] − iqF µν Φ µ = 0 , δ Φ µ = D µ λ, (4.50)which reduces precisely to our system (4.40)–(4.41) in the Lorenz gauge: D µ Φ µ = 0. This section explores the massless limit of the involutive system of a massive higher-spin fermion. In this limit, as we will see, the flat-space involutive system (3.7) acquiresa gauge symmetry with an “unfree” gauge parameter governed by the same involutivesystem as the field. This is nothing but the description of a massless higher-spin fermionin the transverse-traceless gauge. We confirm, along the line of Appendix A.2.2, thatthe resulting involutive system describes the correct number of physical DoF’s of a gaugefermion. Given this reformulation, we go on to studying the consistent free propagationof higher-spin gauge fermions in nontrivial backgrounds.25 .1 Minkowski Background
Let us consider, for the massive rank- n fermion Ψ of Eqs. (3.7), the transformation: δ Ψ = g − ε, ε = n − ε µ ··· µ n − ( x ) u µ · · · u µ n − , (5.1)where g − is the symmetrized gradient operator, already introduced in Eq. (4.2). In viewof the commutation relations for the symmetrized gradient:[ f , g − ] = 0 , [ g , g − ] = ( f + m ) , [ f , g − ] = f + m, (5.2)it is easy to see that the left-hand sides of Eqs. (3.7) transform as: δ ( f Ψ) = g − ( f ε ) ,δ ( g Ψ) = g − ( g ε ) + ( f + m ) ε, (5.3) δ ( f Ψ) = g − ( f ε ) + ( f + m ) ε. To see if transformations of the type (5.1) may become a symmetry of the Fierz-Paulisystem (3.7), let us require that ε itself be governed by the following involutive equations: f ε = 0 , g ε = 0 , f ε = 0 . (5.4)Then, then the variations (5.4) vanish if and only if: mε = 0 . (5.5)Clearly, in the zero mass limit: m →
0, there appears a nontrivial gauge symmetry.The involutive system of a massless fermion therefore enjoys a gauge symmetry (5.1),where the gauge parameter itself is governed by Eqs. (5.4) with zero mass. In this case,the massless Dirac operator f is of relevance, for which we have the following:Massless Dirac: f ≡ /∂ = lim m → f , with [ N, f ] = 0 . (5.6)Note that the set of operators essential for the massless case is given by: F = { f , g , f , g − } , (5.7)and that the massless counterparts of the commutators (5.2) read:[ f , g − ] = 0 , [ g , g − ] = f , [ f , g − ] = f . (5.8)These commutators close completely within the set F \ { g − } . This ensures transverse-traceless gauge symmetry, which in turn controls the DoF count, as we will now show.26et us recall that the rank- n gauge field Ψ and the rank-( n −
1) gauge parameter ε are both governed by the same involutive set of equations: f " Ψ ε = 0 , g " Ψ ε = 0 , f " Ψ ε = 0 . (5.9)It is easy to see from Section 3.1 that the massless limit does not affect the involutivestructure of the Fierz-Pauli system (3.7). Neither does it alter the DoF count (3.12).However, because of the emergence of (unfree) gauge symmetry, the count (3.12) includespure gauge modes as well. In this case, the analysis of Appendix A.2.2 applies, and onecan easily write down the number of physical DoF for a rank- n gauge fermion. As seenfrom formula (A.29), it must be the difference between the DoF count of a massive rank- n fermion and that of a massive rank-( n −
1) fermion: D (0) f ( n ) = D f ( n ) − D f ( n − . (5.10)From the DoF count formula (3.12), then it follows that D (0) f ( n ) = (cid:18) D + n − n (cid:19) [ D − / . (5.11)This is indeed the correct number of physical DoF’s for a rank- n gauge fermion [11]. We would like to have the deformed counterparts of the operators (5.7) in a gravitationalbackground; they constitute the following set:ˆ F = { ˆ f , ˆ g , ˆ f , ˆ g − } , (5.12)which includes the operators appearing in the zero mass limits of Eqs. (3.26), i.e.,ˆ f = / ∇ − m , ˆ g = d ·∇ , ˆ f = /d, (5.13)where, as we recall from the ansatz (3.16), the mass-like term takes the form: m = µ ( N + δ ) , (5.14)with µ being a mass parameter that vanishes in the zero-curvature limit, and δ a numericalconstant. In order to write down the deformed symmetrized gradient ˆ g − , we recall thatEq. (3.24) sets to infinity the suppression scale Λ of the higher-dimensional operators.This leaves us with the following generic form of ˆ g − :ˆ g − = u ·∇ − ˆ µ /u, (5.15)27here ˆ µ is another constant mass parameter vanishing in the limit of zero curvature.Note that Eq. (5.15) is in contrast with its bosonic counterpart (4.17), where only thezeroth-order deformation could be written down. The higher-spin gauge fermion Ψ andthe gauge parameter ε are governed by the following involutive systems: " ˆ f
00 ˆ f ′ Ψ ε = 0 , ˆ g " Ψ ε = 0 , ˆ f " Ψ ε = 0 , (5.16)where ˆ f ′ = ˆ f + m − m ′ , for some mass-like term m ′ of the form (5.14).We consider gauge transformations of the gauge-fermion involutive system: δ Ψ = ˆ g − ε .In view of Eqs. (5.16), it is easy to obtain the following variations: δ ( ˆ f Ψ) = [ ˆ f , ˆ g − ] ε + ˆ g − ( m ′ − m ) ε,δ (ˆ g Ψ) = [ˆ g , ˆ g − ] ε,δ ( ˆ f Ψ) = [ ˆ f , ˆ g − ] ε. (5.17)In order see how these variations may vanish, we need the commutators of ˆ g − with theother operators: { ˆ f , ˆ g , ˆ f } . The simplest one reads:[ ˆ f , ˆ g − ] = ˆ f ′ + 2ˆ µ /u ˆ f + m ′ − µ ( N + D/ . (5.18)The vanishing of the variation δ ( ˆ f Ψ) therefore requires that m ′ = 2ˆ µ ( N + D/ . (5.19)Next, the computation of [ ˆ f , ˆ g − ] is simplified by noting that, in the limit of m → → ∞ , the hermitian conjugate (in the sense of footnote 5) of Eq. (3.17) provides with[ ˆ f , u ·∇ ], whereas the commutator [ ˆ f , /u ] is easy to compute. The end result is:[ ˆ f , ˆ g − ] = 2ˆ µ /u ˆ f ′ + L u ˆ f − ( µ + 2ˆ µ ) ˆ g − + /u (cid:2) µ ( m ′ − ˆ µ ) − L (cid:0) N − + D/ (cid:1)(cid:3) , (5.20)where we used the maximal symmetry of the background. When plugged into the variation δ ( ˆ f Ψ), the last term of Eq. (5.20) − combined with the result (5.19) − implies:ˆ µ = L . (5.21)The terms containing ˆ g − , on the other hand, justify the mass ansatz (5.14), and give: m = 2ˆ µ ( N − D/ . (5.22)This completely fixes all the parameters in the theory. It is conventional to choose thepositive root of Eq. (5.21) [27, 28], which sets: ˆ µ = + L .28ne still needs to show that the variation δ (ˆ g Ψ) also vanishes. Given the rela-tion (C.4), it is straightforward to cast the commutator [ˆ g , ˆ g − ] into the following form:[ˆ g , ˆ g − ] = ∇ − ˆ µ ( ˆ f ′ + m ′ ) + L h u ˆ f + /u ˆ f − N (cid:0) N + D − (cid:1)i . (5.23)The expression of ∇ in terms of the massless Dirac operator is somewhat subtle. Oneneeds to compute the anti-commutator { ˆ f ′ , ˆ f ′ } to show that: ∇ = ˆ f ′ + 2 m ′ ˆ f ′ + m ′ + L h /u ˆ f − N − D ( D − i . (5.24)Then, the expressions (5.23)–(5.24) indeed renders the variation δ (ˆ g Ψ) vanishing onaccount of the relations (5.19) and (5.21).We are now in a position of presenting our gauge invariant involutive system. Therank- n gauge field Ψ and the rank-( n −
1) gauge parameter ε obey: (cid:0) / ∇ − m (cid:1) Ψ = 0 , d ·∇ Ψ = 0 , /d
Ψ = 0 , (cid:0) / ∇ − m ′ (cid:1) ε = 0 , d ·∇ ε = 0 , /d ε = 0 , (5.25)where the mass-like terms are given by: m L = N − D/ , m ′ L = N + D/ , (5.26)and the gauge transformations read: δ Ψ = (cid:0) u ·∇ − L /u (cid:1) ε. (5.27)This system holds good for an arbitrary-spin gauge fermion in D ≥ L appearing in Eqs. (5.26)–(5.27)is the AdS radius. For dS space, we need to do the analytic continuation: L → iL . Rarita-Schwinger Gauge Field
For s = , the gravitational background will have a weaker constraint, but the involutivesystem (5.25)–(5.27) holds good, with L → p D ( D − / | R | . Let us recall from Section 3.2that the massive involutive system is consistent in Einstein spaces. Going massless in thiscase, by requiring gauge symmetry, does not pose any additional condition. To see this,let us notice how the gauge variations (5.17) could vanish for generic spin. The conditionson the background played rˆole only through Eqs. (5.20), (5.23) and (5.24). An Einsteinmanifold may well be conformally non-flat, i.e., possess a non-vanishing Weyl tensor. Inthis case, the right-hand side of Eq. (5.20) picks up an additional term: W µνρσ γ µ u ν u ρ d σ ,which gives zero contribution in the variation δ ( ˆ f Ψ), since d µ ε = 0. Similarly, Eq. (5.23)would include terms containing a Weyl tensor and at least one d µ , and they do not29ontribute to the variation δ (ˆ g Ψ). Last but not the least, Eq. (5.24) also picks up theterm: W µνρσ γ µν γ ρσ . By using the symmetries of the Weyl tensor, the γ -matrix productcan be rewritten as: γ µνρσ − g µρ g νρ . The latter terms give zero on account of the Bianchiidentity and tracelessness of the Weyl tensor. Therefore, it is necessary and sufficient torequire that the background be an Einstein space.This result makes sense from the perspective of supergravity. The classical solutions ofpure N = 1 supergravity are indeed Einstein spaces, on which fluctuations of the masslessspin- Majorana fermion propagate consistently. However, extended supergravity theoriesadmit more generic classical backgrounds. In particular, Maxwell-Einstein spaces appearin pure N = 2 (un)gauged supergravity, and this seems to contradict our result. Oneof the loopholes lies in the deformed gauge transformation of the gravitino; it involves a U (1) gauge field [29–31] − a possibility we do not consider. Moreover, in the gauged theorythe complex gravitino has a U (1) charge as well. Let us recall that in Section 3.3 we assumed minimal coupling, i.e., a nonzero charge q of the higher-spin fermion. However, it is manifest that the resulting involutive sys-tem (3.44)–(3.45) is ill-defined in the massless limit: m →
0. This can be traced backto Eqs. (3.34), which admit no solutions of the deformed Dirac, divergence and γ -traceoperators as the mass goes to zero for spin s ≥ . Thus, we are lead to a no-go theorem:a charged gauge fermion cannot propagate consistently in a purely EM background. Inother words, there is no consistent theory of a gauge fermion, minimally coupled to a U (1)field, that admits a pure background of the Maxwell field as a classical solution. This isin accordance with the no-go results [5, 32] that forbid in flat space the minimal couplingof a massless fermion with spin s ≥ to a U (1) gauge field.One way to bypass this no-go is to consider additional interactions in the theorysuch that purely U (1) backgrounds are not allowed. This works at least for a masslesscharged Rarita-Schwinger field, which requires a cosmological constant [33] (see also [31]for a cohomological derivation). Indeed, N = 2 gauged supergravity [29, 30] consistentlyincorporates a massless gravitino minimally coupled to a U (1) field (graviphoton) as wellas gravity in the presence of a cosmological constant. Determined by Eq. (5.26), the massparameter in this case is also related to the U (1) charge. In AdS the relations read: m = 1 /L = 2 q M P . (5.28)The classical solutions of pure N = 2 are, of course, Maxwell-Einstein spaces on whichfluctuations of the massless charged gravitino propagate consistently. Whether a similartype of yes-go can be found for higher-spin gauge fermions is an open question.30 Partially Massless Fields
In a constant curvature space, it turns out that gauge symmetries of a higher-spin fieldappear for a discrete series of mass parameters, known as partially massless (PM) points.Originally studied in [34, 35], this phenomenon was further investigated in [36–40]. Inthis section, we consider the involutive system of PM bosons and fermions. Just like amassless system is described by Eqs. (4.33) or (5.25), a PM field and its gauge parameterare also governed by the same type of involutive systems. However, PM fields are moregeneral in that their gauge transformations may include multiple gradients of the gaugeparameters. A PM field is said to have depth ( k + 1) when its gauge transformationcontains ( k + 1) space-time derivatives plus possibly a lower-derivative tail:Boson : δ Φ ( k +1) s = (cid:2) ( u ·∇ ) k +1 + · · · (cid:3) λ s − k − , k = 0 , , · · · , s − , Fermion : δ Ψ ( k +1) n = (cid:2) ( u ·∇ ) k +1 + · · · (cid:3) ε n − k − , k = 0 , , · · · , n − , (6.1)where the subscripts on the fields and gauge parameters denote their respective ranks(unlike that on an operator, which gives the negative of its weight), whereas the superscripton a PM field stands for its depth. Let us denote by ˆ g − k − the weight-( k + 1) operatorsappearing in the PM gauge transformations (6.1):ˆ g − k − = ( u ·∇ ) k +1 + lower-derivative tail . (6.2)Note that the strictly massless case corresponds to depth = 1, i.e., k = 0. We would liketo find the explicit form of ˆ g − k − , i.e., that of the depth-( k +1) gauge transformations (6.1)as well as the PM discrete points of the mass parameters in AdS space.The DoF count works in the following way. As we will see, just like the strictlymassless case, the PM field and its gauge parameter will both be governed by theirrespective involutive systems. Therefore, the analysis of Appendix A.2.2 also applieshere; the number of physical DoF will simply be the difference between the DoF countsof a massive field and a massive gauge parameter:Boson : D ( k ) b ( s ) = D b ( s ) − D b ( s − k − , Fermion : D ( k ) f ( n ) = D f ( n ) − D f ( n − k − . (6.3)Then, the DoF count at depth ( k +1) follows directly from formula (2.10) or (3.12). Belowwe go into the details separately for bosonic and fermionic PM fields. With some abuse of notations, we will denote the discrete mass points by m k and m ′ k respectively forthe PM field and the gauge parameter. These mass parameters are of course w = 0 operators, for whichthe subscript k does not correspond to the weight but to the value of depth minus one. Accordingly, themass parameters in the strictly massless case are denoted by m and m ′ , as in Eqs. (4.32) or (5.26). .1 Bosonic Fields For bosonic PM fields, it will be convenient to define the following operator: (cid:3) ≡ [ d ·∇ , u ·∇ ] , (6.4)which can be written in terms of ∇ through Eq. (C.4) in AdS space. In analogy withthe strictly massless case of Section 4.2, the involutive system of a spin- s depth-( k + 1)PM boson Φ ( k +1) s and its gauge parameter λ s − k − can be written as: " ˆ g ( k +1)0
00 ˆ g ′ ( k +1)0 Φ ( k +1) s λ s − k − = 0 , ˆ g " Φ ( k +1) s λ s − k − = 0 , ˆ g " Φ ( k +1) s λ s − k − = 0 , (6.5)where ˆ g = d · ∇ and ˆ g = d are the usual divergence and trace operators appearingin Eq. (4.16), while the deformed d’Alembertian operators ˆ g ( k +1)0 and ˆ g ′ ( k +1)0 generalizeEqs. (4.32) for arbitrary depth. We will prove that the d’Alembertians are given by:ˆ g ( k +1)0 = (cid:3) − L ( k + 2)( k − N − D + 3) = ∇ − m k + L u d , ˆ g ′ ( k +1)0 = (cid:3) − L k ( k + 2 N + D −
1) = ∇ − m ′ k + L u d , (6.6)where the PM mass parameters at depth ( k + 1) are specified as: m k L ≡ ( N − k − N − k + D − − N, m ′ k L ≡ ( N + k )( N + k + D − − N. (6.7)We will also prove the following explicit form of the gauge transformations:ˆ g − k − = ( u ·∇ ) ǫ k (cid:2) ( u ·∇ ) − L u ( N − s + 1) (cid:3) [ k +1] / , (6.8)where ǫ k = (cid:2) − ) k (cid:3) , which is 1(0) for k even(odd). Note that Eq. (6.8) induces thethe following iterative expression on a spin-( s − k −
1) gauge parameter:ˆ g − k − = (cid:2) ( u ·∇ ) − L c k u (cid:3) ˆ g − k +1 , c k = k , k ≥ . (6.9)In what follows we provide a proof of Eqs. (6.5)–(6.9) by recourse to the method ofinduction. To proceed, let us make the following ans¨atze for the deformed d’Alembertians:ˆ g ( k +1)0 = (cid:3) − L ( a k N + b k ) , ˆ g ′ ( k +1)0 = (cid:3) − L ( a ′ k N + b ′ k ) , (6.10)where a k , b k are their primed counterparts are numerical constants. Therefore, in orderto prove Eqs. (6.6)–(6.7) we ought to show the following: a k = − k + 2) , b k = ( k + 2)( k − D + 3) , a ′ k = 2 k, b ′ k = k ( k + D − . (6.11)Similarly, the PM gauge transformations will also be proved with ans¨atze compatible withEqs. (6.8)–(6.9). Below present our proofs for k = 0 , ,
2, and then for generic k .32 = 0: This is the strictly massless case, for which Eqs. (6.5)–(6.7) have already beenproved in Section 4.2. Indeed, for k = 0 the gauge transformation is given by: ˆ g − = u ·∇ ,whereas the dynamical equations reduce to Eqs. (4.32) given the trace constraints. k = 1: This corresponds to depth 2 − the simplest nontrivial PM gauge symmetry. In thiscase, the most generic form of the PM gauge transformation could be:ˆ g − = ( u ·∇ ) − L c u , c = constant . (6.12)In order to compute the gauge variations of the EoM’s we need the commutator of ˆ g − with { ˆ g (2)0 , ˆ g , ˆ g } , which are given in Eqs. (C.11)–(C.13). Upon making use of the involutivesystem for the gauge parameter λ s − , these variations simplify to Eqs. (C.14)–(C.15).Consequently, gauge invariance requires the following choice of constants: a = − , b = − D − , a ′ = 2 , b ′ = D, c = 1 . (6.13)These are precisely the values given for k = 1 by Eqs. (6.11) and the gauge transforma-tion (6.8), with c being the eigenvalue of ( N − s + 1) corresponding to λ s − . k = 2: Let us make the ansatz that the depth-3 gauge transformation is implemented by:ˆ g − = u ·∇ (cid:2) ( u ·∇ ) − L c u (cid:3) , c = constant . (6.14)The variations of the EoM’s of the PM field Φ (3) s is easy to compute given the basic com-mutation relations (C.10)–(C.13). Again, the involutive system for the gauge parameter λ s − is taken into account in order to simplify these gauge variations. Their explicit formsare given in Eqs. (C.16)–(C.17). In order for the gauge variations to vanish we must have: a = − , b = − D − , a ′ = 4 , b ′ = 2( D + 1) , c = 4 . (6.15)Again, these are the values Eqs. (6.11) and the gauge transformation (6.8) give for k = 2.Here, c is indeed the eigenvalue of ( N − s + 1) corresponding to λ s − . Also, the recursionformula (6.9) works, since setting k = 2 therein reproduces Eq. (6.14) with c = 4.Generic k : Let us assume that the involutive system (6.5)–(6.9) holds good up to andincluding k = j −
2, for some integer j ≥
2. It will then follow that the same systemalso consistently describes the case k = j . To see this, let us make the ansatz that thedepth-( j + 1) PM gauge transformation is implemented by the following operator:ˆ g − j − = (cid:2) ( u ·∇ ) − L c j u (cid:3) ˆ g − j +1 , (6.16)where c j is some constant to be determined. Recall that for the deformed d’Alembertianswe have the ans¨atze (6.10). Then we can compute the gauge variations of the left-hand33ides of the involutive equations for Φ ( j +1) s . They take the following form: δ h ˆ g ( j +1)0 Φ ( j +1) s i = (cid:2) (cid:3) − L ( a j N + b j ) (cid:3) (cid:2) ( u ·∇ ) − L c j u (cid:3) ˆ g − j +1 λ s − j − ,δ h ˆ g Φ ( j +1) s i = d ·∇ (cid:2) ( u ·∇ ) − L c j u (cid:3) ˆ g − j +1 λ s − j − , (6.17) δ h ˆ g Φ ( j +1) s i = d (cid:2) ( u ·∇ ) − L c j u (cid:3) ˆ g − j +1 λ s − j − , with the “unfree” gauge parameter λ s − j − being subject to: (cid:2) (cid:3) − L (cid:0) a ′ j N + b ′ j (cid:1)(cid:3) λ s − j − = 0 , d ·∇ λ s − j − = 0 , d λ s − j − = 0 , (6.18)where a j and b j and their primed counterparts are constants to be determined.In computing the right-hand sides of Eqs. (6.17), one needs to make repeated use ofthe commutators (C.10)–(C.13), and conditions (6.18) on the gauge parameter. After atedious but straightforward calculation, one arrives at the following results: δ h ˆ g ( j +1)0 Φ ( j +1) s i = n L ( u ·∇ ) j +1 L j + · · · o λ s − j − ,δ (cid:2) ˆ g Φ ( j +1) s (cid:3) = n L ( u ·∇ ) j L j + · · · o λ s − j − , (6.19) δ (cid:2) ˆ g Φ ( j +1) s (cid:3) = n L ( u ·∇ ) j − L j + · · · o λ s − j − , where the ellipses stand for lower-derivative terms, and the L j ’s are given by: L j = [ a ′ j − a j − j + 1)] N + [ b ′ j − b j − ( j + 1)( a j + 2( j + D − , L j = ( j + 1)( a ′ j − j ) N + [( j + 1)( b ′ j − D + 1) − c j − ( j − j ( j + D −
1) + D − , L j = [ j ( j + 1) a ′ j − c j − j ( j − N + [ j ( j + 1) b ′ j − D + 2( j − c j − j ( j − j + D − . (6.20)In deriving the above expressions one makes use of the assumption that the involutivesystem (6.5)–(6.8) holds good for k ≤ j −
2. Thus, the expressions (6.9)–(6.11) are validup to and including k = j −
2. Now, in order for the gauge variations (6.19) to vanishit is necessary that the gauge parameter λ s − j − belongs simultaneously to the kernelsof L j , L j and L j . It is however easy to see that, for a nontrivial gauge parameter,such conditions can only be satisfied when the operators themselves vanish. This gives aunique set of solutions for c j , a j , b j , a ′ j and b ′ j ; it coincides with that given by Eqs. (6.9)and (6.11) for k = j . Too see that these values also suffice for the vanishing of thegauge variations (6.19), one needs to compute all the lower-derivative terms omitted inthe ellipses. While one can convince oneself by explicitly working them out for any given j , we choose not to present this tedious exercise, and conclude without further ado.34et us now summarize the results. In AdS space, the involutive system of a spin- s depth-( k + 1) PM boson Φ ( k +1) s and its spin-( s − k −
1) gauge parameter λ s − k − reads: (cid:0) ∇ − m k (cid:1) Φ ( k +1) s = 0 , d ·∇ Φ ( k +1) s = 0 , d Φ ( k +1) s = 0 , (cid:0) ∇ − m ′ k (cid:1) λ s − k − = 0 , d ·∇ λ s − k − = 0 , d λ s − k − = 0 , (6.21)with the mass terms given by Eqs. (6.7) for k ≥
0. Note that the above system hasbeen presented without the u d -terms appearing in the d’Alembertians (6.6). This ispossible because the trace conditions themselves are a part of the involutive system. Thedepth-( k + 1) PM gauge symmetry transformations of the system (6.21) are of the form: δ Φ ( k +1) s = ˆ g − k − λ s − k − , (6.22)where the operator ˆ g − k − contains up to ( k + 1) derivatives, given explicitly in Eq. (6.8). For fermionic PM fields, let us define a deformed covariant derivative ∆ µ as follows:∆ µ ≡ ∇ µ − L γ µ , [∆ µ , ∆ ν ] = − L (cid:0) u [ µ d ν ] (cid:1) . (6.23)In analogy with the strictly massless case of Section 5.2, the involutive system of a rank- n depth-( k + 1) PM fermion Ψ ( k +1) n and its gauge parameter ε n − k − can be written as: " ˆ f ( k +1)0
00 ˆ f ′ ( k +1)0 Ψ ( k +1) n ε n − k − = 0 , ˆ g ′ " Ψ ( k +1) n ε n − k − = 0 , ˆ f " Ψ ( k +1) n ε n − k − = 0 , (6.24)where ˆ f ( k +1)0 and ˆ f ′ ( k +1)0 are the deformed Dirac operators, while ˆ g ′ ≡ d · ∆ = d ·∇ − L /d is a deformed divergence, and ˆ f the usual γ -trace operator. We will show that:ˆ f ( k +1)0 = / ∆ − L ( N − k −
2) = / ∇ − m k , ˆ f ′ ( k +1)0 = / ∆ − L ( N + k ) = / ∇ − m ′ k , (6.25)where the PM mass parameters at depth ( k + 1) generalize Eqs. (5.26), and are given by: m k L ≡ N − k − D/ , m ′ k L ≡ N + k + D/ . (6.26)The gauge transformations will be quite similar to the bosonic ones (6.8). Explicitly,ˆ g − k − = ( u · ∆) ǫ k (cid:2) ( u · ∆) − L u ( N − n + 1) (cid:3) [ k +1] / . (6.27)Again, this induces the following iterative expression on a rank-( n − k −
1) gauge parameter:ˆ g − k − = (cid:2) ( u · ∆) − L δ k u (cid:3) ˆ g − k +1 , δ k = k , k ≥ . (6.28)35n what follows we will employ the method of induction to prove Eqs. (6.24)–(6.28).We start by making the following ans¨atze for the deformed Dirac operators:ˆ f ( k +1)0 = / ∆ − L ( α k N + β k ) , ˆ f ′ ( k +1)0 = / ∆ − L ( α ′ k N + β ′ k ) , (6.29)where α k , β k , α ′ k and β ′ k are numerical constants. Then, the proof of Eqs. (6.25)–(6.26)boils down to finding the following solutions for these constants: α k = 1 , β k = − ( k + 2) , α ′ k = 1 , β ′ k = k. (6.30)With ans¨atze compatible with Eqs. (6.27)–(6.28), the PM gauge transformations will alsobe proved. Below we present the proofs for k = 0 , ,
2, and then for arbitrary k . k = 0: This is the strictly massless case, already considered in Section 5.2. Note thatbecause of the γ -trace conditions, in writing the involutive system one can replace thedeformed divergence d · ∆ by d ·∇ . Clearly, Eqs. (6.24)–(6.27) for k = 0 take the form ofEqs. (5.25)–(5.27). The gauge transformation in this case is given by: ˆ g − = u · ∆. k = 1: This corresponds to the simplest nontrivial PM gauge symmetry with depth 2. Inthis case, the PM gauge transformation can be implemented by an operator of the form:ˆ g − = ( u · ∆) − L δ u , δ = constant . (6.31)The computation of the gauge variations of the EoM’s requires the commutator of ˆ g − with { ˆ f (2)0 , ˆ g ′ , ˆ f } , which are given in Eqs. (C.20)–(C.25). These variations simplify toEqs. (C.26)–(C.27) when the involutive system for the gauge parameter ε n − is taken intoaccount. The following choice of constants is required by gauge invariance: α = 1 , β = − , α ′ = 1 , β ′ = 1 , δ = 1 . (6.32)These coincide with the values given for k = 1 by Eqs. (6.30) and the gauge transforma-tion (6.27), where δ is precisely the eigenvalue of ( N − n + 1) corresponding to ε n − . k = 2: Let us assume that the depth-3 gauge transformation is implemented by:ˆ g − = u · ∆ (cid:2) ( u · ∆) − L δ u (cid:3) , δ = constant . (6.33)It is easy to compute the variations of the EoM’s of the PM field Ψ (3) n given the com-mutation relations (C.20)–(C.25). On account of the involutive system for the gaugeparameter ε n − , these expressions simplify considerably. Their explicit forms are given inEqs. (C.28)–(C.29). The vanishing of the gauge variations then requires that α = 1 , β = − , α ′ = 1 , β ′ = 2 , δ = 4 , (6.34)36hich are precisely the values Eqs. (6.30) and the gauge transformation (6.27) give for k = 2. Note that δ is indeed the eigenvalue of ( N − n + 1) corresponding to ε n − . Therecursion formula (6.28) works too, as it reduces to Eq. (6.33) with δ = 4 for k = 2.Generic k : Suppose the involutive system (6.24)–(6.28) is consistent up to and including k = j −
2, for some j ≥
2. Then, the same system holds good also for k = j . This can beproven with the following ansatz for the depth-( j + 1) PM gauge transformation:ˆ g − j − = (cid:2) ( u · ∆) − L δ j u (cid:3) ˆ g − j +1 , (6.35)where δ j is some constant to be determined. Given the ans¨atze (6.29) for the deformedDirac operators, it is straightforward to compute the gauge variations of the left-handsides of the involutive equations for Ψ ( j +1) n . These variations can be written as: δ h ˆ f ( j +1)0 Ψ ( j +1) n i = (cid:2) / ∆ − L ( α j N + β j ) (cid:3) (cid:2) ( u · ∆) − L δ j u (cid:3) ˆ g − j +1 ε n − j − ,δ h ˆ g ′ Ψ ( j +1) n i = d · ∆ (cid:2) ( u · ∆) − L δ j u (cid:3) ˆ g − j +1 ε n − j − , (6.36) δ h ˆ f Ψ ( j +1) n i = /d (cid:2) ( u ·∇ ) − L δ j u (cid:3) ˆ g − j +1 ε n − j − , where the “unfree” gauge parameter ε n − j − will be governed by: (cid:2) / ∆ − L (cid:0) α ′ j N + β ′ j (cid:1)(cid:3) ε n − j − = 0 , d · ∆ ε n − j − = 0 , /d ε n − j − = 0 , (6.37)given that α j and β j and their primed counterparts are some numerical constants.The right-hand sides of Eqs. (6.36) can be computed by making repeated use of thecommutators (C.20)–(C.25), as well as the conditions (6.37) on the gauge parameter. Oneobtains the following results after a tedious but straightforward calculation: δ h ˆ f ( j +1)0 Ψ ( j +1) n i = n L ( u ·∇ ) j +1 P j + · · · o ε n − j − ,δ (cid:2) ˆ g ′ Ψ ( j +1) n (cid:3) = n L ( u ·∇ ) j P j + · · · o ε n − j − , (6.38) δ h ˆ f Ψ ( j +1) n i = n L ( u ·∇ ) j P j + · · · o ε n − j − , where the ellipses contain lower-derivative terms, and the P j ’s are given by: P j = ( α ′ j − α j ) N + [ β ′ j − β j − ( j + 1)( α j + 1) ] , P j = ( j + 1)( α ′ j − N + ( j + 1)[ α ′ j (2 β ′ j + D − − j − D + 1] N + [( j + 1)( β ′ j + j + D − β ′ j − j ) − δ j − j )] , P j = ( j + 1)[( α ′ j − N + ( β ′ j − j )] . (6.39)The derivation of the above expressions relies the assumption that the involutive sys-tem (6.24)–(6.27), and therefore the expressions (6.28)–(6.30) hold good up to and includ-ing k = j −
2. Now, vanishing of the gauge variations (6.38) necessarily requires that the37auge parameter ε n − j − belongs simultaneously to the kernels of P j , P j and P j . Fora non-trivial gauge parameter, however, such conditions can be satisfied iff the operatorsthemselves vanish. This leads to a unique set of solutions for δ j , α j , β j , α ′ j and β ′ j , whichcoincides with that spelled out by Eqs. (6.28) and (6.30) for k = j . That these valuesare also sufficient for the gauge variations (6.38) to vanish can be proved by explicitlyshowing that all the lower-derivative terms vanish. It is not difficult to convince oneselfof this fact for any given j , but we conclude without presenting this tedious exercise.We now summarize our results. In AdS space, the involutive system of a rank- n depth-( k + 1) PM fermion Ψ ( k +1) n and its rank-( n − k −
1) gauge parameter ε n − k − reads: (cid:0) / ∇ − m k (cid:1) Ψ ( k +1) n = 0 , d ·∇ Ψ ( k +1) n = 0 , /d Ψ ( k +1) n = 0 , (cid:0) / ∇ − m ′ k (cid:1) ε n − k − = 0 , d ·∇ ε n − k − = 0 , /d ε n − k − = 0 , (6.40)with the mass terms given for k ≥ /d -piece appearing in the deformed divergence d · ∆; this possiblebecause the γ -trace conditions themselves are included in the system (6.40). The depth-( k + 1) PM gauge transformations of this involutive system are of the form: δ Ψ ( k +1) n = ˆ g − k − ε n − k − , (6.41)where ˆ g − k − is spelled out in Eq. (6.27), and it contains up to ( k + 1) derivatives. This section studies the Lie superalgebra formed by the various operators acting on sym-metric tensor(-spinor)s in maximally symmetric spaces. Section 7.1 presents the flat-spacealgebra, while Section 7.1 shows how in AdS space the algebra closes only nonlinearly witha central extension. In this regard, let us note that nonlinear Lie algebras are general-izations of ordinary Lie algebras containing different order products of the generators onthe right-hand side of the defining brackets without violating Jacobi identities. In AdSspace, the nonlinear bosonic subalgebra of operators has been studied in [44–48], whilethe full supersymmetric algebra was considered in [44, 46, 49]. In flat space, the Lie superalgebra of all the operators on symmetric tensor(-spinor)s turnsout to be a subalgebra of osp (4 | sp (4) [46]. In order to present the Lie algebras, let us firstlist all the flat-space operators, along with their various properties (Table 3). They appear in Physics as Higgs algebra [41] and W algebra [42], in quantum optics [43], and so on. w ) Typed’Alembertian g ∂ g d · ∂ − g − u · ∂ +1 bosonicTrace g d − g − u +2Massless Dirac f /∂ f /d − f − /u +1Index Operator N u · d { g , g , g , f , f } .However, it also includes the hermitian conjugates (in the sense of footnote 5) of theseoperators as well: { g , g − , g − , f , f − } . The positive-weight operators appear not in theEoM’s, but in the hermitian conjugates thereof; their inclusion is tantamount to admittinga Lagrangian formulation, e.g., via BRST approach [45, 49]. Last but not the least, theindex operator N is added as it provides a grading to all the operators.The graded commutators of all these operators are given in Table 4. The computationis easy because ordinary derivatives commute: [ ∂ µ , ∂ ν ]Φ = 0 = [ ∂ µ , ∂ ν ]Ψ. In particular,[ ∂ µ , ∂ ν ] is blind to the statistical nature of the field. As we will see in the next section, thisseemingly naive observation provides valuable input when it comes to curved backgrounds. In a curved background, the deformed counterparts of the flat-space operators in Table 3do not form an algebra in general because of non-commutativity of covariant derivatives.It can be shown that the bosonic subalgebra may close, perhaps nonlinearly, only inconstant curvature manifolds [50], or in Freund-Rubin type backgrounds AdS p × S q withequal radii [48], in which case the algebra is simply a covariant uplift of the AdS p algebra.In the supersymmetric case, however, there is an immediate puzzle in deforming theflat-space generators: the commutator of covariant derivatives acts differently on bosonicand fermionic fields, as wee see from Eq. (C.3). Then, how can the same operator algebrabe realized on states with different statistics? The resolution of the puzzle lies in that acentral charge Z must be introduced in the following way. In AdS space, when Eq. (C.3)39able 4: Graded Commutators of Flat-Space Operators[ ↓ , →} N g g g − g g − f f f − N − − − g g g g − f g − − g − f g N +2 D f g − − f f g g g − f g N + D f − g − is compared with Eqs. (6.23), the following possibility immediately comes to one’s mind:∆ µ ≡ ∇ µ + Zγ µ , such that [∆ µ , ∆ ν ] = ( − L u [ µ d ν ] , for bosons; − L u [ µ d ν ] , for fermions, (7.1)where Z is a bosonic operator of mass dimension 1 that commutes with all the othergenerators. A bosonic state Φ and a fermionic state Ψ carry different charges under Z : Z Φ = 0 , Z
Ψ = − L Ψ . (7.2)In other words, deformed covariant derivative ∆ µ in the supersymmetric case reduces to ∇ µ and (cid:0) ∇ µ − L γ µ (cid:1) respectively for bosons and fermions. As a supersymmetric gener-alization of (C.19), one has the commutation relation: [ γ µ , ∆ ν ] = [∆ µ , γ ν ] = 2 Zγ µν .In what follows we will set the AdS radius to unity: L = 1. One can start by definingthe following deformed bosonic operators:Divergence : g ≡ d · ∆ , Symmetrized Gradient : g − ≡ u · ∆ , d’Alembertian : g ≡ [ g , g − ] . (7.3)40n view of Eqs. (7.1), the deformed d’Alembertian g can also be expressed as: g = ∆ − N ( N + D −
2) + u d . (7.4)Furthermore, the deformed Dirac operator can be chosen such that its anti-commutationrelations with the other fermionic operators mimic their flat-space counterparts. It is easyto check that the following choices achieve the desired feat:Dirac : f ≡ / ∆ − ( D − Z, Gamma Trace : f ≡ γ · d, Symmetrized Gamma : f − ≡ γ · u. (7.5)The remaining three bosonic operators include the index operator N ≡ u · d , andTrace : g ≡ d , Symmetrized Metric : g − ≡ u . (7.6)This exhausts the list of operators. It is straightforward to calculate all the graded commu-tators. While many of them close linearly like their flat-space counterparts, nonlinearityarises in some of the commutators. The results are summarized below in Table 5.In particular, the deformed d’Alembertian g has nonlinear commutation relationswith the divergence and gradient as well as with all the fermionic operators:[ g , g ] = 2(2 N + D − g − g − g ≡ c , [ g , g − ] = − g − (2 N + D −
1) + 4 g − g ≡ c , [ g , f ] = 2 ( f − g − g − f ) ≡ c , (7.7)[ g , f ] = (2 N + D − f − f − g + 4 Z ( g − f f ) ≡ c , [ g , f − ] = − f − (2 N + D −
1) + 2 g − f − Z ( g − − f f − ) ≡ c . The Dirac operator f also closes nonlinearly with the divergence, gradient, and itself:[ f , g ] = ( N − D + 1) f − D − Z f − f − g + 2 Z ( g − f f ) ≡ c , [ f , g − ] = − f − ( N − D + 1) + 2( D − Z f − + g − f − Z ( g − − f f − ) ≡ c , (7.8) { f , f } = 2 g + 2 N ( N + D − − g − g + f − f ) + 2( D − Z ≡ c . Last but not the least, we have nonlinear closure of the following commutators:[ g , f − ] = − [ g − , f ] = f + Z (2 N + D − − f − f ) ≡ c . (7.9)Some comments are in order at this point. First, the AdS nonlinear superalgebra(Table 5) contains a bosonic central charge Z , which does not show up in the flat-space41able 5: Graded Commutators of Operators in AdS[ ↓ , →} N g g g − g g − f f f − ZN − − − g c c c c c g g g − c c g − − g c − c g N +2 D f g − − f f c g g − f g N + D f − g − Z { g , g , g − , g , g − , N } . Second, one can perform a covariant upliftof the AdS D -superalgebra to render it consistent for any Freund-Rubin type backgroundAdS p × S q with equal radii, exactly the same way the bosonic algebra can be [48]. In thiscase, the AdS p × S q -superalgebra will be non-analytic in the neighborhood of flat space. In this article, we have studied the involutive systems of equations describing the freepropagation of massive, massless and partially massless symmetric tensors and tensor-spinors. For massive and massless fields, we have employed the involutive deformationmethod to find consistent dynamical equations and constraints/gauge-fixing conditions,42ompatible with gauge symmetries if present, in gravitational and electromagnetic back-grounds. For partially massless fields, we have given explicit expressions for the gaugetransformations and mass parameters at arbitrary depth. We have also shown that theLie superalgebra of operators acting on symmetric tensor(-spinor)s in AdS space closesnonlinearly as an extension of the flat-space algebra by a bosonic central charge.As pointed out in the Introduction, in the involutive approach, all the consistencyissues regarding the propagation of higher-spin fields are under proper control. The mu-tual compatibility and possible gauge invariance of the equations describing the systemare taken care of by the involutive structure itself, which thereby preserves the degreesof freedom count. On the other hand, higher-derivative terms may inflict Ostrograd-sky instability [57], while non-canonical kinetic terms may affect hyperbolicity or causalpropagation. The latter issues become manifest in the involutive approach, unlike inthe Lagrangian formulation, so much so that avoiding them simply becomes a matter ofchoice. More importantly, the involutive deformation method can also be employed toconstruct consistent interactions [22]. This goes beyond the scope of our present work.The various deformed involutive systems presented throughout this article could beviewed as the infrared limits of some effective-field-theory equations. Let us recall fromSections 2 and 3 that, for higher-curvature and higher-derivative terms in the equations,the suppression scales Λ and ¯Λ were introduced. For a given system, such a scale oughtto be parametrically larger than other mass scales in order for an effective field theory de-scription to be valid. Eventually, for the sake of simplicity, we considered only the infraredlimit by sending these scales to infinity. This also rids the systems of higher derivativesand/or kinetic deformations that might otherwise jeopardize causal propagation. Onecould however keep these scales finite, and move on to searching for the deformed in-volutive systems. Thus, one would find higher-curvature corrections to the equations ofmotion, e.g., those for massive higher-spin fields in string theory [13–15, 51].Throughout this article, we only considered the propagation of a single higher-spinfield in a pure gravitational or electromagnetic background. One could generalize theanalysis for interactions with more generic backgrounds [19], and thus find yes-go results.For example, as already mentioned in Section 5.3, Einstein-Maxwell backgrounds do admitthe propagation of a charged spin- gauge field. On the other hand, the assumption of afield in isolation is a strong one since, in a nontrivial background, various fluctuations ofdifferent spins may mix in the EoM’s even at the linear level. Relaxing this assumptionwould again lead to yes-go results by weakening the constraints on the backgrounds,otherwise required by consistency. One obvious example includes the graviton fluctuationin any geometry sourced by a nontrivial stress-energy tensor. Surely, its propagation willbe consistent, thanks to General Relativity, but the linearized equations will inevitably43ix the graviton with the fluctuations of the fields contributing to the stress-energy tensor.By construction, the involutive deformations we obtained have smooth flat limits.Accordingly, so do the deformed masses chosen in Sections 2 and 3; the deformationshowever are non-unique in that they could be arbitrary polynomials of the index opera-tor N . For gravitational backgrounds, these ambiguities could be removed by requiringsmooth massless limits. However, the non-uniqueness of mass deformations persists in thecase of electromagnetic backgrounds. In fact, it is even consistent to start with flat-spacemasses that are polynomials of N , generalizing the Regge law in string theory.What rˆole would mixed-symmetry fields play if included in the spectrum? Let us recallthat even a massive higher-spin fermion calls for an AdS background, whereas AdS isnot a solution of superstring theory. On the other hand, string theory admits an AdS × S background. As noted in Section 7.2, one can perform a covariant uplift of the higher-spin involutive systems to make them consistent even in such a background [48]. In thelatter case, however, the deformations will not be analytic in the neighborhood of flatspace [48]. This is in sharp contrast with string theory. While our analysis is restrictedto symmetric tensor(-spinors)s only, it is the mixed-symmetry fields in string theory thatensure analyticity in the background curvature. This point could be further justified byconsidering the theory of charged open bosonic strings in a background gauge field [13,15].The full Virasoro algebra ensures consistent propagation of the string fields. However, ifthe subleading Regge trajectories are excluded by switching off some of the oscillators,the remaining non-trivial generators no longer form an algebra [48].44 cknowledgments We would like to thank S. Biswas for initial collaboration, and I. Cortese, K. Mkrtchyan,M. Sivakumar, Z. Skvortsov, and M. Taronna for valuable comments. RR acknowledgesthe kind hospitality and support of the Erwin Schr¨odinger International Institute forMathematics and Physics and the organizers of the scientific activity “Higher Spins andHolography” (March 11–April 05, 2019), during which part of this work was presented.
A Involutive System of Equations
Involutive systems of partial differential equations (PDE) and how they control the num-ber of DoF’s of a dynamical system are well studied in the literature [21]. Related tothe count of Cauchy data [21], the DoF count can be made by relying on the notion of“strength” of an involutive system. This direction was first explored by Einstein [52], andfurther developed by subsequent authors [53–56]. In this appendix, we explain the basicsof involution and derive some necessary formulae for DoF count. For technical details,which we will skip, readers may resort to Ref. [22] and references therein.Let us work with the convention that repeated indices appearing all as either covariantor contravariant ones are symmetrized with minimum number of terms. This gives usthe rules: µ ( k ) µ = µµ ( k ) = ( k + 1) µ ( k + 1), µ ( k ) µ (2) = µ (2) µ ( k ) = (cid:0) k +22 (cid:1) µ ( k + 2), µ ( k ) µ ( k ′ ) = µ ( k ′ ) µ ( k ) = (cid:0) k + k ′ k (cid:1) µ ( k + k ′ ), and so on, where µ ( k ) has a unit weight byconvention, and so the proportionality coefficient gives the weight of the right hand side. A.1 Involution Basics
We consider a set of fields Φ A , with A = 1 , , ..., f , and denote their k -th space-timederivative by Φ Aµ ( k ) . Let their dynamics be described by the following system of PDE’s: T a [Φ A , Φ Aµ , . . . , Φ Aµ ( p ) ] = 0 , with a = 1 , , ..., t. (A.1)The maximal derivative order p is called the order of the system. Consider any order- p ′ subsystem: T b [Φ A , ∂ µ Φ A , . . . , Φ Aµ ( p ′ ) ] = 0 , b ⊂ a, p ′ ≤ p . The system (A.1) is involutive ifit contains all the differential consequences of order ≤ p ′ derivable from the subsystem.If the system (A.1) is involutive, it may possess nontrivial identities of the form: X a L ia T a = 0 , i = 1 , , . . . , l, (A.2)with L ia being local differential operators. These are called the gauge identities . The(total) order of a gauge identity is again the maximal derivative order appearing therein.45ote that gauge identities are more generic than Noether identities, and may exist evenwithout gauge symmetries. The two coincide only for a set of Lagrangian equations thatis involutive to begin with [22]. Gauge identities play an important rˆole in that theyreflect algebraic consistency of the involutive system, and control the DoF count.In general, the involutive system (A.1) may also enjoy local gauge symmetries : δ ε Φ A = X α R Aα ε α , δ ε T a | T =0 = 0 , α = 1 , , . . . , r, (A.3)where ε α are the gauge parameters, while R Aα are differential operators of finite order. Itmay happen that the gauge parameters are not arbitrary (as is often the case with partialgauge fixing), but they themselves are governed by an involutive system of equations. Inthe bulk of the article, we only have to deal with gauge symmetries of the latter kind. A.2 DoF Count
Let us assume that Φ A ( x ) are analytic functions of the space-time coordinates x µ . Onemay write down a Taylor series expansion of Φ A ( x ) around some point x µ :Φ A ( x ) = ¯Φ A + ∞ X k =1 k ! ¯Φ Aµ ··· µ k ( x − x ) µ · · · ( x − x ) µ k , (A.4)where a “bar” stands for the corresponding unbarred quantity evaluated at x = x . Here,the Taylor coefficients at O ( k ) are furnished by the quantities ¯Φ Aµ ( k ) , which constitute a setof monomials. Because of the EoM’s (A.1), however, not all of these monomials remainundetermined. Moreover, if the system enjoys gauge symmetries, some of the monomialswill be physically equivalent. Let us define the following quantities: n k = Total number of monomials at O ( k ) , ˆ n k = Number of undetermined gauge-inequivalent monomials at O ( k ) . Then, the number of physical DoF per point in D dimensions will be given by: D = f D −
1) lim k →∞ (cid:18) k ˆ n k n k (cid:19) . (A.5)This formula measures the number of physical DoF’s as the proliferation of the physicalmonomials relative to the unconstrained ones, `a la Einstein [52]. For large k , we will seebelow that ˆ n k ∼ n k /k , and so the above limit yields a finite number. The dimension-dependent proportionality factor can be obtained, for example, by matching with theDoF count for a scalar field. Note that the formula (A.5) gives the number of physicalpolarizations, i.e., the number of physical DoF’s in configuration space.46e will make use of Eq. (A.5) for a system of free-field equations. In other words, theEoM’s (A.1) are assumed to be linear in the fields. At x = x , they can be written as: J a, ν ( p ) A ¯Φ Aν ( p ) + b a = 0 , J a, ν ( p ) A ≡ δT a δ Φ Aν ( p ) , (A.6)where b a will be linear in ¯Φ Aν ( p ′ ) with p ′ < p . Note that the quantity J a, ν ( p ) A is called the zeroth-order symbol matrix . In general, one may have the m th -order symbol matrix : J a, ν ( k ) A, µ ( m ) ≡ δT aµ ( m ) δ Φ Aν ( k ) , m ≡ k − p ≥ , (A.7)where T aµ ( m ) denotes the m -th gradient of the EoM’s. Then, the m -th gradient of Eq. (A.1)evaluated at x = x gives a straightforward generalization of (A.6), which is J a, ν ( k ) A, µ ( m ) ¯Φ Aν ( k ) + · · · = 0 , (A.8)where the ellipses stand for linear terms in the monomials ¯Φ Aν ( k ′ ) at order k ′ < k = p + m .The above equation involves monomials at order k ≥ p ; their total number is given by: n k = f (cid:18) k + D − k (cid:19) . (A.9)The space of these monomials is determined by the finite system (A.8) of linear inhomo-geneous equations, whose total number amounts to n Tk = t (cid:18) m + D − m (cid:19) = t (cid:18) k − p + D − k − p (cid:19) . (A.10)Note that in order for the system (A.8) to be compatible, a left null vector of the symbolmatrix must annihilate the inhomogeneous term, and vice versa. This compatibilitycriterion is automatically satisfied by any involutive system (since otherwise the systemwould not be involutive in the first place). Existence of a left null vector of the m th -order symbol matrix then gives rise to an identity at O ( k ). Such an identity must be aconsequence of the gauge identities (A.2). If q is the total order of the gauge identities,then taking ( k − q )-th gradient of Eq. (A.2) leads us to an identity of the following form:Θ i, ν ( m ) a, µ ( k − q ) J a, ρ ( k ) A, ν ( m ) ¯Φ Aρ ( k ) + · · · = 0 , k ≥ q ≥ p, m = k − p ≥ , (A.11)where the ellipses contain terms linear in ¯Φ Aν ( k ′ ) with k ′ < k . Because k can be madearbitrarily large, in order for identity (A.11) to hold good, it is necessary thatΘ i, ν ( m ) a, µ ( k − q ) J a, ρ ( k ) A, ν ( m ) = 0 , for large k. (A.12)47herefore, Θ i, ν ( m ) a, µ ( k − q ) serves as a set of left null vectors of the symbol matrix J a, ρ ( k ) A, ν ( m ) forlarge k . The total number of these null vectors is equal to n Lk = l (cid:18) k − q + D − k − q (cid:19) . (A.13)They will be linearly independent if the original gauge identities (A.2) are irreducible.The number of O ( k )-monomials determined by the system is given by the rank ofthe symbol matrix of order m = k − p . The rank, in turn, is the difference betweenthe number (A.10) of O ( k )-equations and the number of independent left null vectorsof the symbol matrix. Once these quantities are known, one can count the number ofundetermined O ( k )-monomials. The DoF count further requires modding out gauge-equivalent monomials if gauge symmetries are present in the system.Let us Taylor expand the local gauge symmetry parameters appearing in Eq. (A.3): ε α ( x ) = ¯ ε α + ∞ X k =1 k ! ¯ ε αµ ··· µ k ( x − x ) µ · · · ( x − x ) µ k . (A.14)If s is the order of the gauge transformation (maximal order of R Aα ), then taking m -thgradient of the equation: δ ε T a | T =0 = 0, leads us to the following schematic form: J a, ν ( k ) A, µ ( m ) n Ξ A, ρ ( k + s ) α, ν ( k ) ¯ ε αρ ( k + s ) o + · · · = 0 , m = k − p ≥ , (A.15)where the ellipses contain terms linear in ¯ ε αν ( k ′ ) with k ′ < k + s . Again, since k can bearbitrarily large, Eq. (A.15) necessarily implies the following : J a, ν ( k ) A, µ ( m ) Ξ A, ρ ( k + s ) α, ν ( k ) = 0 , for large k = p + m. (A.16)Therefore, Ξ A, ρ ( k + s ) α, ν ( k ) furnishes a set of right null vectors of the m th -order symbol matrixfor large k . The total number of such right null vectors is given by: n Rk = r (cid:18) k + s + D − k + s (cid:19) . (A.17)These vectors will all be nontrivial and linearly independent for irreducible gauge sym-metries with unconstrained parameters. If it is otherwise, the DoF count becomes moreinvolved. This is also the case when the gauge identities are reducible. Taking such casesinto account, we will now derive some formulae for DoF count. If the gauge parameters are completely arbitrary, which is not the case we deal with in this article,the relation would be true for any k = p + m . .2.1 No Gauge Symmetries In general, the system (A.1) may contain equations of various orders. Suppose the numberof equations at order p is given by t p . The generalization of the count (A.10) would read: n Tk = X p t p (cid:18) k − p + D − k − p (cid:19) . (A.18)The gauge identities may come at different orders as well. Moreover, the gauge identitiesmay not be irreducible. Suppose there are l q,j number of gauge identities at total order q and reducibility order j . It is not difficult to convince oneself that the generalizationof (A.13) to the total count of independent gauge identities will be given by: n Lk = X q,j ( − ) j l q, j (cid:18) k − q + D − k − q (cid:19) . (A.19)In the absence of gauge symmetries, the number of undetermined physical monomials at O ( k ) will be given by: ˆ n k = n k − ( n Tk − n Lk ), which is equal toˆ n k = f (cid:18) k + D − k (cid:19) − X n t n − X j ( − ) j l n, j ! (cid:18) k − n + D − k − n (cid:19) . (A.20)We can make use of the following asymptotic expansion for binomial coefficients [53, 56]: (cid:18) k ± n + D − k ± n (cid:19) = (cid:18) k + D − k (cid:19) (cid:26) ± nk ( D −
1) + O (cid:18) k (cid:19)(cid:27) , k → ∞ . (A.21)Now, plugging the above expansion into Eq. (A.20) and dividing by Eq. (A.9), we obtain: f ˆ n k n k = c + (cid:18) D − k (cid:19) X n n t n − X j ( − ) j l n, j ! + O (cid:18) k (cid:19) , (A.22)where c is called the compatibility coefficient , given by: c ≡ f − X n t n − X j ( − ) j l n, j ! . (A.23)We will assume that the system (A.1) is absolutely compatible , i.e., c = 0. In this case,the DoF count (A.5) can be computed by taking a limit of Eq. (A.22), which gives: D = X n n t n − X j ( − ) j l n, j ! . (A.24)This is the formula for physical DoF count of an absolutely compatible involutive systemof with reducible gauge identities, but no gauge symmetries.49 .2.2 Irreducible Gauge Symmetries with Constrained Parameters Now we will take into account the presence of irreducible gauge symmetries of the system.Let us consider the case when the gauge symmetry parameters are not arbitrary, but obeysome differential constraints. In other words, we have a set of gauge parameters ε α , with α = 1 , , ..., r , governed by the following order- ˜ p system of PDE’s: T a [ ε α , ε αµ , . . . , ε αµ (˜ p ) ] = 0 , with a = 1 , , ..., ˜ t. (A.25)We further assume that the system (A.25) is involutive, and that the gauge symmetriesappear in a single finite order s . The k -th derivatives of the gauge parameters evaluated at x = x constitute a set of monomials ¯ ε αµ ( k ) . Because the gauge symmetries are irreducible,the number of undetermined monomials at O ( k + s ) follows directly from Eq. (A.20):ˆ n Rk = r (cid:18) k + s + D − k + s (cid:19) − X n ˜ t n − X j ( − ) j ˜ l n, j ! (cid:18) k + s − n + D − k + s − n (cid:19) , (A.26)for large k , where ˜ t n is the number of equations at order n , and ˜ l n, j number of gaugeidentities at total order n and reducibility order j . This count generalizes Eq. (A.17) tothe case when the gauge parameters are governed by an involutive system of equations.In order to find the number of O ( k ) monomials ¯Φ Aµ ( k ) that are undetermined as wellas gauge inequivalent, we must subtract the count (A.26) from the gauge-redundantcount (A.20). To simplify the exercise we first note that the expansion (A.21) gives:ˆ n Rk = (cid:18) k + D − k (cid:19) (cid:26) ˜∆ + 2 k ( D − (cid:16) ˜ D + s ˜ c (cid:17) + O (cid:18) k (cid:19)(cid:27) , k → ∞ , (A.27)where ˜ c and ˜ D are respectively the compatibility coefficient and the DoF count of theinvolutive system (A.25) of the gauge parameters; they are given by:˜ c = r − X n ˜ t n − X j ( − ) j ˜ l n, j ! , ˜ D = X n n ˜ t n − X j ( − ) j ˜ l n, j ! . (A.28)While ˜ c = 0 by the assumption of absolute compatibility, ˜ D counts the number ofpure gauge DoF of the original system (A.1) that enjoys the local gauge symmetry underconsideration. A straightforward calculation now leads to the physical DoF count: D = X n n t n − X j ( − ) j l n, j ! − ˜ D . (A.29)This is an intuitively-clear generalization of Eq. (A.24): the physical DoF count is obtainedsimply by subtracting the pure-gauge DoF count from the dynamical DoF count (includinggauge modes). When gauge symmetries are absent, ˜ D = 0, and we recover Eq. (A.24).50 Involutive Deformations
Given a set of free field equations in the involutive form − with all the gauge identitiesand symmetries identified − it is possible to systematically deform the theory and therebyintroduce consistent of interactions [22]. The algebraic consistency and the correct DoFcount are obtained, even for the deformed system, by strictly preserving the involutivestructure. The same approach can be taken also for the problem of writing down consistentEoM’s for fields propagating freely in nontrivial backgrounds [17–19]. To see how thisworks, let us first enumerate the consistency conditions to be taken into account:1. Algebraic Consistency:
The dynamical equations and constraints/gauge-fixingconditions ought to be mutually compatible. They should not give rise to any newconditions on the fields that cease to exist when the background is switched off [6].2.
Gauge Invariance:
When placed in a nontrivial background, the gauge symmetriesof a dynamical system should be preserved in order to eliminate unphysical modes.3.
No Higher Derivatives:
Constraint equations must not contain more than onetime-derivatives of the field, i.e., they cannot be promoted to dynamical ones. Onthe other hand, dynamical equations ought to include two time-derivatives at most.Otherwise, the system will generically be plagued with Ostrogradsky instability [57](see also [58] for a recent discussion).4.
Hyperbolicity:
Even when the dynamical equations contain only up to two time-derivatives, non-canonical kinetic terms may ruin the hyperbolicity of the system.In other words, such terms may render the Cauchy problem ill posed [7].5.
Causality:
A hyperbolic system of PDE’s describing the dynamics of some fieldshould also have a propagation speed not exceeding the speed of light. When non-canonical kinetic terms are present in the dynamical equations of a Lorentz-invarianttheory, this feature cannot be taken for granted (see [11] for a recent review).6.
DoF Count:
Last but not the least, the count of physical DoF’s of a dynamicalsystem should be correct. In other words, consistent free propagation in a nontrivialbackground implies that the DoF count does not alter by turning off the background.In the involutive deformation method conditions 1 , , • The flat-space free system of equations is written down in an involutive form. • All the gauge identities and gauge symmetries of the system are identified. • Zeroth-order deformation of the system, in the presence of a nontrivial background,is obtained by replacing ordinary derivatives by covariant ones (minimal coupling). • Because covariant derivatives do not commute, zeroth-order deformations will notbe self sufficient in general. Higher-order deformations of the equations, gaugeidentities/symmetries will cast Eqs. (A.1)–(A.3) into the following schematic form: T a = T a + gT a + g T a + · · · ,L ia = L ia, + gL ia, + g L ia, + · · · , (B.1) R Aα = R Aα, + gR Aα, + g R Aα, + · · · , where the numerical subscript denotes the deformation order in some dimensionlessparameter g . In fact, the deformation parameter g is just a book-keeping device totrack the power of background curvature. For example, linear terms in the curvaturewill be O ( g ), quadratic-curvature terms will be O ( g ), and so on. • The deformations (B.1) are chosen in such a way that the gauge identities and gaugesymmetries hold good order by order in g , and that the number equations and gaugeidentities/symmetries at a given derivative order do no change . • Because derivatives and curvatures are dimensionful quantities, their higher powersmust come with suppression by a relevant mass scale Λ. Accordingly, the respectivemass dimensions of the deformations (B.1) remain the same at any order. In orderfor an effective field theory description to make sense, Λ should be parametricallylarger than any other mass scale in the system.This method ensures that the system remains involutive and absolutely compatible,and contains the same number of physical DoF’s before and after the deformation. Whilealgebraic consistency of the system is guaranteed by the involutive structure, causal prop-agation is maintained by avoiding non-canonical kinetic terms in the dynamical equations. In principle, the derivative orders of the equations and gauge identities/symmetries may increase atany order in g . We, however, do not explore this possibility in order to make sure that the consistencyconditions involving higher derivatives, hyperbolicity and causality (3 , Technical Details
Here we provide some technical details omitted in the bulk of the article for the sake ofreadability. Appendix C.1 deals with gravitational backgrounds, whereas C.2 with EMbackgrounds. They present some useful formulae and elaborate on important technicalsteps leading to some of the derivations for both bosonic and fermionic fields.
C.1 Gravitational Background
The Riemann tensor can be decomposed into the following irreducible pieces: R µνρσ = W µνρσ + (cid:0) D − (cid:1) (cid:0) g µ [ ρ S σ ] ν − g ν [ ρ S σ ] µ (cid:1) + D ( D − Rg µ [ ρ g σ ] ν , (C.1)where D is the space-time dimensionality. Note that a conformally flat Einstein manifoldis a maximally symmetric space. For a maximally symmetric space, one can write: R µνρσ = − L ( g µρ g νσ − g µσ g νρ ) , R µν = − (cid:0) D − L (cid:1) g µν , R = − D ( D − L , (C.2)where L is the AdS radius (for dS space, we make the substitution: L → − L ). Then,the commutator of covariant derivatives (1.4)–(1.5) reduces to the following form:[ ∇ µ , ∇ ν ] = ( − L (cid:0) u [ µ d ν ] (cid:1) , for bosons; − L (cid:0) u [ µ d ν ] + γ µν (cid:1) , for fermions. (C.3)The commutator of divergence and symmetrized gradient in this case reads:[ d ·∇ , u ·∇ ] = ( ∇ − L N ( N + D −
2) + L u d , for bosons; ∇ − L N (cid:0) N + D − (cid:1) + L (cid:0) u d + /u /d (cid:1) , for fermions. (C.4) Computations with Bosonic Fields
The derivation of the explicit form of Eq. (2.12) relies on the following commutators:[ d ·∇ , ∇ ] = − R µνρσ ∇ µ u ρ d ν d σ + R µν ∇ µ d ν + ( ∇ µ R µν ) d ν − ∇ [ µ R ν ] ρ u µ d ν d ρ , [ d ·∇ , R µν u µ d ν ] = ( ∇ ρ R µν ) u µ d ν d ρ + ( ∇ µ R µν ) d ν + R µν ∇ µ d ν , (C.5)[ d ·∇ , R µνρσ u µ u ρ d ν d σ ] = ( ∇ α R µνρσ ) u µ u ρ d α d ν d σ + 4 ∇ [ µ R ν ] ρ u µ d ν d ρ + 2 R µνρσ ∇ µ u ρ d ν d σ . With the help of these commutators, it is easy to obtain the following:[ˆ g , ˆ g ] = 2( α − R µνρσ ∇ µ u ρ d ν d σ + ( α + 1) R µν ∇ µ d ν − R µνρσ u µ u ρ d ν d σ [ α , d ·∇ ] − R µν u µ [ α , d ·∇ ] − R [ α , d ·∇ ] + [ M , d ·∇ ] + α ( ∇ α R µνρσ ) u µ u ρ d α d ν d σ + 2(2 α − ∇ [ µ R ν ] ρ ) u µ d ν d ρ + α ( ∇ µ R νρ ) u ρ d µ d ν + α ( ∇ µ R ) d µ + O (cid:0) (cid:1) . (C.6)53q. (2.12) then follows from the decomposition (C.1). Terms containing gradients of thecurvature can be further massaged with the decomposition given in Eqs. (2.13)–(2.14).In order to prove Eq. (2.21), let us note the combination ( R µνρσ u µ u ρ d ν d σ − R µν u µ d ν )commutes with the trace operator, which is easy to show. Then, with the choices (2.15)the commutator [ˆ g , ˆ g ] reduces to the following:[ˆ g , ˆ g ] = − R [ α , d ] + [ M , d ] + O (cid:0) (cid:1) , (C.7)which gives rise the relation (2.21) for the choices and (2.17)–(2.18).In (2.25) we used the operators ˆ O i , i = 1 , ,
3, without spelling out their explicit forms;these operators are defined as follows: ˆ O = [ˆ g , ˆ g ] , ˆ O = [ˆ g , ˆ g ] + D − D +2) ˆ g R (2 N + D −
2) + ˆ g P ( N ) , (C.8)ˆ O = [ˆ g , ˆ g ] − D − D +2) [2ˆ g R (2 N + D − − ˆ g ( u d · U + u · U )] − ˆ g Q ( N ) . Next, we move on to the massless case and give the explicit expression of the weight-0operator X appearing in Eq. (4.22); it reads: X = − R µνρσ u µ u ρ d ν d σ − R µν u µ d ν ) − N − N + D − D − D +2) R + M ′ . (C.9)Then, we consider the details of partially-massless bosons in Section 6.1. To avoidclumsiness in the expressions, in what follows we will set the AdS radius to unity: L = 1.The following commutation relations involving the d’Alembertian operator are useful:[ (cid:3) − aN − b, u ·∇ ] = − u ·∇ { a + 2(2 N + D − } + 4 u d ·∇ , (C.10)[ (cid:3) − aN − b, ( u ·∇ ) − cu ] = − u ·∇ ) ( a +4 N +2 D )+8 u u ·∇ d ·∇ +4 u ( (cid:3) + ac ) , (C.11)where a, b and c are numerical constants. For the divergence operator, note from Eq. (6.4)that, by definition: [ d ·∇ , u ·∇ ] = (cid:3) . We also have the following important commutator: (cid:2) d ·∇ , ( u ·∇ ) − cu (cid:3) = 2 u ·∇ { (cid:3) − c − (2 N + D − } + 4 u d ·∇ . (C.12)Last but not the least, the trace operator has the commutation relations: (cid:2) d , u ·∇ (cid:3) = 2 d ·∇ , (cid:2) d , ( u ·∇ ) − cu (cid:3) = 4 u ·∇ d ·∇ + 2 { (cid:3) − c (2 N + D ) } . (C.13)The variations of the left-hand sides of EoM’s for the case k = 1 are given by: δ h ˆ g (2)0 Φ (2) s i = (cid:8) ( u ·∇ ) L + u Q (cid:9) λ s − ,δ (cid:2) ˆ g Φ (2) s (cid:3) = { u ·∇L } λ s − , δ (cid:2) ˆ g Φ (2) s (cid:3) = {L } λ s − , (C.14)54here we recall that L = 1, and the L ’s and Q ’s are the following linear functions of N : L = ( a ′ − a − N + [ b ′ − b − a + 2 D )] , Q = [4( a ′ − − c ( a ′ − a − N + [4 b ′ − c ( b ′ − a − b )] , L = 2( a ′ − N + 2( b ′ − c − D + 1) , L = 2( a ′ − c ) N + 2( b ′ − Dc ) . (C.15)Similarly, the variations for a depth-3 PM field, corresponding to k = 2, read: δ h ˆ g (3)0 Φ (3) s i = { ( u ·∇ ) L + u u ·∇Q } λ s − ,δ h ˆ g Φ (3) s i = { ( u ·∇ ) L + u Q } λ s − , (C.16) δ h ˆ g Φ (3) s i = { u ·∇L } λ s − , where again the L i ’s and Q i ’s are linear functions of N , given by: L = ( a ′ − a − N + [ b ′ − b − a + 2( D + 1))] , Q = [4(3 a ′ − c − − c ( a ′ − a − N + [ c (3 a + b + 2 D − − b ′ ( c − − D − , L = 3( a ′ − N + (3 b ′ − c − D + 2) , Q = − ( c − a ′ N + b ′ ) , L = 2(3 a ′ − c − N + [(3 b ′ − ( D + 2)( c + 2) + 6] . (C.17)Next, we elaborate on the computations with fermionic fields in gravitational backgrounds. Computations with Fermionic Fields
In deriving Eq. (3.14), one can first make use of the commutator (1.5) to write:[ / ∇ , d ·∇ ] = R µνρσ γ µ u ρ d ν d σ − R µν γ µ d ν − R µνρσ d µ ( γ ν γ ρσ ) . (C.18)Thanks to the γ -matrix identity: γ ν γ ρσ = γ νρσ + 2 η ν [ ρ γ σ ] , and the properties of theRiemann tensor, the last term in the above equation simplifies to R µν γ µ d ν . Then, onecan plug in the Riemann-tensor decomposition (C.1) to arrive at Eq. (3.14).Next, we give the technical details of PM fermions in Section 6.2. Here, the AdS radiusis set to unity: L = 1. It is important to note that, unlike the usual covariant derivative ∇ µ , the deformed one ∆ µ does not commute with γ -matrices. To be explicit:[ γ µ , ∆ ν ] = [∆ µ , γ ν ] = − γ µν . (C.19)55ome commutators involving the Dirac operator that will be useful for our purpose are: (cid:2) / ∆ − aN − b, u · ∆ (cid:3) = − ( a + 1) u · ∆ + /u (cid:0) / ∆ − N + /u /d (cid:1) , (C.20) (cid:2) / ∆ − aN − b, ( u · ∆) − cu (cid:3) = − a + 1)( u · ∆) + 2 u (cid:0) / ∆ − N + ac + /u /d (cid:1) +2 /u u · ∆ (cid:0) / ∆ − N − /u /d (cid:1) , (C.21)where a, b and c are numerical constants. Similarly, for the divergence operator:[ d · ∆ , u · ∆] = (cid:0) / ∆ + N + D − (cid:1) (cid:0) / ∆ − N (cid:1) + (cid:0) /u + u /d (cid:1) /d, (C.22) (cid:2) d · ∆ , ( u · ∆) − cu (cid:3) = 2 u · ∆ (cid:0) / ∆ + N + D (cid:1) (cid:0) / ∆ − N − (cid:1) − c − u · ∆+4 u d · ∆ + 2 u · ∆ (cid:0) /u + u /d (cid:1) /d. (C.23)The gamma-trace operator, on the other hand, has the commutation relations: (cid:2) /d, u · ∆ (cid:3) = (cid:0) / ∆ − N (cid:1) + /u /d, (C.24) (cid:2) /d, ( u · ∆) − cu (cid:3) = 2 ( u · ∆ + /u ) (cid:0) / ∆ − N − /u /d (cid:1) − c − /u. (C.25)First, we compute the variations of the left-hand sides of EoM’s for a depth-2 PMfermion, which corresponds to k = 1. Given the commutation relations (C.21), (C.23)and (C.25), and the involutive system of the gauge parameter, they reduce to: δ h ˆ f (2)0 Ψ (2) n i = { ( u · ∆) P + /u u · ∆ M + u N } ε n − ,δ h ˆ g ′ Ψ (2) n i = { u · ∆ P + /u M } ε n − , (C.26) δ h ˆ f Ψ (2) n i = { u · ∆ P + /u M } ε n − , where we set L = 1, and the P , M and N ’s are the following polynomial functions of N : P = ( α ′ − α ) N + [ β ′ − β − α + 1) ] , P = 2( α ′ − N + 2[ α ′ (2 β ′ + D − − D − N + [2( β ′ + D )( β ′ − − δ − , P = M = 2( α ′ − N + 2( β ′ − , (C.27) M = 0 , M = 2( α ′ − N + [2( β ′ − − δ − , N = [2( α ′ − − ( α ′ − α ) δ ] N + [2( α ′ + 1) − ( β ′ − β ) δ ] . Similarly, the variations for a depth-3 PM field, corresponding to k = 2, are given by: δ h ˆ f (2)0 Ψ (3) n i = { ( u · ∆) P + /u ( u · ∆) M + u u · ∆ N + u /u R } ε n − ,δ h ˆ g ′ Ψ (3) n i = { ( u · ∆) P + /u u · ∆ M + u N } ε n − , (C.28) δ h ˆ f Ψ (3) n i = { ( u · ∆) P + /u u · ∆ M + u N } ε n − , P , M , N and R ’s are polynomial functions of N , given by: P = ( α ′ − α ) N + [ β ′ − β − α + 1) ] , P = 3( α ′ − N + 3[ α ′ (2 β ′ + D − − D − N + [3( β ′ + D +1)( β ′ − − δ − , P = M = 3( α ′ − N + 3( β ′ − , M = 0 , M = 6( α ′ − N + [6( β ′ − − δ − , (C.29) N = [6( α ′ − − ( α ′ − α ) δ ] N + [6 β ′ − − ( β ′ − β − α − δ ] , N = (4 − δ )[( α ′ (2 β ′ + D − − D + 1) N + β ′ ( β ′ + D − , N = R = (4 − δ )[( α ′ − N + β ′ ] . This finishes our exposition of the computational details for gravitational backgrounds.
C.2 Electromagnetic Background
Let us emphasize that minimal coupling to the EM background has been assumed. Here,the commutator of covariant derivatives acts the same way on bosons and fermions:[ D µ , D ν ]Φ = iqF µν Φ , [ D µ , D ν ]Ψ = iqF µν Ψ . (C.30)Below we elaborate on some computations involving bosonic and fermionic fields. Computations with Bosonic Fields
The derivation Eq. (2.37) makes use of the following commutation relation:[ d ·D , D ] = − iqF µν D µ d ν − iqd µ V µ , (C.31)which simplifies the commutator [¯ g , ¯ g ] to the following form:[¯ g , ¯ g ] = iq ( α − F µν D µ d ν − iqα∂ ( µ F ν ) ρ u ρ d µ d ν + iq ( α − d · V − iq [ α, d ·D ] F µν u µ d ν + [ ¯ M , d ·D ] + O (cid:0) (cid:1) . (C.32)Then, one can easily arrive at Eq. (2.37) from the definition of A µνρ given in Eq. (2.36). Computations with Fermionic Fields
We will now provide justification for the ans¨atze (3.32)–(3.33). At first order in F µν , thenon-minimal deformation A of the Dirac operator may contain five independent terms: A = iq (cid:0) a + F + µν + a − F − µν + a F µρ γ ρ γ ν + a F νρ γ ρ γ µ (cid:1) u µ d ν + iqa F ρσ γ ρσ + · · · , (C.33)where the a ’s are weight-0 operators of mass dimension −
1, and the ellipses stand forterms containing derivatives or higher powers of the field strength. The third term on the57ight-hand side of Eq. (C.33) is however redundant since it is proportional to ¯ f , underthe assumption (3.31). Without any loss of generality therefore one can set: a = 0.Given this, if one further requires that the Dirac operator be hermitian in the sense offootnote 5, one must also set: a = 0. This justifies our ansatz (3.32). Similarly, thenon-minimal deformation B of the divergence operator takes the generic form: B = iq (cid:0) b F µν γ µ d ν + b F ρσ γ ρσ /d (cid:1) + · · · , (C.34)with b and b being weight-0 operators of dimension −
1, and the ellipses contain deriva-tives and higher powers of the field strength. Again, without any loss if generality, onecan set: b = 0. This leads us to the ansatz (3.33).Next, we compute the graded commutators of Section 3.3, which are eventually ex-pressed in Eq. (3.37). Starting from Eq. (3.28), a straightforward computation gives:[ ¯ f , ¯ g ] = iq [1 − m ( a + − a − + 2 b )] F µν γ µ d ν + 2 iq ( a − − b ) F µν d µ D ν + iq ( a + − a − ) (cid:2) F µν γ µ D ν ¯ f + F ρσ γ ρσ (cid:0) ¯ g − / D ¯ f (cid:1)(cid:3) − iq ( a + − a − + 2 b ) F µν γ µ d ν ¯ f + · · · , (C.35)where the ellipses stand for terms containing derivatives or higher powers of the fieldstrength, and commutators involving the weight-0 operators a ± , a and b . In derivingthe above result, we have used a number of γ -matrix identities, in particular: F + µν = ( γ µ γ ρσ γ ν − γ ν γ ρσ γ µ ) F ρσ , F − µν = − ( γ µν γ ρσ + γ ρσ γ µν ) F ρσ . (C.36)On the other hand, Eq. (3.29) leads rather easily to the following result:[¯ g , ¯ f ] = 2 b F µν γ µ d ν ¯ f + · · · . (C.37)Finally, in order to work out { ¯ f , ¯ f } from Eq. (3.30), we need to compute the anti-commutator { /d, A} with the help of the following γ -matrix identities: γ µνρσ γ λ + γ λ γ µνρσ = 2 γ µνρσλ , γ µ γ µνρσ = ( D − γ νρσ ,γ µν γ ρ + γ ρ γ µν = 2 γ µνρ , γ µνρ = γ µν γ ρ + 2 η ρ [ µ γ ν ] . (C.38)After a straightforward calculation, one arrives at the following expression: { ¯ f , ¯ f } = 2¯ g − m ¯ f − iq [( D − a + − ( D − a − + 4 a + 2 b ] F µν γ µ d ν + iqF µν (cid:2) a + + a − ) u ν d ν + { a + ( D − a + − a − ) } γ µν (cid:3) ¯ f + · · · . (C.39)Clearly, the F µν γ µ d ν - and F µν γ µ D ν -terms appearing in the first lines of Eqs. (C.35)and (C.39) obstruct the closure of these commutators, for spin s ≥ . Their coefficientsmust therefore be set to zero, which results in the choice (3.35). At O ( q ), other offending58erms may appear through derivatives of the field strength. Omitted in the ellipses ofEqs. (C.35), (C.37) and (C.39), such terms can be eliminated by the condition (3.36).We finish with the derivation of Eq. (3.40). Because the non-minimal corrections tothe gauge-identity operators (3.38) are proportional to the EoM’s, it is easy to see whythe schematic form (3.40) should appear. After a somewhat tedious computation, onefinds that the operators ¯ O , ¯ O ′ and ¯ O are given by:¯ O = ¯ h − ( iǫq/m ) F µν γ µ d ν ¯ f , ¯ O ′ = ¯ h ′ + (2 iq/m ) F µν (cid:0) ǫγ µ d ν + u µ d ν ¯ f (cid:1) , (C.40)¯ O = ¯ h +(2 iq/m ) F µν (cid:2) u µ d ν ¯ g + ǫγ µ D ν ¯ f + ǫγ µν ¯ f / D + (cid:0) − ǫ (cid:1) γ µ d ν ¯ f − (2 − ǫ ) / D γ µ d ν (cid:3) . This marks the end of the necessary technical details.
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