PoS(Modave VIII)004 http://pos.sissa.it/ a fact that we demonstrate. Given that any inter- − ∗ [email protected] Speaker. These notes comprise a part ofEighth the Modave introductory Summer lectures on School Higher inries Spin Mathematical and Theory Physics. turn presented on in interactions We the to constructfields find free in that higher-spin flat inconsistencies space theo- show are up in inWhile tension general. massive with fields Interacting gauge exhibit massless invariance superluminal and propagation, this appropriatesuch leads non-minimal pathologies to terms as various may no-go they cure theorems. do in String Theory acting massive higher-spin particle is described byindependent an upper effective bound field on theory, the we ultraviolet computespace cutoff a and in model discuss the its case implications. of Finally, wetheorems electromagnetic consider coupling for various in massless possibilities flat of fields, evading among theploy which no-go the Vasiliev’s higher-spin BRST-antifield method gauge for theory a isnon-abelian simple one. cubic but coupling We and non-trivial em- to gauge explore system its in higher-order flat consistency. space to find a ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Eighth Modave Summer School in Mathematical26th Physics, august - 1st september 2012 Modave, Belgium Rakibur Rahman Higher Spin Theory - Part I Physique Théorique et Mathématique & InternationalUniversité Solvay Libre Institues de Bruxelles, Campus PlaineE-mail: C.P. 231, B-1050 Bruxelles, Belgium PoS(Modave VIII)004 , ) 3 ]. 3 9 N 3.1 ex- = ( , s 8 O D , (1690), 7 3 , ρ 6 ]. This duality Rakibur Rahman 12 (1670), ] that forbid them 2 5 π , 4 , 3 , 2 , 1 ]. Appropriate non-minimal terms may 13 we show how String Theory provides rem- 4 ] for interacting HS massless fields in flat space, 2 5 ] is conjectured to be holographically dual to , 4 10 , , 3 8 [ , 4 2 , 1 , we consider the fact that interacting massive HS particles are 5 ], and the first non-trivial checks of the duality appear in [ shows that massive HS fields exhibit superluminal propagation in external 11 we study interactions of HS gauge fields by considering the different possi- 3.2 6 the notorious Velo-Zwanziger problem [ − non-abelain cubic coupling in flat space by using the BRST-antifield formalism and 3 2 (2040) etc. These particles are composite, and their interactions are described by com- − 4 3 2 The underlying HS theory may provide a better understanding of what String Theory is. may be regarded as the simplesthelp us non-trivial understand example some of aspects AdS/CFT of correspondence, gauge/gravity and dualities may in general. that some of the spectacular features of Stringand Theory, open-closed e.g., duality, planar rest duality, modular heavily invariance on their presence. from interacting in flat spaceceeds with certain electromagnetism value. (EM) One would or liketheories gravity to for when evade massless their these HS theorems spin fields. in order write down interacting a plicated form factors. However, whenmasses, the one exchanged should momenta be are able small to describe compared their to dynamics their by consistent local actions. Vector Models [ For massless particles, there exist powerful no-go theorems [ , we start with construction of free HS theories for massive and massless fields. In Section In what follows we will present an introduction to HS fields and their interactions. In Sec- In Quantum Field Theory fundamental particles carry irreducible unitary representations of 2 − 5. Vasiliev’s HS gauge theory in AdS 4. It is conjectured that String Theory describes a broken phase of a HS gauge theory [ 2. Massive HS particles do exist in the form of hadronic resonances, e.g., 3. Massive HS excitations show up in (open) string spectra. In fact, they play a crucial role in 1. While free HS fields are fine, severe problems show up as soon as interactions are turned on. explore its higher-order consistency. tion all of which essentiallyance, derive while Section from the fact that interactions arecure in the tension Velo-Zwanziger acausality, with and gauge in invari- Section described by effective field theories, computecutoff a for model their independent electromagnetic upper coupling boundFinally, in on in the flat Section ultraviolet space, and discussbilities of the evading implications the of aforementioned no-go thisor theorems, result. e.g., in adding AdS a space, mass or term,a considering working 1 in higher-derivative interactions. We work out a simple example of presents the various no-go theorems [ backgrounds edy to this pathology. In Section we discuss the difficulties that one faces when interactions are turned on. In particular, Section the Poincaré group, and thereforeleast can in have principle. arbitrary (integer The or motivations for half-integer) studying values higher-spin of (HS) the particles spin, are at manifold. Higher Spin Theory - Part I 1. Motivations & Outline PoS(Modave VIII)004 ) of 1 ) − ) (2.4) (2.6) (2.5) (2.1) (2.2) (2.3) 2 3 s , − − s s s , ]. + , s s 2 15 + + − 3 Rakibur Rahman 4 D − ( − . The field is then
D C D s ( 2 ( , C . In this case, we have s 2 C s space-time dimensions? D ) + D 1 tensor is given by s − s and spin , s . m , ! + µ ! ) 2 ) s W ) should eliminate 1 µ . − + . 0 − 1 2.4 0 D W tensor that removes . ( D = − 0 ) ( = = C s ! 2 . s µ D 2 s = ) µ − ( ... − C s 1 ) 1 ... µ s requires that all lower spin values be eliminated; 2 µ 2 ( 3 − µ ... 2 φ 1 1 ) = . Then the total number of propagating DoFs are s ) − µ µ C s ) 2 , s φ , φ , 3 s a field may possess. Let us consider the first Wigner , 2 m 1 1 µ s µ µ − − + P 1 − − ∂ s µ µ ] and of Bargmann and Wigner on relativistic wave equa- + 4 , which is traceless: s , s P  η 3 ]. A systematic study of HS particles, though, was initiated s µ − 16 ( + ... − = + 14 D 1 D ( 1 µ 4 D ( ( φ C C C − 2 C ]. Their approach was field theoretic that focused on the physi- . The transformation properties under the Lorentz group is uniquely D is the Pauli-Lubanski pseudovector. They define respectively mass − ) ( 15 ) ) + 1 is however reducible under the rotation subgroup, because it contains C σ s s P tensor, equation [ , ,
+ s s s 2 s s 2 1 νρ ( , + + J s s 2 2 down to 0. The Casimir 1 4 m s − − D µνρσ = D D ε 2 ( ( then demands that the Klein-Gordon equation be satisfied: 2 1 C C C 1 C = = ≡ − and the two basic quantum numbers 0 µ 2 ], however, made it clear that one could replace positivity of energy by the requirement 6= − m m W , 17 b − How many degrees of freedom (DoF) does this field propagate in The two Casimir invariants of the Poincaré group are The task of constructing a theory of HS fields dates back to 1936, when Dirac tried to gener- D = 1 these components. Similarly, the divergence constraint ( this is done by imposing the divergence/transversality condition: The latter is a necessary condition in order for the total energy to be positive definite [ many, but its trace part hasthe already actual been number incorporated is in less the by tracelessness of the field itself, so that all spin values from and spin determined for a bosonic field bya the totally usual symmetric rank- choice of the representation Note that the number of independent components of symmetric rank- The tracelessness condition is a symmetric rank- by Fierz and Paulical in requirements 1939 of [ Lorentz invariancerepresentations and of positivity the of Poincaré energy. group [ Thetions works [ of Wigner onthat unitary a one-particle state carries an irreducible unitary representation of2.1 the Poincaré Massive group. Fields class, which corresponds to a physical massive particle of mass alize his celebrated spin- Higher Spin Theory - Part I 2. Construction of Free Higher Spin Theories where The Casimir The representation C PoS(Modave VIII)004 . - n  0. × γ 1 ) 2 (2.7) (2.8) (2.9) − ,..., (2.10) (2.12) (2.11) − D 3 ) 0, into a n 2 1 − , / ] = n 3 − D , ( [  2 − 2 µν 2 1 n φ − Rakibur Rahman × µ n + + )] ∂ D 2 D ) should be counted. ( ≡ − C , ] n 2.9 0 , D [ . Let us try to incorporate 3 = . To avoid such difficulties tensor-spinors, each elimi- , 1 µν  − 2 ) φ ) n 1 ρσ φ µν + ) by directly replacing ordinary − dimensions, a symmetric rank- is therefore σ φ D n µ ∂ − ( ( D 2.9 ρ ∂ C ( ∂ )-( α , . µν 2 1 ) + 1, and doublets of rank 2.7 0 0 η 1 . . − + 3 , the Lagrangian incorporates symmetric 2 D 0 − = = 1 as expected. n 2 1 2 n µν ] achieved the feat in 1974 by writing down n n − = + , + φ µ µ ρ n s ) and ( 2 2 20 ) n ...... µ 1 1 ν m 4 − ... µ µ φ = 1 2 2.4 1 n µ ψ ψ µ s ( 1 ) − )-( representation of the Lorentz group , and satisfies 0, and the transversality condition, ψ ∂ µ + 2 1 m 1 as expected. ρ .
∂ ) µ -traceless: D = 2 2.2 ∂ 1 γ − γ ( + / independent components, where  ] 2 νρ + , on the other hand, is represented by a totally symmetric s ) are satisfied, all the lower spin fields are forced to vanish. ∂ C 6 n 2 D φ µν 1 2 [ 2 α ( / 2 , ] µ φ , another of rank 2 1 2 n 2 ) 2.9 D ∂ n + = [ − 2 ( × − 2 n ) ] spelled out a procedure for introducing these auxiliary fields 2 2 1 ] that attempts to introduce electromagnetic interaction at the of them. But there is an over-counting of )-( m ) fields and conjugate momenta D n is written in terms of a set of symmetric traceless tensor fields of n × 2 = − µν , + 19 − 15 / − ) , s , which is ] 2.7 1 φ L s n n n n ) D = [ 
µ , 2 − ( n 2 2 + 1 ... m n , 1 L 3 µ × 1 − − + ) and ( 2 ) + n ψ − 4, this number reduces to 2 n 1 , D  is to be determined. While varying the above action one must keep in mind + D ( 2.4 n ( = 0. For a half-integer spin ( − D C D α n C )-( ( D , ] and Chang [ C ,..., 2 = = [ 3 2.2 18 0 − − 6= n s m , , + 2 f is symmetric traceless. One therefore obtains the following EoMs: D − D 4 in particular, this is 2 ( s constraints, since only the traceless part of the divergence condition ( , -trace and divergence constraints, being symmetric rank- µν C = s Fierz and Pauli noted in [ We will work out an example of this inconsistency in Section Fronsdal [ To understand the salient features of the Singh-Hagen construction, we consider the massive A fermionic field of spin The counting of DoFs is analogous to that for bosons. In γ 2 φ 1 / ] D D [ In derivatives with covariant ones, they are noit longer mutually was compatible suggested thatis all equations possible involving only derivatives at beinteractions the obtainable are cost absent. from of a introducing Lagrangian. lower-spin This auxiliary fields, which must vanish when traceless tensor-spinors: one of rank the Klein-Gordon equation, spin-2 field, which is described by a symmetric traceless rank-2 tensor to construct HS Lagrangians. Singh and Hagen [ It transforms as the level of the equations ofgreater motion than (EoM) 1. and constraints If lead one to modifies algebraic the inconsistencies Eqs. for ( spin an explicit form ofLagrangian the for Lagrangian an for integer spin a free massive field of arbitrary spin. TheWhen Singh-Hagen the Eqs. ( Lagrangian as where the constant that rank Higher Spin Theory - Part I In particular when 2 The number of dynamical DoFs tensor-spinor of rank tensor-spinor has The nates PoS(Modave VIII)004 0. = . At . The s (2.15) (2.16) (2.19) (2.13) (2.14) (2.18) (2.17) . Then µν φ n . φ ]. 0, can be i ,..., ν 2 3 20 ∂ ,..., and tensor field. , φ = µ 3 2 ) ) 2 n , , provided that 1 ∂ ) . k µ µν 1 2 2 Rakibur Rahman − 2 = − φ ... − n D − β 1 ν = ( µ k D 2 µ s 0 follows from the ( D ∂ k ... ( ψ 1 µ 1 = µ − Dm µ ∂ . 2 ψ ∂ 0 µν 2 µν . 0, for − 0, for 1, 0. These can be achieved φ . φ 2 = ν 0 h = 0, so that the Klein-Gordon ) gives −  = = ∂ 2 ] > s n and hence non-zero if ) tensor-spinors. The explicit = 2 ρσ 1 1 µ , = µ m k − − φ ): β D 2 n ∂ 2 1 n φ ... ) 2.12 φ σ µ µν 1 µ m − 2 µ φ ∂ ... − ... . n 1 φ 1 ν ρ 2.11 −  0 k µ µ ) ∂ ∂ 2 as well as requiring µν µ ψ D ψ 1 ν µ = φ 0 and ∂ k ( EoMs now gives 1 ∂ = ν , µ µ − φ ) ...
− = ∂ ∂ ) ∂ 1 φ ∂ 2 1 α µν 2 ) D D 2 µ 1 µ − 2 φ ... β β + µν ∂ D 1 − 2 − ( 2 µ φ + ( φ ∂ D − D 1 ( ν φ ∂ 2 2 0 and 2 ∂  m µν Dm + [ β 1 µ α φ , proportional to 5 2 = 2 β -traceless rank-( ν ) ∂ = + (  n + γ ∂ φ + 2 µ 2 2 0 and ) µ µ β , so that the condition ... µν , the transversality condition, 1 ∂ − φ ∂ = φ φ ] 2 1 ( µ µ  n 2 ) µ ) ∂ ψ µ µν 2 )] 1 + 2 ( ∂ φ − ... ] 1 µ − 1 1 n -traceless tensor-spinor of rank ν 2 Dm D D β µ ( ∂ γ ( − ∂ 2 1 m , − = ψ µ µ ) , on the other hand, reduces to 2 = ) k D 2 s 1 ∂ ∂ µ ( φ +  − + ) − ∂ 2 add D − D ( ) D  ( ... 2 1 ) L 1 β µν − 2 µ ) = φ 2 ∂ 2 − µ 1 [( β ∂ − α 0 and the vanishing of the auxiliary field must follow if the associated deter- [( , one must successively obtain D s ( + ( = 2 2 ) ) comprise a linear homogeneous system in the variables µν one needs to introduce an auxiliary symmetric traceless rank- = [ φ νρ ν k ∆ φ ∂ 2.16 µ becomes algebraic, not containing µ ∂ are constants. The double divergence of the one must introduce two symmetric ( )-( ∂ ∆ 2 2 1 is the space-time dimensionality. The divergence of Eq. ( , k 1 − β D 2.15 One can carry out this procedure for higher spins to find that the following pattern emerges: = L one also satisfies the conditions: form of the Lagrangian for an arbitrary spin is rather complicated and is given in Ref. [ obtained by introducing a symmetric by successively obtaining for each The EoM for the auxiliary scalar Eqs. ( One can now check that this Lagrangian yields The transversality condition can be recoveredThe by latter setting condition, however, doesof not by introducing follow an from auxiliary the spin-0 EoMs. field This problem can be taken care where Thus one constructs the following Lagrangian for a massive spin-2 field For an integer spin Higher Spin Theory - Part I where Lagrangian. Let us add the following terms to the Lagrangian ( condition equation and the transversality condition follow from it. minant does not vanish. The latter is given by Note that each value of Similarly for a half-integer spin PoS(Modave VIII)004 , s ≡ µ s ... µ 1 µ (2.21) (2.20) (2.22) (2.24) (2.25) (2.27) (2.23) (2.26) ... 2 Φ µ Φ · s µ ∂ ... 3 Rakibur Rahman µ ] 0 Φ 21 ) is nothing but the .
s φ . µ · h ... 2.25 ∂ · 3 · , µ ∂ 2 . ) · Φ s 0 · φ ∂ µ ∂ 0 ∂ ... = h 4 · + 1 µ 0 2 ∂ − + s ) , Φ 2. Furthermore, the two symmetric denotes divergence: µ · 0 2 φ ) ) ... · 1 − µ 0 ∂ 3 ) reduces to . = ∂ 0 ∂ s h µ − ( )( φ ) µ λ s ) 2 s ) . ∂ µ 2 ( 1 µν 2.19 . ( ] by considering again the spin-2 case. In φ s − ... ) − − 2 1 η 3 1 2 2 ν D s 0 D µ 21 ( , and 2 ( − λ D s 2 + )( 1 − + µ D µ F ( 1 D 2 2 2 ... + , ∂ ) ) µ 1 ) = s 1 − 6 s µ µ 2 + 0. The Lagrangian reduces to [ µ µ µ s ) = µ h ( µ Φ ( ... · µ ... : = 2 2 µν η s h 2 φ µ µ s µν 2 1 1 8 ∂ φ µ 1 · identified as the metric perturbation around Minkowski µν µ h ≡ µ λ Φ h ... ∂ − + 1 ≡ 0 δ · 1 η + ( µ s µν h 2 µ ( ∂ µ ) h 2 µν ( ∂ s + ( ≡ ) Φ ... s µ h 1 4 s 2 1 2 µ = ] considered the massless limit of the Singh-Hagen Lagrangian µ ... νρ ) µ ) has acquired a gauge invariance with a symmetric traceless 3 3 s ) to h µ µ 0 ... + F µ 21 tensors can be combined into a single symmetric tensor, νρ 3 µ 2 η ... Φ µ 0 φ ) ∂ ) 2 1 2.20 ρ ( s 2 µ µ µ Φ 2.22 µ ∂ 1 2 1 ∂ µ Φ ( − ... )( − 1 2 1 s δ η 1 µ ( : then gives − = 1 is symmetric but not traceless. The Lagrangian ( Φ − into a single field − s ρ s = µν µ φ ( ∂ L µν φ ( s ... h 1 1 2 4 1 L µ and − + ): λ and rank- s µν = φ 2.20 L . The Lagrangian ( s µ ... 1 µ Let us illustrate the Fronsdal-Fang formulation [ In 1978, Fronsdal and Fang [ Gauge invariance and the trace conditions on the field and the gauge parameter play a crucial Φ 1 µ ∂ which is double traceless: linearized Einstein-Hilbert action with background. This describes a masslessjust the spin-2 infinitesimal particle, version and of the coordinate corresponding transformations: gauge symmetry is where prime denotes trace: the massless limit, the Singh-Hagen spin-2 Lagrangian ( We combine This reduces the Lagrangian ( Here the new field gauge parameter role in the Fronsdal-Fang constructionnumber in of that propagating they DoFs. ensure To absence doLagrangian the of ( DoF ghosts, count, and we give write the down correct the EoMs that follow from the Higher Spin Theory - Part I 2.2 Massless Fields to find considerable simplification ofcouple the in theory. this limit, For except the fortraceless the bosonic one rank- case, with all the highest the rank auxiliary fields de- The tracelessness of PoS(Modave VIII)004 D (2.32) (2.33) (2.30) (2.35) (2.28) (2.31) (2.34) (2.29) 2. -traceless: = tensor with γ n . s components µ s  ) ... 3 3 Rakibur Rahman µ h.c. 0 − s Ψ − , n s ∂ 6 µ . n ... + ) µ s 3 4 µ µ ... . , 3 ) ... − µ 0 0 Ψ 3 -traceless tensor-spinors of 1 , which is triple µ 0 · 6 ¯ γ n ) to D Ψ , = µ − ( ∂ ) Φ . 0 n ) 2 independent components in s 1 ... C ), the symmetric traceless rank- n  µ µ 1 , ) µ ]: ) 2.29 µ = s ∂ ... n − − 4 3 1 ... µ ) ) n 3 Ψ µ s 0 µ 21 4. + 0 2.21 ( ... ( µ components. Thus the total number 1 µ − ¯ 2 5 Ψ ∂ n 1 and the de Donder gauge for ) ... µ s =  S − 4 1 3 2 3 ) however does not completely fix the , 2 − ) + = s µ 0 6Ψ s µ D ) 1 1 s , − 1 s − D Φ µ s µ n µ 2 ( + ( s . in ( − µ 2.30 µ . ∂ ... , 0 are still allowed. Therefore, we can gauge 0 + C 5 ( η s n 2 0 ... s is vanishing. A symmetric rank- ∂ ( 1 2 2 2 2 µ − s = = − µ 1 2 n + = n µ Φ s − 1 1 2 − µ s · 6Ψ 4 µ D ... − ) + − ) tensor-spinor, s µ 5 7 ( ... D n s s ... 00 + µ ∂ 1 µ ∂ ( 6 µ − 1 ... n µ , 1 C µ  1 ... µ s n µ C ... 1 ... F µ is given by ( D 2 µ 0. This reduces Eq. ( 2 Ψ − µ Φ ( ∂ µ n + µ ... ) F λ h.c. ∂ µ 6 2 C  s = 5 Φ −  µ 6S ...  , s − · i s 1 − s ¯ − ]: µ 1 µ 6Ψ n µ ) ∂ µ µ ... n ≡ ( ... D 1 + 5 22 S 1 because γ ... ( n µ 00 ≡ µ 2 1 + 1 2 s µ − C s µ n µ Φ Φ ... s µ µ − 1 − ... Ψ , = ...  1 µ ) enjoys the gauge invariance ( ... n · s 2 D 1 0 µ µ µ ( µ ∂ S = ≡ + ... Φ n G C 1 m s Ψ µ 2 2.28 , µ µ b ... ∂ 6 ... 2 − S 1 D µ n µ µ D ¯ 6Ψ 2 respectively; the other auxiliary fields decouple completely. These three ( ... F  1 C 0. This leads to the following Lagrangian [ µ n − , one is left in this case only with three symmetric ¯ n − Ψ = 1 2 n µ = is the Fronsdal tensor defined as + is traceless because indeed describes a massless field. Eq. ( ... s n 1 s µ s µ µ 1 and L µ ... i = ... 1 Ψ ... 2 − s µ 1 3 µ µ µ n gauge parameter enables us to remove G F , γ Φ ) 3 n This is the arbitrary-spin generalization of the Lorenz gauge for For the fermionic case too we can take the massless limit of the Singh-Hagen Lagrangian. µ 1 2 γ 1 − µ s by imposing an appropriate covariant gauge condition, e.g., γ In particular, this leaves us exactly with 2 DoF for all The resulting EoMs read For spin can be combined into a single symmetric rank- where gauge since gauge parameters satisfying away another set of of propagating DoFs turns out to be [ rank Another equivalent form of the EoMs is the so-called Fronsdal form: ( Thus where the fermionic Fronsdal tensor Higher Spin Theory - Part I where dimensions. Now that Eq. ( This contains precisely the correct numbermetric of double-traceless conditions field to determine thevanishing components double of the trace sym- has PoS(Modave VIII)004 ) 1 × the + )] − ]. 2.38 2 D (2.40) (2.37) (2.38) (2.39) (2.36) ]. This 24 is triple − n 24 n -traceless µ fields and , γ ... 3 − 1 tensor-spinor µ − n Rakibur Rahman S n + D ( Lagrangian in is. The gauge condi- C s ) enjoy a gauge symme- n − µ . ) 0 0. Then the field equations dimensions. However, the ... 1 , 3 2 2.36 0 µ = = − D ) shows. Therefore, we have ε )-( . 1 n -traceless rank- = n a hierarchy of generalized Christoffel justifies the choice of auxiliary − µ 0 . γ , ) n : − 2 n ... µ − 1 ] 2 / 2.36 3 µ = ] − 2.34 ... µ 0 n 22 ) D ... 1 − [ µ n 3 µ , massless HS theories are interesting Ψ n 2 µ ) renders the non-dynamical components µ 0 ... ε 4 1 1 ... Ψ + µ × 3 µ 2 ε µ ) 2.30 γ µ D ε ( n ( · . γ , 0 C n n ∂ [ components in ), to set constraints. Now, the symmetric 2 2 = µ + + 2 2 ( 1 4 / ). / n ] 4 γ ] -traceless because 8 µ − 0. This allows further gauge transformations for , 2.39 D n D [ ) γ [ D − ... 2 n ) and the EoMs ( 2 1 2.31 2 = µ + µ D 2 − 4 n ... × ( × n 2 µ − S ) is µ µ C 2.33 D )] ... )] ε 4 ... 3 2 1 2 = µ 0 − µ µ 2.39 ( − 0 ] for geometric formulations of HS gauge fields. n − 6Ψ ∂ µ Ψ = n n 23 m , ... , , = 2 ≡ 6 4 f 3 µ n n ) to the Dirac equation. However, one can use part of the residual µ µ − D − dimensions to a theory containing symmetric tensor fields of spin ... ∂ ε ... n 6 n 1 2 µ is actually non-dynamical as Eq. ( µ D + 2.36 + ]: G n Ψ enables us to choose, for example, the gauge D µ D δ ( 22 1 ( 1)-dimensional gauge invariance gives rise to the Stückelberg symmetry ... − 2 C n C -traceless tensor-spinor parameter µ dynamical as seen from Eq. ( + µ is [ is. The Lagrangian ( γ 0 0, which indeed describe a massless fermion. − − − ... n s − ) D 1 Ψ ) µ µ -traceless because = µ n 1 ...... γ ε , n 2 1 µ µ 1 − µ 0 is ... much simpler than the original Singh-Hagen one n 1 Φ − Ψ n , µ − µ n . The ( 2 s Ψ ... ) and the residual gauge fixing eliminate + 2 − ∂ µ 6 n 4 in particular, this is 4 as expected. We parenthetically remark that the gauge fixing ( D G ( components each. In this way, one finds that the number of dynamical DoFs dimensions. The tracelessness of the higher dimensional gauge parameter has non-trivial 2.38 + = C To get the DoF count let us note that a symmetric triple Apart from their gauge theoretic and geometric aspects Here the gauge condition does not involve derivatives, so that it cannot convert constraints into evolution equations. One can construct HS analogs of the spin-2 Christoffel connection [ 2 [ 3 4 / D ] D D ( D [ -traceless part of -traceless if C In γ 2 conjugate momenta has in consequences; one can gaugegauge-fixed away Lagrangian some, is but then equivalent not to all, the Singh-Hagen of one the up auxiliary to field (Stückelberg) redefinitions fields. [ The dimensions KK reduces in 0 through This is in contrast with the bosonic case where the gauge condition ( where which the gauge parameter satisfies tion ( does not reduce the EoMs ( become since they give rise to the massive theoriesconstruction through proper Kaluza-Klein (KK) reductions [ fields in the latter. For the bosonic case, for example, a massless spin- traceless part of symbols linear in derivativesvery of simple the in field terms with ofunderstood these simple in objects. gauge this approach. The transformation trace See properties. also conditions Ref. on The [ the wave HS equations field and are the also gauge parameter can also be easily gauge invariance, which satisfies the condition ( [ gauge parameter Note that the left-hand side of Eq. ( Higher Spin Theory - Part I One can also rewrite the EoMs in the Fronsdal form: try with a symmetric This equation contains the appropriate number ofγ independent components because PoS(Modave VIII)004 ) 3.1 (3.5) (3.4) (3.1) (3.3) (3.2) (3.6) -matrix S particle of s Rakibur Rahman , ) N p , 0 ,..., 1 p = ] to give an overview of some 0 as: . ( ]. Let us consider an 0 5 0 S 1 φ → and a massless spin-  2 [ ) ) for massive spin 2 coupled to EM q ) = , q N q ) ( > ( s . 2.4 , q s s µ ( 0 0 µ .. ) ,..., i )-( .. 1 s . 3 1 p µ µ = = 0 0 µ .. ε · particle; it is transverse and traceless: 2 ε s 2.2 µ = q = µ µ i µν φ s i 2 p φ · α , 1  ρν µ ... D µ 2 i φ ( 1 , p 9 µ µ m q i 0 p D − ≡  2 ) , µρ i ) = q 0 D g q ( , ( 1 s ) , one obtains s µ = ieF µ N = µ ∑ i , the Lagrangian approach is not free of difficulties either; D ... µν ... 1 spur  µν 2 F ( µ 2 has been observed in Nature. Neither is there any known → µ φ 3.2 ε 1 )
µ > ], by taking recourse to the Lagrangian formulation. However, 2 ε ε , s 1 -matrix-theoretic approach, that there are obstructions to consistent m and 15 q µ S , q − ]. Below we will closely follow Ref. [ N 2 5 3.1 p D , 4 , ,..., . Now the Klein-Gordon equation and the transversality condition give 3 1 µ , p 2 ( , S ieA 1 external particles of momenta is the polarization tensor of the spin- + . The matrix element factorizes in the soft limit ) µ N q µ ∂ ( q s µ = .. 1 µ µ ε D No massless particle with One can show, in a purely As we already mentioned, turning on interactions for HS fields at the level of EoMs and con- The polarization tensor is redundant inof that it the has massless more components particle. thanvanish the for physical This spurious polarizations polarizations, redundancy i.e., is for eliminated by demanding that the matrix element momentum string compactification that gives atwo. Minkowski space Indeed, with interactions massless ofno-go particles theorems HS [ of gauge spin fields larger in than flat space are severely constrained by powerful element with does not describe the sameas number Fierz of and DoFs Pauli asas suggested the we [ free will theory. see Such ininconsistencies difficulties show Sections can up be for both avoided, massless and massive HS fields3.1 when interactions No-Go are Theorems present. for Massless Fields of the most important no-go theorems. 3.1.1 Weinberg (1964) long-range interactions mediated by massless bosons with where straints, by replacing ordinary derivativestake, for with example, covariant the ones, naïve covariant results version in of Eqs. inconsistencies. ( Let us Higher Spin Theory - Part I 3. Turning on Interactions where This constraint does not exist when the interaction is turned off, so that the system of equations ( The non-commutativity of covariant derivatives then resultsple, in for unwarranted a constraints. constant For EM exam- field strength PoS(Modave VIII)004 2, = (3.8) (3.9) (3.7) ). All s (3.10) . 0. 3.9 0, and (ii) i minimally → = − q h.c. µ i µν p Rakibur Rahman − ψ ] for half-integer ∑ µ 2 ψ · ]. 3 ∇ gauge field to find that µ 5 2 ) that the spurious polar- ¯ 6ψ 2 , 3.4  − ψ . · 6 h.c. 0 ∇ + 0 = ¯ ψ µ  ε . ) transforms as µανβ 0 µ + γ R 0 . These theorems leave open the possibility = 3.8 2 5 ψ 1 αβ − ) has no solution for generic momenta. ∇ 6 s ψ ≥ 0 µ i ν described by the tensor-spinor ¯ s p ψ γ 3.7 with the massless spin 2 (graviton). The latter is − 2 1 µ 10 , ... ¯ , only if ε κ ) 1 i µ i ν − µ i  ε p µ p field µ g i ( 6ψ 5 2 g 3, Eq. ( − ∇ 1 ∇ 6 N ≥ = √ ∑ i 0. This is nothing but the conservation of charge. For µ = -matrix language by considering, for example, scattering of x s ¯ S 6ψ D = 2 µν d i g + should be eliminated, as in the free case, by the gauge invariance Z δψ ∑ µν ∼ µν 0, which can be satisfied for generic momenta if (i) ψ S ψ = ] studied the gravitational coupling of a spin- δ ∇ 6 3 µ i p µν i ¯ ψ g h ∑ g − is also transverse and traceless. One finds from Eq. ( √ 1 x − s D µ d ... 1 Z µ 1, this is possible when ) shows that the graviton interacts universally with all matter in the soft limit ) reduces to i α . The first condition imposes energy-momentum conservation, while the second one requires − = 3.7 3.7 κ The Aragone-Deser no-go theorem is based on Lagrangian formalism, and therefore has a In order to completely rule out massless HS particles, one should consider a truly universal This argument does not rule out massless bosons with spin larger than 2, but say that they Aragone and Deser [ s = = S i The redundancy in the field Under this gauge transformation, however, the action ( these issues can be addressedmassless in HS the particles off soft gravitons. very important implicit assumption: locality.example, in Some the non-locality guise in of the aalso Lagrangian form possible appearing, factor to for for have gravitational a couplings non-minimal may description involving remove a the larger difficulties. gauge It invariance is than ( izations decouple, for any generic momenta g all particles to interact withsimply the the same equivalence strength principle. For such a theory isin fraught the with free grave theory. inconsistencies:coupled The the to system gravity, unphysical is of gauge governed a by modes spin- the decouple action: only interaction which no particleEq. can ( avoid. Gravitational coupling is one such interaction. In fact, that massless HS particles may very well interact but can mediate3.1.2 only short-range Aragone-Deser interactions. (1979) cannot give rise to long-range interactions. Similar arguments were used in Ref. [ Higher Spin Theory - Part I where For Eq. ( spins to forbid interacting massless fermions with i.e., the action is invariantthe only gauge in modes flat do space not decouple whereterms, for the regular an in Riemann interacting the tensor theory. neighborhood vanishes. It of is flat In easy space, other to do see words, not that come local to non-minimal the rescue [ PoS(Modave VIII)004 q ]. ]: θ 4 + − 25 p (3.12) (3.13) (3.11) (3.14) ]. Un- 5 and p -direction are q ~ Rakibur Rahman implies, through 0, one can choose µν 0. This is in direct . 6= i Θ 2 s particle are 6= i q s s + 2 , + q , let us consider the effect p , ) | p q m . ~ | , amounts to a rotation of i l m − s , ]. l , ) Θ , | 5 + , 0 θ s Θ ν 2. Now rotational invariance and , i | ( s p q s + ± R µ , = ( , + + + q p , 1 , µ p p q 1 is that the matrix element must van- q ± | | + = . The off-shell graviton has a space-like , ) ) p s + s s ) gives a non-vanishing matrix element. i 0 > h s has been used. For p θ θ s + s i i and h = ) θ + i 0 , ) m ± ∓ 2 3.11 p ( ( m θ p ~ 2 decoupling of the unphysical states is incom- q i ± ~ | decomposes into two real scalars, a vector and a 1 2 e e − , > µν | 11 exp exp p ~ s µν = = q ~ ( with Θ direction. The one-particle states transform as | | 3 Θ i = = 2 1 s q s m ~ δ ]. Therefore, no massless HS particle with spin larger , i i 0 2 + 5 s s + , p , Θ = ( q + + 2 p , , 3 | µ ) + p q p R | π p ) m -matrix approach to prove the following important theorem [ + , h 2 l S θ 1 and 0 p ( | Θ around the ) give us → ± ) † R lim = ( q , θ θ R i 0 | ( 0 s 3.13 p R = | + p , )-( ], in which m is the energy-momentum four-vector cannot contain massless particles of h q 4 x + 3 3.12 d p can have minimal EM coupling in flat space [ with ν h . Under spatial rotations, 0 by an angle 3 2 p ~ m , Θ ) 0. The matrix element is completely determined by the equivalence principle: 1 − ≥ θ R Θ ( s → = , 2, the only solution to this equation for R 0 q q , ~ 0 ), that the matrix element vanishes in all frames as long as + | ≤ matrix elements are Lorentz covariant. The spurious states however must decouple from Θ 1.” They considered in 4D a particular matrix element: the elastic scattering of a massless p ~ m particle off a soft graviton. The initial and final momenta of the spin- | > µν 3.14 s for which s Note that the theorem does not apply to theories which do not have a Lorentz covariant energy- Weinberg and Witten took the Θ µν patible with the principle of equivalence [ The same holds true for HS gaugeto fields. such At theories any rate, by the introducing Weinberg-Witten theorem unphysical cander states be that Lorentz extended correspond transformations, to these spurious polarizations unphysicalthe [ polarizations mix with theall physical physical ones matrix such elements. that Bygravitationally considering interacting one-graviton massless fields matrix with elements one can show that for than 2 can be consistently coupled toparticles gravity with in flat space. Similarly, one can show that no massless momentum tensor, e.g., supersymmetric theories and generalvariant relativity. stress-energy In tensor order in to these have theories a gauge one in- has to sacrifice manifest Lorentz covariance [ respectively, while the polarizations aremomentum identical, say the “brick wall” frame [ The difference of thearound signs in the exponents arises because the commuting variables. Combining the two realtensors: scalars into a complex one, we have the spherical contradiction with the equivalence principle as Eq. ( spin Higher Spin Theory - Part I 3.1.3 Weinberg-Witten (1980) & Its Generalization “A theory that allows the constructionΘ of a conserved Lorentz covariant energy-momentum tensor spin- where the normalization of a rotation symmetric traceless rank-2 tensor. These fieldsin will the be basis represented by in spherical which tensors the if total we angular work momentum and projection thereof along the the transformations ( Since ish. Because the helicitiesEq. are Lorentz ( invariant, the Lorentz covariance of PoS(Modave VIII)004 (3.23) (3.21) (3.22) (3.15) (3.17) (3.18) (3.20) (3.24) (3.16) (3.19) ) 0 ϕ 0∗ , or from a . ϕ 0 µν . − ] = ϕ 2 Rakibur Rahman 0 µν ϕ αβ ϕ ϕ − · ∗ µν ) by the field redefini- 2 ∂ µν ϕ . · ( ϕ 0 2 [ ∂ 2 m 2.19 = m µν 0 − 2 1 η ϕ ) , ) in order to minimally couple ] 0 + − ]: 2 , 0 ) c.c. 0 m ν 15 ϕ ) 3.15 ϕ ) = ν + 1 0 · 0 ∂ ) = ) are the simplest instance of a Fierz– ϕ 0 − ∂ ϕ µν µ ν  ϕ . D ( , ) in arbitrary space-time dimensions. ϕ ν D 0 ∂ into a single traceful field − 0 3.23 µ ∂ µ ∂ − φ ϕ = = )-( − , D 0 · 3.15 ]): 0 . ) + [( − 0 ν 0 ϕ ∗ ϕ ∂ µν µν 0 2 ] ϕ 26 ν ϕ · = and ) ϕ 2 · 3.21 ϕ ∂ 0  ∂ ) µ ) with a single dimension compactified on a circle µ 12 m ∂ = ϕ ( 2 ) µν ( φ ∂ + ( 2 µ − 2 · 1 m φ ) leads to 2 ∂ ϕ m | + ϕ ( m ∂ 0 2.25 0 − − · 1 2 ϕ ϕ D ∂  3.16 µ ) ≡ − · ( + 2 (see, e.g., [ [( ≡ − D 2 ∂ | m g ) µν µν )( + − µν 2 R 2 R | ν ϕ  µ ) one arrives at an interesting consequence, − µ ( ∂ µ . ∂ µ ∂ ϕ D ) µν · ( ∗ ∂ , that combines 3.19 η ϕ φ D + ( ≡ | · 1 one is led to the dynamical trace constraint: 2 − 2 µν µ ) µ F > η · ) reduces to the Klein-Gordon equation + µν νρ R 2 ) and ( 2 D ϕ ϕ ϕ | 1 − ) ( ) one gets the transversality condition: µ 2 D 3.16 νρ ∂ Tr m ( 3.18 ϕ + 1 2 3.17 µ 0 and − ieg D µν − 2 . The corresponding EoMs read  6= φ ( m 2 −| + = / = m ≡ FP = µν L µν ϕ L Now let us complexify the spin-2 field in the Lagrangian ( For massive HS particles, gauge transformation is no longer a symmetry of the action. So they For simplicity, let us consider the massive spin-2 case. We begin by reviewing the key proper- R which is manifestly hyperbolic and causal. Eqs. ( it to a constant EM background.do The not minimal commute, coupling is so ambiguous that becausewhich we covariant we are derivatives call actually the led gyromagnetic to ratio a family of Lagrangians containing one parameter, tion: Then from Eq. ( so that finally Eq. ( Pauli system, which originates from the Lagrangian ( are not subject to suchpresence algebraic of interactions. inconsistencies However, as they are their plagued massless with counterparts other experience pathologies as in weties the will of see the below. free theory, described by the Fierz-Pauli Lagrangian [ Higher Spin Theory - Part I 3.2 Pathologies of Interacting Massive Higher Spins Kaluza-Klein reduction of the Lagrangian ( of radius 1 Note that this Lagrangian can be obtained either from the Lagrangian ( Thus, for Taking divergences and the trace of Eq. ( Combining Eqs. ( PoS(Modave VIII)004 . ) 1 2 . . ) ρ ∞ = ν 3.27 (3.29) (3.30) (3.31) (3.32) (3.26) (3.27) (3.28) (3.25) ϕ g → µ t ( . . On the , ρ | 0 has real ) ) 0 x n ~ | · ϕ / . 2 0 ) = iegF 0 itn n D ( n ϕ Rakibur Rahman  ( + ) 0 , − ∆ µν exp ϕ  ϕ F 0 · ϕ ρ . µν · ϕ ˆ D µν ∂ ϕ i , ρ 0 · D F · =  − D . ϕ 2 D 0  ν e ρ D 0 νρ µν ϕ ∂ ( 3 4 of the resulting coefficient µν ϕ 2 F ϕ µ ϕ · η ) 2 F ∂ µν − + ], would appear. The very ex- n ∂ 2 F η + . ( ( Tr ρ F ) ) ) µν 29 ∆ 2 1 Tr + ν ν ϕ ϕ 2 F 2 1 · Tr ∂ ϕ · ν ϕ − 1 2 F µ · ) ρ ( D ) − ϕ ν F ρ D Tr − ) · F ) gives νρ F D µ 2 · F µρ ( F F ) µ ), while the constraint equations ( ) that contain second-order derivatives: · ρλ ( ( + 2 ϕ F D ]. With this choice, one obtains − ϕ · 2 D 3.25 σ ϕ m Tr ν ) e 1 2 · − / F ν 2 ) 27 ∂ 3.28 0 3.25 σν e ( ) 2 F µ ϕ ( − ϕ 2 ϕ ( ∂ , the normal to the characteristic hypersurfaces, ) µ Tr
 ρ ( − m ν µ λ ) 2 1  Tr ∂ / D σ g n ) 13 − − ν D  ρ e F 2 D D µ ( 2 ∂ ρσ . The determinant ( ) m + (
5 ↔ ρσ 2 F D / 2 1 4 3 ρσ ν with  F 1 2 µ 1 m − − ϕ F ) h ( in those units, a number of new phenomena, including / µ − D D ·  2 2 3 2 e + ∂ ) 1 ) ) ( i m D
2 2 1 3 2 0 ) + (
/ = µ ( m 2 1 m 0 ϕ ϕ ie − D / / O − − ν ( ϕ e µν D D 3 2 2 ie µν D ( = ( η 0 D
3 2 F = 2 3 − ] and Nielsen-Olesen instabilities [ 2 1 ) ϕ − 0 − ν 1 − − − − 28 ϕ ϕ D D ϕ µν − ν · · = µν , the system is hyperbolic, with maximum wave speed ϕ g D 0 n 2 3 ϕ D ~ D 2 ( ϕ ) · ( − +  µ 2 ie ν D m D = ϕ  = · − ) 0 2 − 2 for any D ( µν ϕ D 0 4 µν ] one can employ the method of characteristic determinant to investigate the causal R ( n m ϕ 2 13
≡ 1 2 D ) can be substituted in the second and third terms to obtain − − . If some invariant were µν D D = e R ) / , and is precisely the Federbush Lagrangian of [ 2 3.29 2 This procedure is akin to solving the EoMs in the eikonal approximation letting One still needs to see whether the dynamical DoFs propagate in the correct number and Unlike in the free theory, the trace does not vanish in the presence of the EM background. 2 1 ( µν = 5 m R 0 = matrix determines the causal propertiessolutions of the for system: if the algebraic equation properties of the system. Onein replaces the highest-derivative terms of the EoMs causally. To this end, let us isolate the terms in Eqs. ( istence of these instabilities implies thatwith any EM effective Lagrangian can for be a reliable, charged even particle well interacting below its own cutoff scale, only for small EM field invariants. and ( Schwinger pair production [ This expression, however, makes perfect senseof for small values of the EM field invariants in units The trace constraint can also be rewritten as The last term can be actually dropped in view of ( Following [ Higher Spin Theory - Part I The resulting EoMs are The first term ondivergence the constraint right-hand of side the signals freeThe a theory unique potential is minimally breakdown turned coupled of into modelg the a that propagating does DoF not field count, give equation since rise unless the to a wrong DoF count has therefore Combining the trace and the double divergence of Eq. ( PoS(Modave VIII)004 , ]. B ~ ar- ] for 31 ε 30 (3.35) (3.34) (3.33) i 2 (bro- αβ = η , with ρ ε N n ) ρ Rakibur Rahman − ν 0, which entails F 2
µ so that one expects > ( e n , perpendicular to 2 / − , ] 2 2 n ~ E ~ i rows and columns are 2 m + 29 ) ] incorporates appropri- −
2 3 , ) 4 its determinant reads 1 n 2 β · 36 B ~ n 28 = + ˜ = F α . . ) ( D 0, D 2 2 ν ( ) 2 i . F E ~
= 6 B ~ D µ 2 · ( 2 1 αβ e m E 3 ~ n n ~ 2 η · ( 2 − , B ~ 0, the system admits faster-than-light F − ) ]. Thus, field theoretically it is quite ) 2 + β 2 ν 2 Tr B ~ ) 35 δ n 2 1 2 ) = E ~ µ ,
( n ) takes the form 2 − ih ( n × e 34 2 m 3 α ∆ β n 2
~ ( 1 , ρ . In particular, in 2 since it originates from the very existence of n ρ 3.32 ) F ]. More importantly, it persists for a wide class + · / 33 F for 14 − 3 ˜ B ~ ρ F αρ 32 0 αβ 1 µ n ( F n ≥ h n 2 ( h q )
s ). Therefore, superluminal propagation can take place ν 2 2 . Moreover, the propagation itself ceases to occur when- ≡ e n and 2 m = m 2 µ far away from the instabilities [ B ~ ) | / ) 3.34 0 n n − n ~ n ie | 2 − · ( ) µν 2 ˜ 2 3 4 ( F n ( m h + / 8 ) e ) β ( 2 ν n δ
2 1 α ( µ − − , so that δ D D 2 ) = ( n ρσ n 2 3 1 2 ( ]: it contains a massive gravitino that propagates consistently, even when the F . One can always find a Lorentz frame where the pathology shows up: the ∆ 2 − − can reach the critical value in a frame where 38
e , 2 / µνρσ B ~ 2 37 ε ) = 2 1 m n ( . From a canonical point of view, in an EM background, the equal time commutation relations become 2 3 ], which as we see arises already at the classical level ) 3 2 ≡ ) 13 αβ ( ≥ µν µν is non-vanishing in all Lorentz frames. One can always find a vector ˜ field in a constant external EM background. A more well-known example is ( F 2 2 2 3 B M ~ Note that the pathology is not a special property of the spin-2 case; it generically shows up The pathology in the corresponding quantum theory was found much earlier by Johnson and Sudarshan [ Let us now consider 4D EM field invariants such that The superluminal propagation of the physical modes of a massive HS field in an external field B ~ 6 thanks to the last factor appearing in ( ken) SUGRA [ massive spin for all charged massive HS particles with of non-minimal generalizations of the theory [ ill-defined. That the Johnson-Sudarshan and Velo-Zwanziger problems have a common origin was later shown in [ magnetic field ever longitudinal modes for massive HS particles [ is a serious problem.come However, addition to of the non-minimal rescue. terms and/or For new dynamical example, DoFs the may Lagrangian proposed in Ref. [ labeled by pairs of Lorentz indices even for infinitesimally small values of bitrarily small. This is thesmall values most of serious the EM aspect field of invariants to the have problem: well-behaved it and persists long-livedproblem even propagating [ for particles. infinitesimally This is the so-called Velo-Zwanziger challenging to construct consistent interactions of massive HS particles. 4. Remedy for the Velo-Zwanziger Problem ate non-minimal terms that propagatespin- only within the light cone the physical modes of a massive Higher Spin Theory - Part I other hand, if there are time-like solutions where propagation. The coefficient matrix appearing in ( Note that this expression is to be regarded as a matrix whose that for which the characteristic determinant vanishes provided that PoS(Modave VIII)004 ]. ]. ]. 39 43 43 .A (4.3) (4.2) (4.4) (4.1) (4.5) , 7 at the , -model 41 π 41 e σ , the EoMs -brane and µν D F 0 Rakibur Rahman e , n . − , , under the graviphoton [  . ) m ) P δ µν F ) M π ]: m / iG e 40 m π , − − [ ( ( 3 1 2 0 ν cov 1 − m , √ L p ] = ( 2 ν -brane couples to charges ν D G = µ D tanh D 1 , e 12 Tr  µ 4 1 + ) + eF D -model for bosonic open string in a constant EM + [ n / F 15 0 σ + 0 EM and gravity, along with non-minimal terms. G e L m ]. Like for free strings, there is an infinite set of cre- is the covariant momentum only up to a rotation: L π p ) ( → 43 µ  0 1 n both , ], and perform a careful analysis of the mode expansion 0 − α , = L − 41 40 µ 0 µ 0 m α tanh α i  ≡ 1 . π ] = ( 1 2 n µ L = ]. With a careful definition of the mode expansion, it is possible to = D , 0 m G 42 α L , [ , it is convenient to define a different covariant derivative: 41 µν , 40 ieF is the total charge of the string, and living in the world-volume of the = π µ  e A ), however, has an important effect since it deforms the masses of the open-string + ν cov 0 p e 4.4 , = µ cov p e  We set the string tension to The commutation relations obeyed by the Virasoro generators can be worked out as usual. The We consider an open bosonic string whose endpoints lie on a space-filling In this Section we will see how String Theory bypasses the Velo-Zwanziger problem. To An important point is that the mode 7 end result is the emergence of an additive contribution to excitations. With Up to this shift, the Virasoro algebra remains precisely the same as in the free theory: Causality is preserved by the presence of background, originally considered in [ 4.1 Open Strings in Constant EM Background string endpoints. For anare electromagnetic those background of with the usual constant free fieldis strings while strength exactly the solvable boundary [ conditions arehave deformed, commuting so center-of-mass that coordinates, the which obey canonical(non-commuting) commutation covariant relations momenta with the [ where The shift ( this end, one starts with the (exactly solvable) and the Virasoro generators. Thethe language physical of state string conditions fields for toVelo-Zwanziger string problem obtain is states a absent Fierz-Pauli can for system, be the which translated symmetric explicitly tensors into shows of that the indeed first the Regge trajectory [ Higher Spin Theory - Part I cosmological constant vanishes, if it has a charge, Maxwell field ation and annihilation operators that are well-defined in the physically interesting regimes [ PoS(Modave VIII)004 (4.6) (4.7) (4.8) (4.9) (4.12) (4.14) (4.11) (4.13) (4.10) (4.15) (4.16) is called i , Φ | , ν m a ν m Rakibur Rahman 2 a − µ 1 † m − µ a † m a µν i µν ) i ) iG , iG + i will be defined in such a way 0 2 + | s 1 ν 1 , − s µ N a 0 m † − m µ 1 a = , a m )( , ν ν m ··· 0 )( µν . iG A 1 a 2 i i 1 . µ , , , ν µ = iG 2 µ 0 n m † + † m 0 0 0 G | a a µ G a + µ · m iG )
Tr † 1 ( = = = † n 2 x + m 4 1 A µν a ( ( i i i · 1 ) ) p − s G na + x ]: h Φ Φ Φ 1 m s 1 p µ ( | | | p iG 3 16 ν m µ ) = h 1 2 ∞ ∞ = ... µ h A 42 ∑ ∑ = ∞ 1 a n 2 1 m ... L L ∑ +  , 1 2 A m 1 i m µ = 2 ∞ ) translate into the string field theory language. ( ) give µ 1 † m ∑ − ≡ m 41 = a G + 0 ψ + + √ , i 4.9
1 4.8 + L Tr 2 ν 2 µν ( N = Φ ν 1 ∞ 1 2 40 µ i ) a )-( | ∑ G a m µ D µ 1 − iG Tr 4.7 ∞ D 2 = ∑ D 1 4 ) and ( s + D µν  = + m µν 4.7 i ( i i 1 iG = ∞ N Ψ iG ). Notice that the number operator ∑ | m + + + + 2 4.3 1 2 2 √ √ are integers, just as for free strings: h D D h i 2 1 2 1 i N − − − coefficient tensor is interpreted as a string field, and as such is a function of the 1 s = ≡ − = = ) is empty, Eqs. ( = 0 1 N L L 2 4.9 L defined in Eq. ( µ D Because the Virasoro algebra is the same as in the free theory, a string state The generic state at this level is given by A generic string state in the presence of a constant EM background is constructed the same way “physical” if it satisfies the conditions [ string center-of-mass coordinates. where the rank- with While Eq. ( that its eigenvalues We would like to see how the Eqs. ( 4.2.1 Level where now as in the free theory, sinceIt the is two given cases by: have identical sets of creation and annihilation operators. Higher Spin Theory - Part I 4.2 Physical State Conditions PoS(Modave VIII)004 ) ) is 4.19 (4.20) (4.21) (4.25) (4.22) (4.24) (4.26) (4.27) (4.17) (4.18) (4.23) (4.19) (4.28) (4.29) 4.29 . ) )-( 2 − to obtain 4.27 D µ Rakibur Rahman )( B 1 + D , ( 0 , 2 1 0 , that possesses a suitable =
. = 2 i µν 0 , G | ) µν 0 , ) µ Tr , G † 2 4 1 · 0 = G µν a , ν )  + 0 = x H H 1 ( B ν , = iG ν µ − − . µ µ A ) ρ 0 0 , − ν  iB µ iG µ α 2 1 B A H H = 2 , / . · ρ ·  iG ν √ √ G 0 1 0 µ . ) 2 − ( · B G i 0 iG · + ( = h µ B = 2 i − iG i · = ( µ G + 2 = µ µ µ 0 B − i 2 | 2 0 17 + 1 D
iG − A A H ν 1 2 2 † 1 √ · µν + µ (  H + a G 2 µν  1 + D 1 − µ H D G µ † 1 Tr µ H
≡ q √ a 2 1 2 26, one can gauge away the vector field µν Tr
µ )   2 ) enjoys an on-shell gauge invariance with a vector gauge 1 2 H G ) leave us with x − = H A ( G ≡ ≡ Tr µ − 2 D 4.9 Tr 2 µν 1 2 4.26 µ D − h 1 2 µν D )-( 2 − )-( B  = ) can be cast in the form − H 2 D 4.7 i 2 − 4.23 Φ 2 − | 4.16 2 D ]. Thus, in )-( D 43 4.15 26 [ 2 = = D N is a massive spin-2 field, with mass µν H A generic state at this mass level is written as After the field redefinitions parameter in non-minimal coupling to the background EMalgebraically field. consistent It and is preserves easy the to right see number that of the DoFs, system namely ( ensures that the number of dynamical DoFs is not affected. 4.2.2 Level the physical state conditions ( One finds that the system ( Higher Spin Theory - Part I thanks the commutation relations of the creationof and the annihilation operators, oscillator and vacuum. the By usual the definition field redefinition, the field equations ( The algebraic consistency of this system can be easily verified. The divergence constraint ( Thus PoS(Modave VIII)004 , ) . Let (4.36) (4.37) (4.38) (4.33) (4.30) (4.35) (4.31) (4.32) (4.34) ,...... 3.2 3 F ), one finds , 2 F . , Rakibur Rahman 4.35  1 tensor. The string F , it is clear that the ( s ]: µ , 0 ) and ( . It is manifest that the 43 D ,......
diag 2 = ) , 3 3 G 4.34 α = b ) ) of ! s ( , eb Tr will be block skew-diagonal ) are [ ν , µ g . µ 1 4 )] 4.3 , ... G F ! 1 0 a )] 2 4.9 i 0 1 + ) µ π b − 3 3 e 1 )-( Φ π b 1 0 π 1 e (

eb 0 1 − ( µ g ) − ( π 1 4.7 s i ( , − b 1 α .

( ) ) i 0 0 ], we will now show that these equations − 2 g 2 b iG , b α 2 = tanh 43 ( 0 eb = / tan . = , g ν 1 1 0 + 1 , = n s 6= ) + 41 i 6= s ) + µ µ ) i = a i = ( µ 2 2 G 0 n ... F s b 2 1 ... b e µ 0 18 2 µ ) are algebraically consistent. They form a Fierz- ( ν eb e π ... µ g µ 3 ( Φ π ) 0 , µ 1 Φ (  · , 1 , − 2 ) 4.32 Φ is block skew-diagonal: eF − a G ! ! D / ( )-( ea F , with f ) G Tr tanh tan ( , [ [ 1 2 0 1 1 0 0 1 1 0 ) 4.30 1 1 π π − a

s ( ) ea ) a ,...... f field, with mass ≡ ≡ 1 a = 3 s ( ) ) =  f ) that the highest-derivative terms appearing in the EoMs boil down i − G a N 1 b s , ( are given as ( ( F f = 2 g 2 g diag acting on the field. From the definition ( 1 4.32 G − 2 , G = )-( , the first Regge trajectory contains a symmetric rank- 2 1 and D s ν µ D G f  ( =  4.30 N G eF diag ’s are real-valued functions of the EM field invariants whose values are always small  i = b ) is in fact a diagonal matrix: ν µ and G eF / a ]. At level G We emphasize that if the EM field invariants are small, these functions are always well-defined We employ the method of characteristic determinants, already discussed in Section As one goes higher in the mass level, subleading Regge trajectories show up. It turns out that 43 and their absolute values are much smaller than unity. Given the forms ( One can go to a Lorentz frame in which to the scalar operator us note from Eqs. ( vanishing of the characteristic determinant boils down to the condition: where the functions that ( Pauli system for a massive spin- Higher Spin Theory - Part I 4.2.3 Arbitrary mass level: field equations that result from the physical state conditions ( indeed propagate the physical DoFs within the light cone. 4.3 Proof of Causal Propagation the lower Regge trajectories areis not [ consistent in isolation, whereas the leading Regge trajectory system gives the correct count of DoFs. Following [ with the blocks given by as well: One can easily show that the Eqs. ( where in physically interesting cases. In the same Lorentz frame, clearly PoS(Modave VIII)004 2 ]. ]. = 43 44 g (4.39) (4.40) ]. 3 [ 45 = s ] on EM and ) for a massive 5 Rakibur Rahman ] and , 4 41 3.15 ]. , 3 2 [ ]. 43 , 48 2 = , 2 [ , s 1 47 = , g 32 ) contain non-standard kinetic ) only in the critical dimension . 4.32 4.32 1 )-( )-( ≤ ) ] and ends up having a consistent model, i i b 4.30 4.30 ( eb 41 g ) must be space-like: , 26, one may perform a consistent dimensional < 0 0 < ≥ 4.33 19 2 D n , ]. This value can simply be read from the non-minimal 1 ) so that any field belonging to the first Regge trajectory of Eq. ( 46 ≥ µ ) n 4.32 a ( ea f )-( ), however, satisfy the inequalities 4.30 ). At the Lagrangian level, this can be seen by removing the kinetic-like 4.37 4.30 ), any solution ¨ ckelberg formalism has been employed, for example, in [ ) and ( 4.38 ]. In fact, explicit Lagrangians have been worked out for 4.36 43 . Note that gauge invariance can be restored in the massive theory by introducing the ]. To obtain a consistent theory in 8 43 26 [ For this field, the Stu The existence of a finite cutoff and the origin of the acausality problem are both due to a Last but not the least, let us notice that Open String Theory requires a universal value of Can an interacting massive HS particle be described by a local Lagrangian up to arbitrarily We finish this Section with a few remarks. We have seen that String Theory gives a consistent Note that the generalized Fierz-Pauli conditions ( 8 = spin-2 field simple fact: the free kineticIn term the presence of of a interactions, HSpropagate some field acausally. of exhibits On these gauge the modes other invariance, hand, may i.e.,scattering in acquire it amplitudes the non-canonical diverge. has massless kinetic zero limit All term the this modes. and HSunderstand is these propagator not issues is manifest is singular, at the so Stückelberg all that formalism,To in understand which the this focuses formalism, unitary precisely let on gauge. us the The consider gauge again modes. best the way Fierz-Pauli to Lagrangian ( contributions. One may wonder whether thecan flat-space show that no-ghost indeed theorem the extends no-ghost to theorem this continues to case. hold One in the regime of physical interest [ gravitationally coupled massless particles implythe that massless the limit. cutoff One of would like thatcoupling to Lagrangian constant find must of the vanish the explicit dependence theory in of to the see cutoff if on the the cutoff mass can and be the parametrically larger than the mass. high energy scales? The answer is no, because the no-go theorems [ They, however, give rise to theD Fierz-Pauli conditions ( say in 4D, that contains spin-2 plus a spin-0 field both having the same mass and charge [ for the gyromagnetic ratio forterm all appearing spin in Eq. [ ( cubic interactions by suitable field redefinitions, which uniquely give 5. Intrinsic Cutoff for Interacting Massive Higher Spins propagates causally in a constant EM background.a It Lagrangian is [ also guaranteed that this system comes from reduction of the string-theory LagrangianIn by this keeping way, only one the may singlets start of with the the internal spin-2 coordinates. Lagrangian [ set of EoMs and constraints ( Higher Spin Theory - Part I The functions ( Then, in view of ( which is a Lorentz invariantcausal statement. in We all therefore Lorentz conclude frames. that the propagation of the field is PoS(Modave VIII)004 ]: (5.8) (5.9) (5.5) (5.6) (5.7) (5.1) (5.2) (5.3) (5.4) in the (5.10) 49 n , Λ 47 , involving the 32 9 , Rakibur Rahman  are killed. φ φ µ ∂ m 1 ). However, these fields 2 , − 2 3.15 µν µ F B 4 1 , 4  through the substitution + − ν n µ ∂ φ O n 1 m W , ∗ − µ , 2 ) ) is that this is the unique linear combi- . + | n µ W φ φ 2 λ  Λ µ , ( m m m φ 3.15 , 0 , minimally coupled to EM: . D λ ) ν > µ − − . ν − ∑ µ λ ∂ n 1 m 2 λ λ W | V m D λ 4. Some of these operators may be eliminated m , in ( 1 µ + · µ µ m ( 2 −
20 + ∂ ∂ 2 2 = = D W 0 µ − n ν V ϕ µ = = = −| ν D V δφ = B − µ δ = − µν µ  B δφ ν µν gf µ δ W 4. ϕ W ∂ δϕ renormalizable L µ = 1 µν m L D D | ϕ = + 1 2 2 − µν L m ϕ = , through the field redefinition: = for which 4-derivative bad kinetic terms of the field φ L 2 0 µν ϕ ˜ and ϕ µ → B and µν µν ϕ ϕ denotes operators of dimension µν 4 + ϕ n O In this Section, we will work only in We start with a complex massive spin-1 field, One can instead choose a gauge fixing such that all the fields acquire canonical kinetic terms. 9 ¨ ckelberg fields With the addition of the gauge-fixing term where Contrary to naïve power counting,the we higher know dimensional that operators this appear, Lagrangian we is introduce non-renormalizable. a scalar To make so that one has the gauged Stückelberg symmetry: so that the following transformations become a gauge symmetry: may help us to understandunderstanding the some mass otherwise term, obscure phenomena. For example, one slick way of by field redefinitions or byremaining adding terms non-minimal defines the terms ultimate to cutoffexamples: the of massive Lagrangian. the particles effective coupled field The to theory. smallest EM Let in us flat now space. consider some Spin 1 Higher Spin Theory - Part I Stu nation of In the interacting theory, one thus ends up having non-renormalizable interactions Note that this Stückelberg symmetryelberg is fields a to fake go one back in to the that unitary-gauge one Fierz-Pauli can Lagrangian always ( gauge away the Stück- Stückelberg fields, weighted by coupling constants with negative mass dimensions [ PoS(Modave VIII)004 ), . . In int 5.1 e / L (5.20) (5.11) (5.13) (5.14) (5.15) (5.16) (5.17) (5.18) (5.12) (5.19) ]. m + , ] 0 49 ∼ 2 ˜ µν , ϕ F Λ 0∗ 1 4 32 ˜ ϕ Rakibur Rahman − 1 [ . 270GeV. Below − . the neutral Higgs φ φ ) ∗ ∼  µν −  2 φ ˜ e ϕ φ Λ m 2 µ µν ∗ µν − F D ˜ ϕ . 2 [  m 2 1 2 ( Λ m 2 e ∗ , 2 m 2 µν in the absence of other DoFs. 2 φ F −

e 3 2 − − 4 1 µ µ ] / .  + B 2 m − | mB µ  c.c. ) φ 1, for any given operator dimension- ∼ c.c. B ν φ +
) + 0 3 2 , Λ D , 0 2  ˜ ϕ µ φ ) + m boson has a cutoff ϕ m − 1 m e µ λ µ 0 ν − µν ± − V D m ϕ D , + 2 ν η ) ) W  )( 2 ν 1 2 − D  D µν . ( 2 / λ ∗ φ µ ∗ λ λ / 1 ˜ − − ∗ µ ν ϕ ( µ ( B m m ν φ µ 21 2 ), introduce the Stückelberg fields through ( D D D 2 µν V − D + µ ϕ [ m + µ 1 + ( 0 = = = µ 2 D ϕ 3.15 B − ( ϕ → V · · µ ∗ 2 −
|
µν B 0 δφ 2 φ D ν 2 D µν ˜

| δ ϕ B m m ϕ δϕ µν 2 2 . Parametrically in µ  e − F − − − D µ | 2  m D ie = = D + 2 2 1 0∗ m  | ∗ µ gf1 gf2 ϕ − V + µν 2 1 L L ˜ ϕ = µν − µ ϕ D gf µν | = 2 L ϕ
+ + µν 2 2 ˜ ϕ | m L νρ − -terms are more dangerous than the others because they are weighted by the inverse ˜ ) ϕ e  µ ( contains all interaction operators; they appear with canonical dimension ranging from 4 D O ∗ µν has a mass of 126GeV, which is indeed below the cutoff scale int −| ϕ − L = Now we diagonalize the kinetic operators and thereby make sure that the propagators are Thus the cutoff of a charged massive spin-1 particle is We take the Pauli-Fierz Lagrangian ( = L L Here through 8, and at least oneality, power the of smooth in the massless limit. This is done by the field redefinition: The presence of non-renormalizable interactions implies the existence of a UV cutoff make the fields complex, and then replace ordinary derivatives with covariant ones. Thus result is This system enjoys the covariant Stückelberg symmetry: Higher Spin Theory - Part I one is left with the following operators: along with the addition of the gauge-fixing terms (thereby exhausting all gauge freedom): a local theory addition of non-minimalwithout terms introducing and new field ones redefinitions that cannot instead remove lower these the operators cutoff, parametrically in This means in particular that thethis electrodynamics scale of some new physics mustWe appear. now know This could that be thescalar strong latter coupling or possibility new is dynamical realized DoFs. in nature.Spin 2 The new DoF with One is thus left with PoS(Modave VIII)004 , - ) = 3 43 , m add / (5.26) (5.21) (5.22) (5.24) (5.25) (5.23) 41 e L ( constant, O = 3 ]: m Rakibur Rahman / 49 e , dimension-8 ones: , ) . e c.c. ν ( J c.c. + O µν  + φ F  µ dimension-7 operators:  0, such that φ ] ∂ ) ∗ ν  term. Indeed the term 4 e ] → B e ( ∗ ν ) m µ e 2 [ B O e ∂ µ ≡ , the higher the dimension, the more ( [ 1, coupled to EM is [ e ] , ∂ − O ν ρ ≥ φ − 0 and ˜ c.c. ν . ϕ s φ ) ∂ 1 + ν → ρ ∗ µρ ∂ − ˜ . φ ∂ s ϕ removes this operator, it also yields another . To improve the degree of divergence, one ρ ] m 3 2 ν 2 ]. Attempts along these lines would therefore ∗ ρ ( / ∂ ) µ ∂ µν / ] 1 m B J 1 . These dimension-11 operators could be elimi- µ ρ ∗ ρ e 49 ) µ 7 J − [ ∂ ), completely eliminates the dimension-8 opera- 22 B 4 ν ieF ∗ ∂ m = µ ∂ [ m 2 / φ me ∂ / = 2 Λ ρ  − 5.12 e 2 e ∂ = (  ν ) µν J 2 Λ − that can cancel the F µ / dipole µν i µ ∂ 3 1, states that massive HS particles admit a local description F ( L [( A µν 2 m ]: ie 3 ˜ ) 2 ϕ µν i  4 Λ 49 → , which cannot be improved further. 2 F e m , = ) µ µ 2 7 + ∂ / A / 32 2 e 4 e ( achieves this feat. Thus the possible cutoff is pushed higher: the most e kin dipole / m , 2 7 L m ) ( = µ L ρ = 8 ˜ O ϕ + L 7 L ∗ νρ ˜ ϕ L − ν ρ ˜ ϕ ). In the resulting non-minimal theory, one does have ∗ µρ ˜ ϕ )( 5.21 However, mere addition of a dipole term in the Lagrangian may leave us only with One can follow the same steps for arbitrary spin to find that the largest cutoff of the local 4 / ]. Conversely, other consistency requirements may even sharpen this bound. 2 e This formula, valid in theeven limit for energies aboveeffective their action. mass, This when is they arequiring further model-independent must consistencies, upper be e.g. bound treated on causal propagation, as the may cutoff true only that dynamical result no in DoFs theory a can in lower beat: cutoff the [ 49 terms, as we will now(counter-)terms. see. Indeed, the dipole This term is an improvement over field redefinition plus addition of local equally bad term, namely, may add some local functions of dangerous operators now are weightednated, by up to total derivatives,a by cohomological adding obstruction local to counter-terms. this possibility However, in [ a local theory there is effective theory describing a massive charged particle of dangerous an operator is. The most dangerous operators are therefore the While the field redefinition: ( when added to thetors ( minimal Lagrangian ( which has an intrinsic cutoff [ Arbitrary Spin suggest a cutoff scale Higher Spin Theory - Part I of a lower mass scale. On the other hand, at a given order in which contain pieces notcannot proportional be to improved any any further. of In the the EoMs. scaling limit Therefore, the degree of divergence the non-minimal Lagrangian reduces to: PoS(Modave VIII)004 ) . 5 2.32 particle and s 4 ] for bosonic ]. For bosonic 56 Rakibur Rahman , which involves 57 3 s -matrix is not well- S as the spin- − ). These theories con- 2 Λ s , do not apply to massive − 2.29 3.1 1 is also going to appear. s − 3, the Weyl tensor is vanishing, and = D ), as we have already mentioned, is only that trilinear vertices constructed in [ s 23 5.26 − , which is at our disposal. However, this also means s ]. In fact, gravitational and EM multipole interactions Λ ] useful for a comprehensive overview. 5 − complementary to this set 55 , − ] to consistent local interactions of massless HS fields is due 54 and 1 3 , s : one expects no operational difference between massless parti- 53 ) − | s ]. Because of this technical reason, lighter DoFs may be essential Λ . The very possibility non-local counter-terms invalidates the coho- | − Λ 49 p ( ] for discussions of HS interactions in AdS space. O 52 , 51 , 50 . Vasiliev’s system is a set of classical non-linear gauge-invariant equations for an 10 ] 10 , )), they are immune from many consistency issues that may arise in higher dimensions. 8 2.40 See also Refs. [ Finally, in flat space another way to bypass the no-go theorems is to have higher-derivative It is possible to have consistent HS gauge theories in AdS space, e.g., Vasiliev’s HS gauge The no-go theorems for interacting HS fields, presented in Section The meaning of the cutoff upper bound ( The Aragone-Deser obstruction [ 10 and ( tain interaction terms that do notnon-local. stop at But a the finite non-locality numberinverse powers of is of derivatives, under the so cosmological control that constant they in are that essentially higher-derivative terms are weighted by fields. These cubic vertices are special cases of the general form non-minimal couplings for HS gauge fields [ that these theories do notdefines allow a sensible mass flat parameter limits. On the other hand, the cosmological constant show up, for example, as the 2 theories [ develops a form factor which isif tantamount one to integrates a out finite the non-zeroterms new charge already light below radius. the DoFs, In scale the the resultingmological second action argument case, would of necessarily Ref. contain [ non-local particles. The reason is trivial: gauge transformationsterm. are no However, we longer a saw symmetry that because these ofdetails the fields of mass have (consistent) other interactions problems of when massive HS interactions fields are have present. been Some presented inthe Sections appearance of the Weyl tensor in theso gauge no variation. such In obstructions exist. Because HS gauge fields in 3DIn carry fact, no 3D local HS DoFs gauge (seenotes theory Eqs. by is ( G. a Lucena very Gómez rich on and this rapidly topic expanding subject. An entire set of lecture infinite tower of HS gauge fieldstensor in fields, AdS the space. linearized When equations expressed in take terms the of standard metric-like Fronsdal symmetric form ( cles and massive particles withseeing Compton how wavelength Vasiliev’s larger theories than avoid the the no-godefined AdS theorems since radius. is asymptotic that Another states in way do AdS of interested not space readers exist. may The find formulation Refs. [ of these theories is rather technical; massless fields of arbitrary spins.derivatives in The these light-cone vertices, formulation and puts thereby restrictions provides on a the way number of of classifying them [ some new physics must happenpling below unitarization, this or scale. in the Thiscase, existence new of the physics new theory could interacting becomes result DoFs essentially lighter in non-local than a at the strong a cutoff. cou- scale In not the higher first than for a complete UV embedding of the effective field theory. 6. Interactions of Higher Spin Gauge Fields Higher Spin Theory - Part I PoS(Modave VIII)004 . C field (6.3) (6.4) (6.2) (6.1) 3 2 system 3 2 , we will and higher. . The set of 6.1 ξ 2 3 , another with Rakibur Rahman λ = s ]. The spin- ] generating functions 65 60 . In Section ,  11 and a massless spin- , in the range ] to systematically construct ρ p µ ] have employed the Noether ψ 61 A ν 59 ∂ . , ε 3 µνρ ]. In Ref. [ s µ γ ∂ ] for some applications. µ 60 + . ¯ 2 ψ 65 = } i s , ξ µ , − + 64 ψ µ , 1 ε 2 s µν ψ δ 63 , F , as “spin” for fermions. For bosonic fields, the ≤ 1 4 C , 56 , we introduce a Grassmann-odd bosonic ghost p 2 1 24 − µ ]. The authors in [ λ  , A ≤ − x 58 { λ s 3 D vertex, closely following Ref. [ µ s d = ∂ 2 3 − A Z , we have a Grassmann-even fermionic ghost 2 = − Φ s ε µ 3 2 ] uses gauge invariance as a consistency condition and by con- ] = + A µ 1 − λ s 61 δ ψ , µ : ε A [ ) 0 ( S ] for general methodology and also [ 62 that no-go theorems rule out minimal EM coupling in flat space for , we explore the second-order consistency of this vertex. is the smallest of the three spins. For a vertex containing two fermions and a boson, the 3.1 6.2 3 s For the Grassmann-even gauge parameter The starting point is the free theory, which contains a photon See also Ref. [ The BRST-antifield method [ , described by the action 11 µ Step 1: Ghosts Corresponding to the Grassmann-odd fields now becomes a fermionic gauge parameter is simple enough to allowturing many an non-trivial easy features implementation that of couldSection serve the as BRST guidelines deformation for scheme HSTo particles. construct while possible Let cap- non-minimal us couplings, recall we from standing will go of step the by procedure. step for the sake of aStep clear 0: under- Free Action ψ which enjoys two abelian gauge invariances: one with a bosonic gauge parameter 6.1 Consistent Couplings via BRST-Antifield Method procedure to explicitly constructString off-shell Theory vertices. gives rise to On the the same other set hand, of cubic the vertices tensionless [ limit of employ this method to constructSection a cubic coupling for a very simple non-trivial system, while in struction throws away trivial interactions.preserving Here off-shell we non-abelian will 1 employ this method to construct a parity- complete list of such vertices was given in [ Higher Spin Theory - Part I fields, there is one vertex for each value of the number of derivatives for off-shell trilinear vertices foralso both employ bosonic the and powerful fermionic machinery of fields theconsistent have BRST-antifield interactions been method using [ presented. the One language can ofcality cohomology. and Poincaré With invariance, the this underlying approach assumptions isand of very manifestly useful lo- in Lorentz-invariant general off-shell for vertices obtaining for gauge-invariant HS gauge fields where formula is the same modulo that one uses PoS(Modave VIII)004 ∗ B Φ (6.8) (6.9) (6.6) (6.7) (6.5) (6.10) which − ) 0. defined as is real, − pgh A the antighost . ) = 0  Φ − ) S Rakibur Rahman , which is real. , µ A B 0 ∗ δ S ( ¯ ξψ = . µ )
∂ ∗ A ∗ B Φ − Φ ( , ξ A . µ for the free theory Φ pgh ∂ s µ agh ∗ = , ¯ A ψ − 0 Y ( L the pure ghost number ( Φ . δ − δ − } in the space of local functionals of the ) = . Explicitly, pgh ∗ ∗ A C A ∗ A s X ¯ ξ µ = R Φ , Φ Φ ∂ ( . δ µ δ µ ) ∗ gh ∗ ¯ − X ψ A . agh , , 0 ∗ A an extension of the original gauge-invariant ac- 0 ∗ ) solves the master equation Y + S L Φ − C ( = ρ 25 , 0 δ δ 6.8 2 ψ S µ ≡ s , A ∗ ν X 1 ∂ A R Φ . { sX δ δ } ) + µνρ = ξ A ≡ , γ ∗ A µ ) Φ is nilpotent: C ¯ ( Φ { ψ Y s i , X − pgh ( 2 µν F ) = 1 4 ∗ A belongs to the cohomology of is the generator of the BRST differential 0 Φ − (  0 S S x ) is another grading, defined as D agh d gh Z which is non-zero only for the antifields = − 0 ) is BRST-closed, as a simple consequence of the master equation. From the properties S by terms involving ghosts and antifields. Explicitly, 0 S − agh ) 6.2 The master action On the space of fields and antifields we define an odd symplectic structure, called the an- Now we construct the free master action For each field, we introduce an antifield with the same algebraic symmetries in its indices but Note that the antifields appear as sources forby the corresponding “gauge” ghosts. variations, with It gauge is parameters replaced easy to see that ( Note that of the antibracket, we also find that Step 5: BRST Differential Then, the master action fields, antifields, and their finite number of derivatives. tibracket, as The ghost number ( Step 3: Antibracket which satisfies the gradedBecause Jacobi a identity. field and This itsmust definition antifield be have purely gives opposite imaginary, and Grassmann vice parity, versa. it follows thatStep if 4: Free Master Action tion ( Higher Spin Theory - Part I In the algebra generated by the fields, we introduce a grading In the algebra generated bynumber the ( fields and antifields, we introduce another grading is non-zero only for the ghost fields opposite Grassmann parity. The set of antifields is Step 2: Antifields PoS(Modave VIII)004 are acts Γ of the (6.13) (6.14) (6.12) (6.15) (6.16) (6.11) Γ 0 S and increase the ∆ s Rakibur Rahman ) and Z ( Γ ε is the solution of the , ∆ S ) : Z 12 1 0 12 0 1 g ( − − − − gh ) . Z 0 . : ( ) Γ 3 = g . ( agh ) 2 ∆Γ O S , 1. It is important to note that + ) , , + 0 . 0 0 Z 2 S ( Γ S ( 0 1 00 1 2 0 2 Γ∆ 2 ) = 2 + s g ) = ) = ( − pgh 0 1 ∆ 26 + S S gh , , 1 , = 0 0 , into the following set ) = decomposes into the sum of the Koszul-Tate differen- 0 ) s S S 1 gS s 2 ( ( S ) = µ αβ µ = g 0. This must be a deformation of the solution , + µ ∗ µν ) ( Γ γ 1 2 ∗ 0 ( A Z β S F ¯ O S ∆ 0 0 0 0 0 0 0 0 0 1 ( ( ¯ ψ µ ν ψ ) = gh µ ∆ ∂ ∂ S = = α ∂ , ∂ − − 2 S i S Γ Properties of the Various Fields & Antifields ( ) = − ∆ ( gh ) ξ C splits, up to 0 0 00 0 00 1 0 0 1 1 0 1 1 0 Z µ µ ( S Table 1: ∂ ∂ Γ on them, are given in the following Table. ∆ µ µ ∗ ∗ µ µ ∗ ∗ Z ξ ¯ C ξ ¯ C A ψ A ψ and Γ , and the longitudinal derivative along the gauge orbits, The solution of the master equation contains compactly all information pertaining to the con- The different gradings and Grassmann parity of the various fields and antifields, along with The various gradings are important as ∆ implements the EoMs by acting only on the antifields, and in so doing it decreases the antighost The master equation for deformed master equation, then master equation of the free gauge theory, in the deformation parameter sistency of the gauge transformations.interactions One in can reformulate a the gauge problem theory of as introducing that consistent of deforming the master action. If An Aside: BRST Deformation Scheme the action of tial, Higher Spin Theory - Part I ghost number by one unit, nilpotent and anticommuting: ∆ number by one unit whileonly keeping on unchanged the the original purenumber fields ghost by number. and one unit produces On without the the changing other gauge the hand, antighost transformations. number. Thus It all increases three the pure ghost PoS(Modave VIII)004 ), the (6.23) (6.19) (6.17) (6.20) (6.21) (6.22) (6.18) -closed Γ 6.11 Rakibur Rahman ]: 63 , 62 , 56 . ) -closed, instead of i 2 [ Γ a is a top-form of ghost number ( , = a on the fields and antifields. It is ¯ pgh ξξ ∆ ∗ is BRST-closed: agh the cohomology of the zeroth-order ) and the decomposition ( = C 1 − 1 i S , , ) and g 0 0 , where d | 6.19 a Γ − s ) = = = ( i R , is the deformation of the Lagrangian, while
0 1 0 a ∗ 0 0 , ( . = H a 0 0 db db ¯ = ) means 1 ξξ S + + = agh = + 27 db 1 0 2 1 6.15 a a ξ a + , at ghost number 0. Now, one can make an antighost- ∗ has been chosen to be sS Γ Γ Γ ¯ d ξ 2 , sa + + a 2 2 1 , a C a a and their derivatives, 0 } is non-trivial, the algebra of the gauge transformations is g + ∆ ∆ ξ } 2 ; this expansion stops at ]). , 1 ∗ ’s would contain no derivative and one derivative respectively. − a a and their derivatives. potentially gives rise to minimal coupling, while the second 0 ¯ a C ξ 62 1, and a , { } = 0 ] + µ g ν 2 + 0 ∗ i a a ¯ ψ ψ µ , [ = ∗ ∂ contains gauge-invariant functions of the fields and antifields. For the ) = , a C i , Γ b Γ ]. Thus, if µν µ ( ) gives the following consistency cascade for allowed deformations: ∗ F be a parity-even Lorentz scalar, the most general possibility for it is 61 A { 2 { pgh a , modulo total derivative 6.18 , s i ) with the knowledge of the action of ) = 6.22 i b ( )-( encode information about the deformations of the gauge transformations and the gauge (this is always possible [ ) is fulfilled by assumption, and Eq. ( 2 d a field, it is isomorphic to the space of functions of agh The undifferentiated ghosts The curvatures The antifields 6.20 2 3 6.14 Requiring that The cohomology of • • • and 1 The first piece withcould coefficient produce dipole interaction.cade This can ( be understood by a derivative counting down the cas- The significance of the various termsa is as follows. algebra respectively [ Step 7: Solution of the Consistency Cascade then easy to see that the respective deformed and becomes non-abelian.deforming/non-abelian vertices. In what Given follows the we expansion are ( interested only in gauge-algebra- Step 6: Cohomology of spin- If the first-order local deformations are given by cocycle condition ( where modulo number expansion of the local form Higher Spin Theory - Part I Eq. ( BRST differential 0, then the following cocycle condition follows which says that non-trivial deformations belong to PoS(Modave VIII)004 1 ]. ): a and 37 2 3 6.21 (6.24) (6.26) (6.29) (6.25) (6.27) (6.28) -closed Γ , Rakibur Rahman 0 o 2 SUGRA [ -exact: β . s = = ψ

1 be ˜ a by solving Eq. ( ) N Γ h.c. 1 1 , µναβ a S ν + γ , -exact. The conclusion is ψ 1 α ξ ∆ is by construction , ¯
S  ψ , ( 1 1 1 2 F . This can happen only when 6 ρσ ˜ a a . h.c. ν d F . ∆ + ) γ -variation of each of these terms 2 ] + 2 S ν Γ ) + M -variation of the most general form 1 − ξ Γ µ Γ ψ D µνρσ Γ i 2 ( beside a complex massless spin µ ; then the Pauli term vanishes because γ ) for our non-abelian vertex. Explicit [ ]: F ¯ 6 ∆ 1 2 − ξψ − − ¯ ∞ . Now ∗ ψ 2 − 65 Γ ¯ −  + 6ψ νρ S 6.28 → ) ]. ν F = ξ ∆ 2 -variation is not P ∂ µν ρ µ 2 − 57 ) 1 γ Γ ¯ M F M ψ D − , µ 2 -exact modulo ( (  5 2 ) with the choice -exact [ Γ µ µ Γ∆ = a graviton ¯ ∗ ∆ ¯ ξψ 28 µν − ψ 2 A − = 1 ) holds, then the + 1 − 6.22 g sS 0 µν g F 2 ξ = F can be consistently lifted to an µ = ∗ A µ . Thus we have a consistent Lagrangian deformation + ν − 6.28 ¯ 2 ¯ ψ Φ  Γ γ ν ψ i a is just the Pauli term appearing in ( µ )] ψ  ∗ 4 1 ν ) = x ¯ h.c. S 1 n µν ψ ψ , D S h x + 1 d , 1 D S µν ) + 1 F d µ + S Z [( ig µ ( ¯ A ) requires it to be F Γ vertex gets obstructed beyond the cubic order. ψ Z + = 1 µ ξ 3 2 ¯ 1 g ˜ ψ = + a 6.22 − ) holds, then one can show that the = 0 C 2 3 ) it is easy to see that their , belongs to the cohomology of 0 = µ 1 ∗ A a -exact. By considering, for example, the fermion bilinears appearing in the -non-trivial pieces coming from the unambiguous piece and the ambiguity a − ψ Φ 6.27 ∆ ( Γ 6.29 )] µ contains too many derivatives. This rules out minimal coupling of a massless 1 ∗ S ¯ 1 ψ , a  1 0, one indeed can satisfy Eq. ( 0 S g = [( 0 = g , but condition ( 1 d a field. It is this new dynamical field that removes obstructions to the vertex while keeping the potential to EM, in accordance with what is known [ ) 1 2 3 1 Notice that the vertex We recall that consistent second-order deformation requires that With It turns out that each of the terms in a ( ∆ U locality intact. One may decouple gravity by taking a where the ambiguity, ˜ modulo This theory, however, contains additional DoFs which is the desired non-abelian vertex. 6.2 Second Order Consistency of this antibracket evaluated at zero antifields is computation gives If the vertex is unobstructedshould so separately that be Eq. ( that the non-abelian 1 second line of Eq. ( Higher Spin Theory - Part I which is of course in the cohomology of spin- in cancel each other. Itsimply is because easy ˜ to see that such a cancelation is impossible for the first term in We can prove byquartic contradiction order. that If there Eq. ( is obstruction for our cubic non-abelian vertex to the It is relatively easier to compute the left hand side of ( PoS(Modave VIII)004 , 139 567 , 407 (1988); Rakibur Rahman , 161 (1979). 86 303 , 065016 (2008) , 225 (1992), 78 285 , 1229 (1988). , 59 (1980). 60 96 , 214006 (2013) [arXiv:1112.4285 46 , B1049 (1964). 135 ] or the tensionless limit of String Theory. ]. On the other hand, one could integrate and spin-1 fields only. The resulting theory 10 37 , 2 3 [ 8 P , 1387 (1991), “More On Equations Of Motion For 8 M 29 / , 1615 (1988), “Can Space-Time Be Probed Below the String 3 , 141 (1987). , 41 (1989). 291 216 , 89 (1987), “Cubic Interaction In Extended Theories Of Massless Higher Spin , 378 (1990), “Properties of equations of motion of interacting gauge fields of all , 323 (1977); M. T. Grisaru, H. N. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D 189 243 67 , 129 (1987), “String Theory Beyond the Planck Scale,” Nucl. Phys. B 197 , 81 (1987), “Classical and Quantum Gravity Effects from Planckian Energy Superstring , 996 (1977). 197 Size?,” Phys. Lett. B (2003) [arXiv:hep-th/0304049]. B Collisions,” Int. J. Mod. Phys. A Phys. Lett. B Fields,” Nucl. Phys. B Phys. Lett. B spins in (3+1)-dimensions,” Class. Quant. Grav. [hep-th]]. D. J. Gross, “High-Energy Symmetries of String Theory,” Phys. Rev. Lett. Interacting Massless Fields Of All Spins“Nonlinear In equations (3+1)-Dimensions,” for Phys. symmetric Lett. massless B higher spin fields in (A)dS(d),” Phys. Lett. B Lett. B [arXiv:0804.4672 [hep-th]]. Phys. Lett. B Equality Of Gravitational And Inertial Mass,” Phys. Rev. 15 We would like to thank A. Campoleoni for useful discussions. RR is a Postdoctoral Fellow of [9] A. Sagnotti, “Notes on Strings and Higher Spins,” J. Phys. A [7] D. Amati, M. Ciafaloni and G. Veneziano, “Superstring Collisions at Planckian Energies,” Phys. Lett. [8] E. S. Fradkin and M. A. Vasiliev, “On The Gravitational Interaction Of Massless Higher Spin Fields,” [2] M. T. Grisaru and H. N. Pendleton, “Soft Spin 3/2 Fermions Require Gravity And Supersymmetry,” [1] S. Weinberg, “Photons And Gravitons In S Matrix Theory: Derivation Of Charge Conservation And [3] C. Aragone and S. Deser, “Consistency Problems Of Hypergravity,” Phys. Lett. B [6] D. J. Gross and P. F. 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