Free geometric equations for higher spins

of 14

Please download to get full document.

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
14 pages
0 downs
We show how allowing non-local 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 ROM2F-02/18 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 non-local 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 curva-tures 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 higher-spin gauge theory, a viewpoint clearly suggested, for instance, by the BRST for-mulation of free String Field Theory, that encodes infinitely many higher-spin symmetriesin the Stueckelberg mode [1]. However, String Theory presents some clear simplificationswith respect to unbroken higher-spin theories, well reflected in the familiar option of associ-ating to large-scale phenomena a low-spin low-energy 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 electro-weak theory reduces to a low-spin 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 tensor-spinors were first con-structed 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 higher-spin 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 flat-space 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 atclosed-form 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 higher-spin gauge transformations dependingon the space-time 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 Fang-Fronsdal 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 tensor-spinors while foregoing the restrictions implicit in the Fang-Fronsdal equations. One can well work in generic space-time 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 non-local 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 non-local 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 Fang-Fronsdal form. This analysis will bring us naturally toconsider, following de Wit and Freedman [11], higher-spin 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 low-order traces will be occasion-ally 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 higher-rank tensors. Let us stress thatall these equations are manifestly invariant under gauge transformations without any con-straints 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 Rarita-Schwinger 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 corre-sponding 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 com-bining 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 gauge-invariant, albeit non-local, forms, and howto relate the latter to local forms involving Stueckelberg fields with higher-derivative 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 corre-sponding Einstein-like tensors. In section 4 we recover these equations from the geometricnotions of connection and curvature for higher-spin gauge fields, srcinally introduced by deWit and Freedman [11]. While in [11] the authors linked the local Fang-Fronsdal equations to traces of one and two-derivative 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 corre-sponding curvature, that for a spin- s field contain, respectively, s − 1 and s derivatives.Whereas unconventional, these are natural ingredients of higher-spin kinetic operators, thatin general should contain both the D’Alembertian operator 2 and additional terms withup to s free derivatives. Hence, the non-local structure exposed here is unavoidable inour fully covariant setting. In addition, it anticipates similar properties of the higher-spininteractions. 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 world-sheetreparametrizations, that lies at the heart of String Field Theory, embodies a massive dy-namics 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 spin-3 case Let us begin by describing the spin-3 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 spin-3 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 spin-2 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 gauge-invariant La-grangian 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)
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks