ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Higher-Spin Triplet Fields and String Theory

D. Sorokin

Abstract

We review basic properties of reducible higher-spin multiplets, called triplets, and demonstrate how they naturally appear
as part of the spectrum of String Field Theory in the tensionless limit. We show how in the frame-like formulation the
triplet fields are endowed with the geometrical meaning of being components of higher-spin vielbeins and connections and
present actions describing their free dynamics.

1 Introduction
It is a pleasure to write this contribution on the occa-
sion of the anniversary of Jiri Niederle. As a topic I
have chosen a subject to whichProfessorNiederle and
his collaborators have contributed with several inter-
esting papers [1, 2, 3]. It concerns higher spin field
theory. In particular, I would like to discuss a sys-
tem of higher spin fields which are called triplets. The
physical states of triplet fields describe massless parti-
cles of decreasing spins. In the bosonic case the phys-
ical states of a triplet have spins s, s − 2, s − 4, . . . ,1
or 0 and the physical states of a fermionic triplet
have spin s, s − 1, s − 2, . . . ,1/2. In other words the
triplets form reducible Poincaré group multiplets of
massless higher-spin particles. As we shall see, they
naturally arise in a tensionless limit of String Field
Theory [4, 5, 6, 7, 8, 9, 10] as sets of three tensor
fields (and this is where their name comes from [8]).
We shall also discuss basic group-theoretical and geo-
metrical properties of the higher-spin triplets revealed
in [11].
Let us startwith a brief generic discussion of prob-

lems of Higher Spin Field Theory. The construction
of a consistent interacting theory of higher-spin fields
is one of the oldest long standing problems in theo-
retical physics of particles and fields. Since the early
1930s, it has been addressed by many distinguished
theorists including Majorana, Dirac, Fierz & Pauli,
Rarita & Schwinger, Bargmann & Wigner, Fang &
Fronsdal, Weinberg, Velo & Zwanziger, Aragon &
Deser, Singh&Hagen, deWit&Freedman,Fradkin&
Vasiliev and (later on) by many others1.
The main problem of Higher Spin Field Theory is

the construction of consistent interactions of higher-
spin fields. Higher-spin interactions, e.g. with electro-
magnetic field and gravity, must respect gauge sym-
metries and be consistent with quantum mechanical
principals such as unitarity, causality, etc.
To solve the problem of higher-spin interactions

one first looks for a most appropriate description of
free higher-spin fields which would allow for general-

ization to an interacting theory. Revealing the group-
theoretical and geometrical structures underlying this
theory is of great importance. Until now, the most
successful general approach to the constructionof non-
linear (i.e. interacting) classical higher-spinfield equa-
tions has been the so-called unfolding techniques put
forward by M. Vasiliev [18, 19] (see [12] for a review
and references).
Great efforts have been made in studying both,

massive and massless, higher-spin field theories. Jiri
Niederle has concentrated, in particular, on study-
ing electro-magnetic interactions of massive higher-
spin fields, which, actually, reveal major issues of the
generic problem [2, 3].
It would be very interesting and very important

for the further development of this subject to compare
the results obtained in the construction of higher-spin
interactions with the structure of a known example
of consistent higher-spin field theory which is String
Theory2. People have worked in this direction since
the early years of String Theory, and this is how, for
example, the system of higher spin triplet fields was
first found [4]. So let usmake a very short overviewof
the place of the higher-spin fields in String Theory.

2 String Theory as a theory of
interacting massive
higher-spin fields

String excitations give rise to an infinite number of
fields of increasing spin andmass, the mass of a string
state being a linear function of its spin (Regge tra-
jectory) with the proportionality coefficient being the
string tension T . For instance, in open string theory
we have

M2s ∼ T(s −1) (2.1)
The infinite tower of higher-spin string states plays a
crucial role in ensuring a smooth ultraviolet behav-
ior (or even UV finiteness) of superstring theory, thus
making it a consistent theoryofquantumgravity, since
gravity is an intrinsic part of String Theory.

1For reviews on various aspects of Higher Spin Field Theory and references see e.g. [12, 13, 14, 15, 16, 17].
2Let us recall that StringTheory emerged as a proposal to explain (among other things) the mass-spin dependence (known as Regge

trajectory) of higher-spin hadronic resonances observed in experiments in the 1960s.

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Acta Polytechnica Vol. 50 No. 3/2010

In String Theory the higher-spin excitations are
massive, but our experience in the quantum field the-
ory of vector fields teaches us that for quantumconsis-
tency themass of the vector fields should be generated
as a result of spontaneous breaking of a symmetry of
massless vector fields. If one tries to extrapolate this
statement to String Theory, then the natural question
arises whether String Theory can be a spontaneously
broken phase of an underlying gauge theory of mass-
less higher-spin fields. From eq. (2.1) we see that the
mass tends to zero in the limit inwhich the string ten-
sion goes to zero. So people have tried to answer this
question by taking a tensionless limit of String The-
ory (see, e.g. [20, 21, 22, 23] for various aspects of the
tensionless string limit and higher-spin theory).
One can try to deduce the structure of higher-spin

field interactions from the action of String Field The-
ory. E.g. the action of Open String Field Theory
has the following schematic Chern-Simons-like form
(see [24] for more details)

Sopen string = 〈Φ|Q|Φ〉+ |Φ〉3 (2.2)

where |Φ〉 is a string field and Q is a BRST operator
associated with the symmetries of String Theory.
Due to the complexity of the problem, only terms

in the Lagrangiandescribing a system of free massless
higher-spin fields have been obtained from the String
Field Theory action so far. As we have already men-
tioned, this systemofmassless fieldswasfirst obtained
by S. Ouvry and J. Stern in 1986 [4]. The simplest
triplet consists of a symmetric tensor or (in the case of
half integer spins) tensor-spinor fields of rank s, s −1
and s −2

φm1···ms(x), Cm1···ms−1(x), Dm1···ms−2(x) (2.3)

where the indices m = 0,1, . . . , D − 1 are indices of
space-time of dimension D. The spectrumof the phys-
ical statesof thesefields consistsofparticlesofdecreas-
ing spin:

s, s −2, s −4, . . . ,1 or 0 – in the case of bosons
s, s −1, s −2, . . . ,1/2 – in the case of fermions.

Since each value of spin corresponds to an irreducible
representation of the Poincaré group, the triplet of the
fields (2.3) describes a reducible multiplet of higher-
spin states.

3 Massless higher-spin triplets
from String Theory

Let us consider how the equations of motion of the
triplet fields (2.3) arise in the tensionless and free field
limit of String Field Theory. In the tensionless limit
T → 0, the nilpotent open bosonic string BRST oper-
ator is (see e.g. [9])

Q = c0pmp
m +

∑
k �=0

(
c−kpma

m
k −

k

2
b0c−kck

)
, (3.4)

k = ±1, ±2, . . . , ±∞ Q2 =0

where pm is the momentum of string center-of-mass,
amk are string creation (for k < 0) andannihilation (for
k > 0) oscillator operators (with |k| = 1,2, . . . , ∞ la-
belling the Regge trajectories), c0 and ck are string
reparametrizaition ghosts, and b0 and bk are anti-
ghosts. The (anti)commutator relations satisfied by
the operators are

[amk , a
n
l ] = k δl+k,0 η

mn, {ck, bl} = δk+l,0 . (3.5)

A string field is a sum of an infinite number of higher-
spin fields generated by acting on the string vacuum
with all possible combinations of creation operators

|Φ〉 =
∑

φmn···p···q···(x) · (3.6)
aman · · · ap · · · aq c · · · b · · ·b |0〉.

Now, one may ask the question, do independent
finite subsets of higher-spin fields exist inside (3.6)
which satisfy (at least linearized) string field equations
ofmotion that follow fromthe action (2.2)? The linear
field equations are

Q|Φ〉 =0 . (3.7)

Because of the nilpotency of the BRST operator, they
are invariant under the BRST gauge transformations

δ|Φ〉 = Q|Λ〉 , Q2 =0 . (3.8)

To extract from (3.6) a finite independent set of fields
satisfying eq. (3.7) let us pick up string states corre-
sponding to a single Regge trajectory (e.g. k=1) and
cut this trajectory at a level of spin s. Then it turns
out that the string field (3.6) reduces and splits into
independent pieces each of which is the sum of three
terms

|Φ〉triplet = φm1···ms(x)a
m1
−1 · · · a

ms
−1 |0〉+ (3.9)

Cm1···ms−1(x)a
m1
−1 · · · a

ms−1
−1 c0b−1 |0〉+

Dm1···ms−2(x)a
m1
−1 · · ·a

ms−2
−1 c−1b−1 |0〉 .

By independent we mean that each set of fields (3.9)
independently satisfies the string field equation (3.7).
Symmetric tensor fields φs(x), Cs−1(x) and Ds−2(x)
form the simplest bosonic higher-spin triplet which
satisfies entangled equations of motion that follow
from (3.7), namely:

� φm1···ms = s ∂(ms Cm1···ms−1) , � ≡ ∂n ∂
n

Cm1···ms−1 = ∂
n φnm1···ms−1 − (s −1) ·

∂(ms−1 Dm1···ms−2) ,

� Dm1···ms−2 = s ∂
n Cnm1···ms−2, (3.10)

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Acta Polytechnica Vol. 50 No. 3/2010

where (m1 · · · mp) denotes the symmetrization of the
indices with weight

1
p!
.

Equations (3.10) are invariant under the gauge
transformations with a local symmetric parameter
Λm1···ms−1(x) that follow from the BRST symmetry
(3.8):

δ φm1···ms = s ∂(ms Λm1···ms−1) ,

δ Cm1···ms−1 = �Λm1···ms−1 ,

δ Dm1···ms−2 = ∂
n Λnm1···ms−2. (3.11)

From the second of eqs. (3.10) we see that Cs−1 is an
auxiliary field which is expressed in terms of deriva-
tives of φs and Ds−1, and from the form of the gauge
variationof Ds−1 we see that this field is a pure gauge.
Thus, all physical degrees of freedom are contained in
the field φm1···ms(x). As we have already mentioned
and as follows from the analysis of eqs. (3.10)–(3.11)
[9], the physical degrees of freedom of φm1···ms(x) are
particle states of spin s, s −2, s −4, . . . , 1 or 0 (de-
pending whether s is even or odd).
A questionwhich can nowbe addressed is whether

the triplet fields may have some geometrical nature?

4 Geometrical meaning of HS
triplet fields. Frame-like
formulation

Higher Spin Field Theory is a gauge theory. Un-
derstanding its underlying group-theoretical and ge-
ometrical structure is of great importance for making
progress in solving the higher-spin interaction prob-
lem. Understanding the geometrical nature of the
triplets is part of this generic problem. As far as
the physical field φm1···ms(x) is concerned, its group-
theoretical properties resemble very much the met-
ric field of General Relativity. The gravitational field
gmn(x) is symmetric and it transformsunder linearized
diffeomorphisms as a gauge field

δgmn = ∂m ξn(x)+ ∂n ξm(x),

which is similar to the transformationproperties (3.11)
of φm1···ms(x).
What about auxiliary fields Cs−1(x) and Ds−2(x)?

Can they be related to other geometrical quantities,
such as a generalizedChristoffel symbol, i.e. a higher-
spin connection associated with the higher-spin local
symmetry? The answer to this question is positive.
The geometrical nature of the triplet fields manifests
itself rather clearly in the frame-like formulation [11]
which is the generalization to higher-spinfields [25, 26]
of the Cartan-Tamm-Weyl formulation of gravity in
terms of vielbeins and spin connections.
In the frame formulation of gravity the gravita-

tional field is described by a vielbein one-form

ea = dxm em
a(x), (4.12)

which carries the tangent space Lorentz index. In this
formulation, in addition to the local diffeomorphisms,
the theoryof gravitypossesses local Lorentz symmetry

δ ea(x) = eb ξb
a(x) , (4.13)

where ξab = −ξba is a parameter of local Lorentz
transformation. The gauge field associated with local
Lorentz symmetry is the spin connection

ωab = dxm ωabm(x)= −ω
ba, (4.14)

which transforms under the infinitesimal local Lorentz
transformations as follows

δ ωab = d ξab − ξac ωcb + ξbc ωca. (4.15)

The metric field gmn is composed of the vielbeins

gmn = e
a
m e

b
n ηab. (4.16)

In the linear approximation, in which em
a = δam +

ẽm
a(x) and gmn = ηmn + g̃mn(x), eq. (4.16) reduces

to
g̃ab = ẽab + ẽba . (4.17)

Note that in the linear approximation there is no dis-
tinction between world indices m, n, . . . and tangent
space indices a, b, . . .
The Einstein gravity is characterized by the so-

called torsion-free constraintwhich relates the vielbein
and spin connection

d ea + eb ωb
a =0 . (4.18)

By analogy with the frame formulation of gravity,
in the frame-like formulationofhigher-spinfield theory
[25, 26] one introduces ahigher-spinvielbein one-form,
which is symmetric in the s −1 tangent-space indices,

ea1···as−1 = dxm em
a1···as−1 (4.19)

and the higher-spin connection

ωa1···as−1,b = dxm ωm
a1···as−1,b (4.20)

which is symmetric in the indices a1 · · · as−1 and sat-
isfies the following properties: i) the symmetrization
of all of its tangent-space indices a and b brings zero

ω(a1···as−1,b) =0 , (4.21)

and ii) the trace of the index b with any of the indices
a is zero

ωa1···as−1,b ηa1b =0 . (4.22)

Theproperties (4.21)and(4.22)of thehigher-spincon-
nection are somewhat analogous to the antisymmetry
propertyof the spinconnection(4.20). Todescribe sin-
gle higher-spinfields, one should impose on thehigher-
spin vielbein and connection stronger trace-less condi-
tions (see [12, 11] for a review and references)

ea1a2···as−1 ηa1a2 = 0 , (4.23)

ωa1a2···as−1b, ηa1a2 = 0 .

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Acta Polytechnica Vol. 50 No. 3/2010

If eqs. (4.23) are satisfied, then the traceless condi-
tion (4.22) for ω follows from (4.23) and (4.21). As
has been shown in [11], the reducible triplet multi-
plets are described by e and ω which are not subject
to the traceless constraints (4.23) and ω satisfies a less
restrictive constraint (4.22).
As in the case of Einstein gravity, we also assume

that the higher-spin vielbein and connection satisfy
the torsion-free condition

d ea1···as−1 − dxm ωa1···as−1,bηmb =0 . (4.24)

The torsion-free condition is preserved under the fol-
lowingvariationof e and ω, which are local gauge sym-
metries of the frame-like formulation

δea1···as−1 =

d ξa1···as−1(x)− (s −1)ξa1···as−1,b(x)ηmb,
δωa1···as−1,b = (4.25)

d ξa1···as−1,b(x)− (s −2)ξa1···as−1,bc(x)ηmc ,

where the parameter ξa1···as−1 is symmetric, the pa-
rameter ξa1···as−1,b(x) has the same symmetry proper-
ties as ω and theparameter ξa1···as−1,bc is symmetric in
the indices a1 · · · as−1 and (separately) in the indices
bc. In addition, the symmetrization in ξa1···as−1,bc of
either b, or c with all a gives zero, and the trace of b
or c with any a also gives zero.
The higher-spin triplet fields (3.9) turn out to be

components of the higher-spin vielbein and connec-
tion [11]. The relation is as follows

φa1···as = em
(a1···as−1 ηas)m ,

Da1···as−2 = em
a1···as−2as−1 δmas−1 (4.26)

Ca1···as−1 = (s −1)ωa1···as−1,mm + ∂
m em

a1···as−1 .

The relation (4.26) between the two sets of fields has
been established in [11] by comparing their gauge
transformations and equations ofmotion in the frame-
like and metric-like formulation. In the frame-like
formulation, for the higher-spin vielbein and connec-
tion the equations of motion are the torsion-free con-
dition (4.24), which allows one to express the com-
ponents of ω in terms of derivatives of e, up to the
Stueckelberg gauge transformations with the param-
eter ξa1···as−1,bc(x) of eq. (4.25), and the following
differential equation on ω

δm(b ∂
cωd;n1···ns−2)[c,d] + ∂d ω(b;n1···ns−2)

[m,d] +

∂(b ω
d;

n1···ns−2)[d,
m] =0 , (4.27)

where the indices separated by ‘;’ are world indices,
i.e. they correspond to the index m in (4.20). This
equation is the dynamical field equation of the higher-
spin vielbein, when ω is expressed in terms of ∂ e in
virtue of the torsion-free conditions.
The equations of motion (4.24) and (4.27) can be

obtained from the action which generalizes to higher-
spin fields the action for gravity in the frame formula-
tion. In any dimension D > 1 the gravity action can

be written as an integral over the differential form

S[ea, ωab] =
1
2

∫
MD

ea1 . . . eaD−2εa1...aD−2bc R
bc =

(4.28)
1
2

∫
MD

ea1 . . . eaD−2εa1...aD−2bc (d ω
bc + ωbd ω

dc) ,

or up to a total derivative

S[ea, ωab] =
1
2

∫
MD

εa1...aD−3dbc e
a1 . . . eaD−3 ·(

(D −2)ded ωbc + ed ωbf ωf c
)
=

(4.29)D −2
2

∫
MD

εa1...aD−3bcd e
a1 . . . eaD−3 ·

(deb ωcd −
1
2

ef ωbf)ω
cd ,

where the wedge product of the differential forms is
assumed.
Thehigher-spingeneralizationof the linearizedgravity
action (4.29) in Minkowski space is

S =
∫

M D
dxa1 . . . dxaD−3 εa1...aD−3pqr · (4.30)

(d en1...ns−2p −
s −1
2

dxm ω
n1...ns−2p, m)ωn1...ns−2

q, r .

This action is a straightforward generalization of the
four-dimensional action of [25]. The torsion-free con-
dition (4.24) and the dynamical equation (4.27) are
obtained by varying eq. (4.30) with respect to ω and
e, respectively.
Let us recall that if in equations (4.24), (4.27) and

(4.30) thehigher-spinvielbeinandconnection together
with corresponding gauge symmetry parameters were
subject to the additional traceless condition in their
symmetric target-space indices (4.23), the frame-like
formulation based on the action (4.30) would describe
irreducible physical states corresponding to massless
particles of single spin s (see [11] for more details).
It is a certain relaxation of the trace constraints on
the higher-spin vielbeins and connections which ex-
tended the irreducible higher-spin system to the re-
ducible (triplet) higher-spin multiplet. It should also
be noted that because of the peculiarity of the form
of the frame-like action (4.30), which is constructed of
the wedge product of one-forms ω and (the differen-
tial) of e, it describes the higher-spin triplets whose
physical states have spins s, s − 1, . . . , 3 or 2. The
scalar and the vector fields can nevertheless be added
to this construction in a conventionalway (see [12, 11]
for more discussion of this point).
An analogous formulation has been constructed in

[11] for the fermionic higher-spin triplets describing
physical states of spin s, s − 1, . . . ,3/2. As in the
bosonic case regarding the scalar and the vector states

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Acta Polytechnica Vol. 50 No. 3/2010

of the triplets, the field of spin 1/2 can be included
into this construction as an independent field.
The action for the fermionic triplets is of the first

order in derivatives of the fermionic one-form

ψa1···as−1/2,α = dxm ψ
a1···as−1/2,α
m , (4.31)

where α is the spinorial index and s is now a half-
integer. The field ψ is symmetric in the indices a. In
a D-dimensional flat space-time the fermionic triplet
action in the frame-like formulation has a very simple
form

S = i
∫

M D
dxa1 . . .dxaD−3 εa1...aD−3pqr ·

(
ψ̄d1...ds−3

2

γpqr dψ
d1...ds−3

2 − (4.32)

6

(
s −
3
2

)
ψ̄d1...ds−5

2

pγq d ψ
d1...ds−5

2
r

)
,

It is invariant under the following local transforma-
tions of the field ψ

δψa1...as− 3
2
= (4.33)

dξa1...as− 3
2
−

(
s −
3
2

)
dxb ξa1...as− 3

2
,b

with the tensor-spinor parameter ξαa1...as− 3
2

,b having

the following symmetry properties

ξa1...as− 3
2

,b = ξ(a1...as− 3
2
) ,b , (4.34)

ξ(a1...as− 3
2

,b) = 0 ,

andbeing subject to the (gamma)-tracelessconstraints
with respect to the index b

γb ξ
a1...as−3

2
,b
=0 ξ

a1...as− 5
2

b,
b =0 . (4.35)

If the fermionic one-form ψ and the fermionic pa-
rameters of the gauge transformations (4.33) were (in
addition) gamma-traceless in the indices a, the action
(4.32)would describe a single (irreducible) higher-spin
field of spin s.
The frame-like formulation of the bosonic and

fermionic higher-spin triplet fields considered above
can be generalized to describe these fields in the anti-
deSitter space [11]. Inparticular, theuse of the frame-
like formulation has allowed us to overcome technical
difficulties encountered in [9, 27] and to obtain in [11]
a description of the fermionic higher-spin triplets in
AdS.This is afirst step towards the studyof consistent
interactions of fermionic triplets, since, as has been
known for a long time, a consistent theory of interact-
ingmassless higher-spin fields should be formulated in
a space-time background of constant curvature, like
the AdS space.

5 Conclusion
We have seen how a triplet system of fields of spin
s, s − 2 . . . ,1 or 0 appears in a truncated action for
StringFieldTheory. Using the frame-like formulation,
we endowed these fields with a geometrical meaning
of higher-spin vielbeins and connections subject to a
torsion-free condition and transforming under higher-
spin local symmetries. The frame-like actions for the
bosonic and fermionic triplet fields have been con-
structed in flat and AdS spaces. It is of great interest
and importance to generalize these results by analyz-
ingpossible interactionsof thehigher-spin tripletfields
(see [17, 28] for a recent discussion of this issue and
references). This would give a new insight into the
structure of Higher Spin Field Theory, String Theory
and AdS/CFT correspondence.

Acknowledgement

This work was partially supported by the INFN Spe-
cial Initiative TV12, the INTAS Project Grant 05-
1000008-7928, and anExcellence Grant of Fondazione
Cariparo.

References

[1] Kotecky, R., Niederle, J.: Conformally Covariant
Field Equations II. First Order Massless Equa-
tions, Rept. Math. Phys. 12 (1977) 237–249.

[2] Niederle, J., Nikitin, A. G.: Relativistic wave
equations for interacting massive particles with
arbitrary half-integer spins, Phys. Rev. D64
(2001) 125013.

[3] Niederle, J., Nikitin, A. G.: Relativistic Coulomb
problem for particles with arbitrary half-integer
spin, J. Phys. A39 (2006) 10931–10944,
arXiv:hep-th/0412214.

[4] Ouvry, S., Stern, J.: Gauge fields of any spin and
symmetry, Phys. Lett. B177 (1986) 335.

[5] Bengtsson, A. K. H.: A unified action for higher
spin gauge bosons from covariant string theory,
Phys. Lett. B182 (1986) 321.

[6] Henneaux, M., Teitelboim, C.: First and second
quantized point particles of any spin. In Santiago
1987, Proceedings, Quantum mechanics of fun-
damental systems 2 113–152. (see Conference In-
dex).

[7] Pashnev, A. I.: Composite systems and field the-
ory for a free Regge trajectory, Theor. Math.
Phys. 78 (1989) 272–277.

[8] Francia, D., Sagnotti, A.: On the geometry of
higher-spin gauge fields, Class. Quant. Grav. 20
(2003) S473–S486.

52



Acta Polytechnica Vol. 50 No. 3/2010

[9] Sagnotti, A., Tsulaia, M.: On higher spins and
the tensionless limit of string theory, Nucl. Phys.
B682 (2004) 83–116.

[10] Barnich, G., Bonelli, G., Grigoriev, M.: From
BRST to light-cone description of higher spin
gauge fields, arXiv:hep-th/0502232.

[11] Sorokin,D. P., Vasiliev,M.A.: Reducible higher-
spin multiplets in flat and AdS spaces and their
geometric frame-like formulation, Nucl. Phys.
B809 (2009) 110–157.

[12] Bekaert, X., Cnockaert, S., Iazeolla, C., Vasi-
liev,M.A.: Nonlinearhigher spin theories in vari-
ous dimensions, arXiv:hep-th/0503128.

[13] Vasiliev, M. A.: Higher spin gauge theories in
various dimensions, Fortsch. Phys. 52 (2004)
702–717.

[14] Sorokin, D.: Introduction to the classical the-
ory of higher spins, AIP Conf. Proc. 767 (2005)
172–202, arXiv:hep-th/0405069.

[15] Bouatta, N., Compere, G., Sagnotti, A.: An in-
troduction to free higher-spin fields,
arXiv:hep-th/0409068.

[16] Sagnotti, A., Sezgin, E., Sundell, P.: On
higher spins with a strong Sp(2,R) condition,
arXiv:hep-th/0501156.

[17] Fotopoulos, A., Tsulaia,M.: Gauge InvariantLa-
grangians for Free and Interacting Higher Spin
Fields. A Review of the BRST formulation, Int.
J. Mod. Phys. A24 (2009) 1–60.

[18] Vasiliev, M. A.: Equations of motion of interact-
ingmassless fields of all spins as a free differential
algebra, Phys. Lett. B209 (1988) 491–497.

[19] Vasiliev, M. A.: Consistent equations for inter-
actingmassless fields of all spins in the first order
in curvatures, Annals Phys. 190 (1989) 59–106.

[20] Gross, D. J.: High-Energy Symmetries of String
Theory, Phys. Rev. Lett. 60 (1988) 1229.

[21] Sundborg, B.: Stringy gravity, interacting ten-
sionless strings and massless higher spins, Nucl.
Phys. Proc. Suppl. 102 (2001) 113–119.

[22] Lindstrom, U., Zabzine, M.: Tensionless strings,
WZWmodels at critical level andmassless higher
spin fields, Phys. Lett. B584 (2004) 178–185.

[23] Bonelli, G.: On the tensionless limit of bosonic
strings, infinite symmetries and higher spins,
Nucl. Phys. B669 (2003) 159–172.

[24] Witten, E.: InteractingFieldTheory ofOpen Su-
perstrings, Nucl. Phys. B276 (1986) 291.

[25] Vasiliev, M. A.: ‘Gauge’ form of description of
massless fields with arbitrary spin. (in Russian),
Yad. Fiz. 32 (1980) 855–861.

[26] Aragone, C., Deser, S.: Higher spin vierbein
gauge fermions and hypergravities, Nucl. Phys.
B170 (1980) 329.

[27] Buchbinder, I. L., Galajinsky, A. V., Krykh-
tin, V. A.: Quartet unconstrained formulation
for massless higher spin fields, Nucl. Phys. B779
(2007) 155–177.

[28] Fotopoulos, A., Tsulaia, M.: Current Exchanges
for Reducible Higher Spin Multiplets and Gauge
Fixing, JHEP 10 (2009) 050.

Dmitri Sorokin
E-mail: dmitri.sorokin@pd.infn.it
INFN, Sezione di Padova
via F. Marzolo 8
35131 Padova, Italia

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