On Higher Spins On Higher Spins Part IIPart II

Augusto SAGNOTTIAugusto SAGNOTTIUniversita’ Universita’ didi Roma “Tor Vergata”Roma “Tor Vergata”

(Based on:(Based on: Francia, AS, hep-th/0207002, hep-th/0212185;AS, Tsulaia, hep-th/0311257; AS, Sezgin,Sundell, to appear)

Modern Trands in String Theory IIPorto, June 21-26 2004

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SummarySummary

Part I:Part I:From Fierz-Pauli to FronsdalFree (non-local) geometric equations

Part II:Part II:Triplets and (local) compensator formLinks with the Vasiliev equations

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Geometric equationsGeometric equations1.1. Odd spins (s=2n+1):Odd spins (s=2n+1):

1[ ] ; ...1 0sn v vn R µ

µ∂ =0F µνµ∂ =

2.2. EvenEven spins (s=2n):spins (s=2n):

1[ ] ; ...1

1 0sn v vn R− =0Rµν =

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FermionsFermions

12

12s

S +

∂− ∂ S

12

s

i +

∂=

( )sFNotice:

Example: spin 3/2 (Rarita-Schwinger)

S iµ ≡ ∂ µ µψ ψ− ∂( ) 0=0µνρν ργ ψ∂ =

12

S µµ

∂− ∂ S i ∂

= vν

µν µη ψ −∂ ∂

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FermionsFermionsOne can again iterate:

The relation to bosons generalizes to:

The Bianchi identity generalizes to:

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BosonicBosonic string: BRSTstring: BRSTThe starting point is the Virasoro algebra:

In the tensionless limit, one is left with:

Virasoro contracts (no c. charge):

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String Field equationString Field equationHigher-spin massive modes:

• massless for 1/α’ 0• Free dynamics can be encoded in:

20 ( 0)Q Q

Q

ψ

δ ψ

= =

= Λ

(Kato and Ogawa, 1982)(Witten, 1985)(Neveu, West et al, 1985)

NONO trace constraints on ϕ or Λ

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LowLow--tension limittension limitSimilar simplifications for BRST charge:

Making zero-modes manifest:

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Symmetric tripletsSymmetric triplets(A. Bengtsson, 1986)(Henneaux,Teitelboim, 1987)(Pashnev, Tsulaia, 1998)(Francia, AS, 2002)Emerge from

The triplets are:The triplets are:

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Symmetric tripletsSymmetric triplets

Can also eliminate C:

Gauge theories of

Physical state conditions:Physical state conditions:Propagate spins s,s-2, …, 0 or 1

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((A)dSA)dS symmetric tripletssymmetric tripletsCan build directly, deforming flat-space triplets, or via BRST

Directly:Directly: insist on relation between C and othersBRST: BRST: gauge non-linear constraint algebra

Basic commutator:

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((A)dSA)dS symmetric tripletssymmetric tripletsCan also deform directly the equations without C:

It is convenient to define

Bianchi identity and first equation then become:

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((A)dSA)dS symmetric tripletssymmetric tripletsThe deformed equations can be derived from

Alternatively: modify momenta in contracted Virasorogauge algebra non linearnon linear (but M,N diagonal on triplets)

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Compensator EquationsCompensator Equations

In the triplet:

spin-(s-3) compensatorcompensator:

The first becomes:

The second becomes:

Combining them:

Finally (also from Bianchi):

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Compensator EquationsCompensator Equations

Summarizing:

Describe a spin-s gauge field with:NONO trace constraints on the gauge parameterNONO trace constraints on the gauge fieldFirst can be reduced to minimal non-local form

BUT:BUT:NOTNOT Lagrangian equations

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((A)dSA)dS Compensator Compensator EqsEqsFlat-space compensator equations can be extended to (A)dS:

Gauge invariant under

The first can be turned into the second via (A)dS Bianchi

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Compensator EquationsCompensator Equations

It is possible to obtain a Lagrangian form of the compensatorequations, using BRST techniquesEssentially in Essentially in PashnevPashnev and and TsulaiaTsulaia (1997)(1997)Formulation involves number of fields ~ sInteresting BRST subtletiesHere we discuss explicitly spin s=3

Fields:

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Compensator EquationsCompensator Equations

Gauge transformations:

Field equations:

Gauge fixing:

Other extra fields: zero by field equations

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FermionicFermionic TripletsTriplets(Francia and AS, 2002)

Counterparts of bosonic triplets

GSO:GSO: not in 10D susy strings

Yes:Yes: mixed sym generalizations

Enter directly typetype--0 models0 models

Propagate s+1/2 and allall lower ½-integer spins

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FermionicFermionic CompensatorsCompensators

Recall:

Spin-(s-2) compensator:

Gauge transformations:

First compensator equation second via Bianchi

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FermionicFermionic CompensatorsCompensators

We couldcould extend the fermionic compensator eqs to (A)dSWe could notcould not extend the fermionic tripletsBRST:BRST: operator extension does not define a closed algebra

First compensator equation second via (A)dS Bianchi identity:

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The The VasilievVasiliev equationsequations(Vasiliev, 1991-2003;Sezgin,Sundell, 1998-2003)

IntegrableIntegrable curvature constraintscurvature constraints on one-forms and zero-formsConceptual basis:Conceptual basis: Cartan integrable systemsKey new addition of Key new addition of VasilievVasiliev:: twisted-adjoint representation

(D’Auria,Fre’, 1983)

Minimal caseMinimal case (only symmetric tensors of even rank)(only symmetric tensors of even rank):zerozero--form form ΦΦ :: Weyl curvaturesoneone--form A :form A : gauge fields

gauge generators oscillator star-algebra Sp(2,R)Sp(2,R)

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The The VasilievVasiliev equationsequations

Curvature constraints:Curvature constraints:

[extra [extra non comm.non comm. CoordsCoords]]

Gauge symmetry:Gauge symmetry:

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The The VasilievVasiliev equationsequations(AS,Sezgin,Sundell, to appear)

“Off-shell”: definitions of Riemann-like curvatures“On-shell”: Riemann-like = Wey-like l Ricci-like = 0

What is the role of Sp(2,R) in this transition?

Sp(2,R) generators:

Key on-shell constraint:

gauge fields NOT constrainedNOT constrained

The strong constraint ensures that the proper scalar massesproper scalar masses emergeAt the interaction level must regulate projectorregulate projector

Gauge fields: extended (Gauge fields: extended (uncostraineduncostrained) gauge symmetry) gauge symmetry

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The The VasilievVasiliev equationsequationsNonNon--linear corrections:linear corrections: from dependence on extra coordinates

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Some ReferencesSome ReferencesTriplets and related matters:Triplets and related matters:A. K. A. K. Bengtsson,``ABengtsson,``A Unified Action For Higher Spin Gauge Bosons From Covariant Unified Action For Higher Spin Gauge Bosons From Covariant String String Theory,''PhysTheory,''Phys. . LettLett. B182 (1986) 321; M. . B182 (1986) 321; M. HenneauxHenneaux and C. and C. TeitelboimTeitelboim, in , in ``Quantum Mechanics of Fundamental Systems, 2'',eds. C. ``Quantum Mechanics of Fundamental Systems, 2'',eds. C. TeitelboimTeitelboim and J. and J. ZanelliZanelli(Plenum Press, New York, 1988), p. 113; (Plenum Press, New York, 1988), p. 113; A. A. PashnevPashnev and M. and M. TsulaiaTsulaia,``Dimensional ,``Dimensional reduction and BRST approach to the description of a Regge trajereduction and BRST approach to the description of a Regge trajectory,''ctory,''Mod.Mod. PhysPhys. . Lett.Lett. A12 (1997) 861 [A12 (1997) 861 [arXivarXiv::hephep--thth/9703010]; C. /9703010]; C. BurdikBurdik, A. , A. PashnevPashnev and M. and M. TsulaiaTsulaia,``The Lagrangian description of representations of the Poincare,``The Lagrangian description of representations of the Poincare group,''group,''NuclNucl. . PhysPhys. . ProcProc. . Suppl.Suppl. 102 (2001) 285[102 (2001) 285[arXivarXiv::hephep--thth/0103143]./0103143].

Irreducible sets:Irreducible sets:A. A. PashnevPashnev and M. and M. Tsulaia,``DescriptionTsulaia,``Description of the higher massless irreducible integer of the higher massless irreducible integer spins in the BRST spins in the BRST approach,''Modapproach,''Mod. Phys. . Phys. LettLett. A13 (1998) 1853 [arXiv:hep. A13 (1998) 1853 [arXiv:hep--th/9803207].th/9803207].

BRST in SFT:BRST in SFT:M. M. KatoKato and K. and K. OgawaOgawa,``Covariant Quantization Of String Based On Brs ,``Covariant Quantization Of String Based On Brs Invariance,''Invariance,''NuclNucl. . PhysPhys. B212 (1983) 443; A. . B212 (1983) 443; A. NeveuNeveu, , H.H. NicolaiNicolai and P.C. and P.C. West,``Gauge Covariant Local Formulation Of Free Strings And SupWest,``Gauge Covariant Local Formulation Of Free Strings And Superstrings,''erstrings,''NuclNucl. . PhysPhys. B264 (1986) 573;A. . B264 (1986) 573;A. NeveuNeveu and P.C. West,``Gauge Covariant Local Formulation and P.C. West,``Gauge Covariant Local Formulation Of Of BosonicBosonic Strings,''NuclStrings,''Nucl. Phys. B268 (1986) 125; E. . Phys. B268 (1986) 125; E. WittenWitten, ``, ``NoncommutativeNoncommutativeGeometry And String Field Geometry And String Field Theory,''NuclTheory,''Nucl. Phys. B268 (1986) 253.. Phys. B268 (1986) 253.

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Some ReferencesSome References

TensionlessTensionless limits (limits (recentrecent):):U. Lindstrom and M. Zabzine,``Tensionless strings, WZW models atU. Lindstrom and M. Zabzine,``Tensionless strings, WZW models at critical level and massless critical level and massless higher spin fields,''arXiv:hephigher spin fields,''arXiv:hep--th/0305098;G. Bonelli,``On the tensionless limit of bosonic strith/0305098;G. Bonelli,``On the tensionless limit of bosonic strings, ngs, infinite symmetries and higher spins,''Nucl. Phys. B669 (2003) infinite symmetries and higher spins,''Nucl. Phys. B669 (2003) 159 [arXiv:hep159 [arXiv:hep--th/0305155], ``On th/0305155], ``On the covariant quantization of tensionless bosonic strings in AdSthe covariant quantization of tensionless bosonic strings in AdS spacetime,''JHEP 0311 (2003) 028 spacetime,''JHEP 0311 (2003) 028 [arXiv:hep[arXiv:hep--th/0309222].th/0309222].

Tensionless limits (previous):Tensionless limits (previous):A. Schild,``Classical Null Strings,''A. Schild,``Classical Null Strings,''PhysPhys. . Rev.Rev. D16} (1977) 1722; A. D16} (1977) 1722; A. KarlhedeKarlhede and U. and U. LindstromLindstrom, , ``The Classical Bosonic String In The Zero Tension Limit,''Class``The Classical Bosonic String In The Zero Tension Limit,''Class. . QuantQuant. . GravGrav. 3, (1986) L73; . 3, (1986) L73; H.H.GustafssonGustafsson, U. , U. LindstromLindstrom, P. , P. SaltsidisSaltsidis, B. , B. SundborgSundborg and R. and R. vonvon Unge,``Hamiltonian BRST Unge,``Hamiltonian BRST quantization of the conformal string,''quantization of the conformal string,''NuclNucl. . PhysPhys. B440, (1995) 495;[. B440, (1995) 495;[arXivarXiv::hephep--thth/9410143]; J. /9410143]; J. IsbergIsberg, U. , U. LindstromLindstrom, B. , B. SundborgSundborg, G. , G. TheodoridisTheodoridis,`` Classical and quantized tensionless ,`` Classical and quantized tensionless strings,''Nucl. strings,''Nucl. PhysPhys. B411 (1994) 122.[. B411 (1994) 122.[arXivarXiv::hephep--thth/9307108]./9307108].

VasilievVasiliev equations:equations:MA.Vasiliev, ``Nonlinear equations for symmetric massless higher spin fields in (A)dS(d),'' Phys. Lett. B567 (2003) 139 [hep-th/0304049],``More on equations of motion for interacting masslessfields of all spins in (3+1)-dimensions,'' Phys. Lett. B285, 225 (1992), ``Consistent Equation ForInteracting Gauge Fields Of All Spins In (3+1)-Dimensions,'' Phys. Lett. B243, 378 (1990); E. Sezgin and P. Sundell, ``Analysis of higher spin field equations in four dimensions,'' JHEP 0207, 055 (2002) [hep-th/0205132].