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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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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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