We show how allowing nonlocal terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann
a r X i v : h e p  t h / 0 2 0 7 0 0 2 v 2 3 1 J u l 2 0 0 2
ROM2F02/18
Free geometric equations for higher spins
Dario Francia
†
and Augusto Sagnotti
Dipartimento di Fisica Universit`a di Roma “Tor Vergata”INFN, Sezione di Roma “Tor Vergata”Via della Ricerca Scientiﬁca 1, 00133 Roma, Italy
Abstract
We show how allowing nonlocal terms in the ﬁeld equations of symmetric tensorsuncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases.The end results can be related to multiple traces of the generalized Riemann curvatures
R
α
1
···
α
s
;
β
1
···
β
s
introduced by de Wit and Freedman, divided by suitable powersof the D’Alembertian operator
2
. The conventional local equations can be recoveredby a partial gauge ﬁxing involving the trace of the gauge parameters Λ
α
1
···
α
s
−
1
, absentin the Fronsdal formulation. The same geometry underlies the fermionic equations,that, for all spins
s
+ 1
/
2, can be linked via the operator
∂
2
to those of the spin
s
bosons.
( June, 2002 )
†
I.N.F.N. Fellow.
–2–
1. Introduction and summary
String Theory has long been held by several authors to correspond to a broken phase of a higherspin gauge theory, a viewpoint clearly suggested, for instance, by the BRST formulation of free String Field Theory, that encodes inﬁnitely many higherspin symmetriesin the Stueckelberg mode [1]. However, String Theory presents some clear simpliﬁcationswith respect to unbroken higherspin theories, well reﬂected in the familiar option of associating to largescale phenomena a lowspin lowenergy eﬀective description. This is a generalfeature of spontaneously broken gauge theories, quite familiar from simpler examples: forinstance, diﬀerently from Q.C.D., at low energies the electroweak theory reduces to a lowspin theory with a local Fermi coupling, that for many years has been at the heart of weakinteraction phenomenology. On the other hand, it is in Q.C.D. that gauge theory comes tofull power, with remarkable infrared phenomena responsible for quark conﬁnement. Evenmore striking dynamics can thus be expected from these complicated systems, and this isby itself an important motivation to try to gain some familiarity with them.Free covariant equations for fully symmetric tensors and tensorspinors were ﬁrst constructed in the late seventies by Fronsdal [2] and Fang and Fronsdal [3], starting from the
massive equations of Singh and Hagen [4]. These are interesting classes of higherspin gaugeﬁelds, that in four dimensions exhaust all available possibilities, up to dualities, and havethe clear advantage of allowing rather simple uniﬁed descriptions. Following an importantobservation of the G¨oteborg group [5], that showed how a proper cubic ﬂatspace vertex
could be found for higher spins, Fradkin and Vasiliev [6] have led for many years the searchfor an extension of the free equations to consistent interacting gauge theories of higher spins.Arguments related to the gauge algebra imply that these are bound to involve inﬁnitelymany gauge ﬁelds of increasing spins, and in the early nineties Vasiliev ﬁnally arrived atclosedform dynamical equations for symmetric tensors
φ
µ
1
···
µ
s
of arbitrary rank in mutualinteraction [7], but an action principle is still lacking for this complicated system. A crucialinput in the constructions of [6,7] was the inclusion of a cosmological term, that allowed to
cancel recursively contributions generated by higherspin gauge transformations dependingon the spacetime Weyl tensor, thus bypassing the diﬃculties met in earlier attempts [8].Various aspects of the work of Vasiliev and collaborators are reviewed in [9], while recent,related work is described in [10].A peculiar feature of the FangFronsdal equations is the need for unusual constraints, sothat, for instance, the bosonic gauge parameters are to be traceless, while the correspondinggauge ﬁelds are to be doubly traceless. These constraints manifest themselves as symmetryconditions in the spinor formalism of [6,7], but appear less natural in the usual component
–3–notation
1
. This letter is thus devoted to showing how one can formulate the dynamics of symmetric tensors and tensorspinors while foregoing the restrictions implicit in the FangFronsdal equations. One can well work in generic spacetime dimensions, with the provisothat for
d >
4 these ﬁelds do not exhaust all available possibilities. The end result is ratheramusing, since
the free equations contain nonlocal terms
whenever the gauge ﬁelds havemore than a pair of symmetric Lorentz indices,
i.e.
in all cases beyond the familiar Maxwelland Einstein examples. However, all nonlocal terms can be eliminated by a partial gaugeﬁxing using the trace (or, for fermions, the
γ
trace) of the gauge parameter, that reducesthe geometric equations to the FangFronsdal form. This analysis will bring us naturally toconsider, following de Wit and Freedman [11], higherspin generalizations of the Christoﬀelconnection, Γ
α
1
···
α
s
−
1
;
β
1
···
β
s
, and of the Riemann curvature,
R
α
1
···
α
s
;
β
1
···
β
s
, that are totallysymmetric under the interchange of any pair of indices within the two sets. In terms of these quantities, the gauge invariant bosonic ﬁeld equations will be1
2
p
∂
· R
[
p
];
α
1
···
α
2
p
+1
= 0 (1)for odd spins
s
= 2
p
+ 1, and1
2
p
−
1
R
[
p
];
α
1
···
α
2
p
= 0 (2)for even spins
s
= 2
p
. Here and in the following, a superscript [
p
] denotes a
p
fold trace,while
∂
·
denotes a divergence, but for the sake of brevity loworder traces will be occasionally denoted by “primes”. Moreover, we shall work throughout with a “mostly positive”Minkowski metric.The analogy with the Maxwell and Einstein cases should be evident, and it is ratherpleasing to see a simple pattern extending to all higherrank tensors. Let us stress thatall these equations are manifestly invariant under gauge transformations without any constraints on the gauge ﬁelds or on the corresponding gauge parameters and that, after apartial gauge ﬁxing, they can all be reduced to the conventional, local, Fronsdal form.This geometric form also results in fermionic equations that are closely related to thebosonic ones. In general, the spin(
s
+ 1
/
2) fermionic equations can be formally recoveredfrom the spin
s
bosonic operators, properly multiplied by
∂
2
, and therefore the geometryunderlying the bosonic equations plays a similar, albeit more indirect, role also in thefermionic case. It is amusing to illustrate right away this fact, obvious for the Diracequation, for a less evident case, the RaritaSchwinger equation for spin 3/2, that is quitefamiliar from supergravity. This is usually written in the form
γ
µνρ
∂
ν
ψ
ρ
= 0
,
(3)
1
The double trace condition, however, can be related to the OSp(
D
−
1
,
1

2) structure of the corresponding system with ghosts. We are grateful to W. Siegel for calling this fact to our attention.
–4–but once combined with its
γ
trace, it becomes
∂ ψ
µ
−
∂
µ
ψ
= 0
.
(4)The connection with the Maxwell equation that we are advertising can be exhibited combining it again with its
γ
trace, now multiplied with
∂
µ
∂
2
, and the end result is indeed
∂
2
(
2
ψ
µ
−
∂
µ
∂
·
ψ
) = 0
.
(5)In section 2 we begin by examining the ﬁeld equation for spin 3, and we show how toextend the Fronsdal formulation to fully gaugeinvariant, albeit nonlocal, forms, and howto relate the latter to local forms involving Stueckelberg ﬁelds with higherderivative terms.In section 3 we show how one can deﬁne via an iterative procedure kinetic operators for allhigher spins, derive their Bianchi identities and, making direct use of them, construct corresponding Einsteinlike tensors. In section 4 we recover these equations from the geometricnotions of connection and curvature for higherspin gauge ﬁelds, srcinally introduced by deWit and Freedman [11]. While in [11] the authors linked the local FangFronsdal equations
to traces of one and twoderivative connections, the full geometric equations presented hereare recovered if one insists on resorting to the connection Γ
α
1
···
α
s
−
1
;
β
1
···
β
s
and to the corresponding curvature, that for a spin
s
ﬁeld contain, respectively,
s
−
1 and
s
derivatives.Whereas unconventional, these are natural ingredients of higherspin kinetic operators, thatin general should contain both the D’Alembertian operator
2
and additional terms withup to
s
free derivatives. Hence, the nonlocal structure exposed here is unavoidable inour fully covariant setting. In addition, it anticipates similar properties of the higherspininteractions. It is conceivable, although by no means clear to the authors at the time of this writing, that corresponding simpliﬁcations could take place if the equations of [7] wereformulated along these lines. A related observation is that the BRST charge of worldsheetreparametrizations, that lies at the heart of String Field Theory, embodies a massive dynamics of the Fronsdal type, some aspects of which are manifest in the constructions of [12],that therefore bear a direct relationship to the present work, although the ﬁeld equationsare presented there in a local form with compensators that does not exhibit their link withthe curvatures.
2. The spin3 case
Let us begin by describing the spin3 Fronsdal equation [2], that for the sake of brevitywe shall write in the form
F
123
= 0
,
(6)
–5–where
F
123
≡
2
φ
123
−
(
∂
1
∂
·
φ
23
+
∂
2
∂
·
φ
13
+
∂
3
∂
·
φ
12
) +
∂
1
∂
2
φ
′
3
+
∂
1
∂
3
φ
′
2
+
∂
2
∂
3
φ
′
1
,
(7)exposing only the subscripts of the three Lorentz indices involved. A gauge transformationof the spin3 ﬁeld
φ
123
,
δφ
123
=
∂
1
Λ
23
+
∂
2
Λ
31
+
∂
3
Λ
12
,
(8)transforms
F
according to
δ
F
123
= 3
∂
1
∂
2
∂
3
Λ
′
,
(9)and therefore, as is well known,
F
is gauge invariant only if the parameter is subject to theconstraintΛ
′
= 0
.
(10)An additional subtlety, already met in the spin2 case, is that (6) does not follow directlyfrom a Lagrangian. In order to proceed, one must therefore introduce an analogue of thelinearized Einstein tensor,
G
123
=
F
123
−
12(
η
12
F
′
3
+
η
23
F
′
1
+
η
31
F
′
2
)
,
(11)where
η
denotes the Minkowski metric. The Bianchi identity
∂
·F
23
=12(
∂
2
F
′
3
+
∂
3
F
′
2
)
,
(12)then implies that
∂
·G
23
=
−
12
η
23
∂
·F
′
,
(13)and together with eq. (10) this result is instrumental in deriving a gaugeinvariant Lagrangian for this system, since
∂
·F
′
= 3
2
∂
·
φ
′
−
2
∂
·
∂
·
∂
·
φ
(14)does not vanish identically. Integrating
δ
L
=
δφ
123
F
123
(15)one can ﬁnally recover the Fronsdal action
L
=
−
12(
∂
µ
φ
123
)
2
+32(
∂
·
φ
12
)
2
+32(
∂
µ
φ
′
1
)
2
+34(
∂
·
φ
′
)
2
+ 3
φ
′
1
∂
·
∂
·
φ
1
.
(16)