An Introduction to Higher-Spin Fields An Introduction An Introduction to Higher to Higher - - Spin Fields Spin Fields Augusto SAGNOTTI Augusto SAGNOTTI Scuola Normale Superiore, Pisa Scuola Normale Superiore, Pisa Eotvos Superstring Workshop, Budapest, Sept. 2007 Eotvos Superstring Workshop, Budapest, Sept. 2007 Some reviews: Some reviews: N. Bouatta, G. Compere, A.S., hep N. Bouatta, G. Compere, A.S., hep - - th/0609068 th/0609068 D. Francia and A.S., hep D. Francia and A.S., hep - - th/0601199 th/0601199 A.S., Sezgin, Sundell, hep A.S., Sezgin, Sundell, hep - - th/0501156 th/0501156 X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep - - th/0503128 th/0503128 A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 200 A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 200 8 (?) 8 (?)
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An Introductionto Higher-Spin Fields
An IntroductionAn Introductionto Higherto Higher--Spin FieldsSpin Fields
Augusto SAGNOTTIAugusto SAGNOTTIScuola Normale Superiore, PisaScuola Normale Superiore, Pisa
Eotvos Superstring Workshop, Budapest, Sept. 2007Eotvos Superstring Workshop, Budapest, Sept. 2007
Some reviews:
N. Bouatta, G. Compere, A.S., hep-th/0609068 D. Francia and A.S., hep-th/0601199
A.S., Sezgin, Sundell, hep-th/0501156 X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep-th/0503128A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 2008 (?)
Some reviews:Some reviews:
N. Bouatta, G. Compere, A.S., hepN. Bouatta, G. Compere, A.S., hep--th/0609068 th/0609068 D. Francia and A.S., hepD. Francia and A.S., hep--th/0601199 th/0601199
A.S., Sezgin, Sundell, hepA.S., Sezgin, Sundell, hep--th/0501156 th/0501156 X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hepX. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep--th/0503128th/0503128A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 200A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 2008 (?) 8 (?)
Budapest, Sept. 2007Budapest, Sept. 2007 22
Some Motivations for HSSome Motivations for HSSome Motivations for HS
Key role in String Theory:Key role in String Theory:(Non) Planar duality of tree amplitudes(Non) Planar duality of tree amplitudes
Modular invariance and soft U.V.Modular invariance and soft U.V.OpenOpen--closed dualityclosed duality……………………………………
Key (old) problem inKey (old) problem in (classical)(classical) Field Theory:Field Theory:Only s=0,1/2,1,3/2,2Only s=0,1/2,1,3/2,2
Budapest, Sept. 2007Budapest, Sept. 2007 33
For instance …For instance For instance ……
(Non(Non--) planar duality) planar duality rests on infinitely many poles rests on infinitely many poles [Actual t (or s) dependence implies a growing sequence of spins[Actual t (or s) dependence implies a growing sequence of spins] ] Similarly for Similarly for modular invariancemodular invariance::
Budapest, Sept. 2007Budapest, Sept. 2007 44
What is “Spin” here ?What is What is ““SpinSpin”” here ?here ?
D=4 :D=4 :Up to dualities, all cases exhausted by fully symmetric (spinor)Up to dualities, all cases exhausted by fully symmetric (spinor)tensors:tensors:
D > 4 :D > 4 :Arbitrary Young tableaux : Arbitrary Young tableaux : ““spinspin”” somehow the number of columns. somehow the number of columns. Less developed, clumsy in general, many general lessons can be dLess developed, clumsy in general, many general lessons can be drawn rawn from the previous special set of fields.from the previous special set of fields.
(See X. Bekaert and N. Boulanger, hp(See X. Bekaert and N. Boulanger, hp--th/0606198 and seminar of A. Campoleoni)th/0606198 and seminar of A. Campoleoni)
FierzFierz--Pauli condition: the s=2 case in detailPauli condition: the s=2 case in detailSinghSingh--Hagen and Fronsdal formulations: Hagen and Fronsdal formulations: trace constraintstrace constraintsKaluzaKaluza--Klein massesKlein massesFangFang--Fronsdal formulation for Fermi fieldsFronsdal formulation for Fermi fields
Constrained vs unconstrained formulationsConstrained vs unconstrained formulationsNonNon--local bosonic formulation and Higherlocal bosonic formulation and Higher--Spin GeometrySpin GeometryNonNon--local fermionic constructionlocal fermionic construction
Relation with String TheoryRelation with String TheoryBRST formulation for the bosonic stringBRST formulation for the bosonic stringLowLow--tension limit and tripletstension limit and triplets
Compensator equations for Higher SpinsCompensator equations for Higher SpinsCompensator equations and minimal LagrangiansCompensator equations and minimal LagrangiansAn alternative offAn alternative off--shell formulationshell formulation
Lessons for the nonLessons for the non--local formulationlocal formulationVDVZ discontinuityVDVZ discontinuity
The Vasiliev construction (and the compensator)The Vasiliev construction (and the compensator)Problems with higherProblems with higher--spin interactionsspin interactionsThe Vasiliev construction and the compensatorThe Vasiliev construction and the compensator
Budapest, Sept. 2007Budapest, Sept. 2007 77
Lecture ILecture ILecture I
Free HigherFree Higher--Spin fieldsSpin fields
FierzFierz--Pauli condition: the s=2 case in detailPauli condition: the s=2 case in detailSinghSingh--Hagen and Fronsdal formulations, Hagen and Fronsdal formulations, trace constraintstrace constraintsKaluzaKaluza--Klein massesKlein massesFangFang--Fronsdal formulation for Fermi fieldsFronsdal formulation for Fermi fields
Correct degrees of freedom:Correct degrees of freedom:
Combine with trace: Only traceless spatial componentsCombine with trace: Combine with trace: OnlyOnly traceless spatial componentstraceless spatial components
SpinSpin--s fermion of mass m:s fermion of mass m:
Correct degrees of freedom, with:Correct degrees of freedom, with:
Combine with γ-trace: only γ-traceless spatial componentsCombine with Combine with γγ--trace: trace: only only γγ--tracelesstraceless spatial componentsspatial components
Not a Lagrangian equationNot a Lagrangian equation
Bianchi identity:Bianchi identity:
The Lagrangian actually yields:The Lagrangian actually yields:
(Fronsdal, 1978)
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Implicit NotationImplicit NotationImplicit NotationFor all spins, one can eliminate all indicesFor all spins, one can eliminate all indicesNeed only some unfamiliar combinatoric rules Need only some unfamiliar combinatoric rules
(Francia, AS, 2002)
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Fronsdal equations: spin sFronsdal equations: spin sFronsdal equations: spin s
33 ''2
ϕ− ∂
We have seen that gauge invariance requires:We have seen that gauge invariance requires:
We can also derive the We can also derive the Bianchi identityBianchi identity: :
(Fronsdal, 1978)
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Fronsdal equations: spin sFronsdal equations: spin sFronsdal equations: spin s
Fronsdal construction:Fronsdal construction:
Constraints:Constraints:
Lagrangians: Lagrangians:
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Massless case: spin 3/2Massless case: spin 3/2Massless case: spin 3/2RaritaRarita--Schwinger equation: familiar from supergravitySchwinger equation: familiar from supergravity
Can extend to all Can extend to all ½½--integer spins:integer spins:
Bianchi identity:Bianchi identity:
Additional constraint:Additional constraint:
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Bianchi identity for spin s+1/2Bianchi identity for spin s+1/2Bianchi identity for spin s+1/2
Now derive the Bianchi identity:Now derive the Bianchi identity:
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Kaluza-Klein mass: BoseKaluzaKaluza--Klein mass: BoseKlein mass: BoseCan extend the KCan extend the K--K construction to spinK construction to spin--s cases case(Stueckelberg gauge symmetries)(Stueckelberg gauge symmetries)
[ (s[ (s--3)3)-- parameter missing due to trace condition]parameter missing due to trace condition][(s[(s--4)4)--field missing due to double trace condition]field missing due to double trace condition]
Gauge fixing the Stueckelberg symmetries one is left with:Gauge fixing the Stueckelberg symmetries one is left with:
In terms of traceless tensors Singh-Hagen fieldsIn terms of traceless tensors In terms of traceless tensors SinghSingh--Hagen fieldsHagen fields
Budapest, Sept. 2007Budapest, Sept. 2007 2525
Kaluza-Klein mass: FermiKaluzaKaluza--Klein mass: FermiKlein mass: FermiCan again extend the KCan again extend the K--K construction to the spinK construction to the spin--(n+1/2) case : (n+1/2) case : (Stueckelberg gauge symmetries) (Stueckelberg gauge symmetries)
[ (s[ (s--2)2)-- parameter missing due to parameter missing due to γγ--trace condition] trace condition] [(s[(s--3)3)--field missing due to triple field missing due to triple γγ--trace condition] trace condition]
Gauge fixing the Stueckelberg symmetries one is left with:Gauge fixing the Stueckelberg symmetries one is left with:
In terms of γ-traceless tensors Singh-Hagen fieldsIn terms of In terms of γγ--traceless tensors traceless tensors SinghSingh--Hagen fieldsHagen fields
Budapest, Sept. 2007Budapest, Sept. 2007 2626
Lecture IILecture IILecture II
Free HigherFree Higher--Spin fieldsSpin fields
Constrained vs unconstrained formulationsConstrained vs unconstrained formulationsNonNon--local bosonic formulation and Higherlocal bosonic formulation and Higher--Spin GeometrySpin GeometryNonNon--local fermionic constructionlocal fermionic construction
Why the unusual constraints:Why the unusual constraints:1.1. Gauge variation of F Gauge variation of F
1 1 2 3 4... ...3 ( ' ... )s s
Fμ μ μ μ μ μ μδ = ∂ ∂ ∂ Λ +
2.2. Gauge invariance of the LagrangianGauge invariance of the Lagrangian•• As in the spinAs in the spin--2 case, F not integrable2 case, F not integrable
•• Bianchi identity:Bianchi identity:
( ) ( )2 3 42 2 3 5... ... ...1 ' ... 3 '' ...2 2s s s
F F μ μ μμ μ μ μμ μ μ ϕ∂⋅ − − ∂ ∂ ∂ +∂ + =
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The Fronsdal actionsThe Fronsdal actionsThe Fronsdal actions
NaiveNaive equations:equations: 1...0 0
sF Rμ μ μν= ↔ =
1 1 2 3... ...1 10 ( ' ...)2 2s s
R R F Fμν μν μ μ μ μ μ μη η− = ↔ − +
Do not follow directly from a LagrangianDo not follow directly from a Lagrangian
But:But: when combined with traces they dowhen combined with traces they do
One can define an EinsteinOne can define an Einstein--like tensor:like tensor:
Now:Now: 0Gνρ∂ ⋅ ≡
Gauge invariance of L with NO constraintsGauge invariance of L with NO constraintsGauge invariance of L with NO constraints
Budapest, Sept. 2007Budapest, Sept. 2007 3434
Spin 3: non-local actionSpin 3: nonSpin 3: non--local actionlocal actionOne can simply associate to the previous nonOne can simply associate to the previous non--locallocalequation the nonequation the non--local actionlocal action
( ) ( ) ( ) ( )2
2
2 221 3 1 3' '2 2 2 2
3 ' 1 13 '
L μ αβγ βγ μ α
α α ϕ
ϕ ϕ ϕ
ϕ
ϕ
ϕϕ ϕ ϕ+
= − ∂ + ∂ ⋅ − ∂
∂ ⋅∂ ⋅∂ ⋅ ∂ ⋅ − ∂ ⋅
⋅ + ∂
+ ∂ ⋅∂ ⋅ ∂ ⋅∂ ⋅ ∂ ⋅∂ ⋅∂ ⋅
αβγ α βγ β γα γ αβδϕ = ∂ Λ + ∂ Λ + ∂ Λ
fullyfully invariant underinvariant under
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Kinetic operators for integer spinKinetic operators for integer spinKinetic operators for integer spin
IndexIndex--free notation:free notation:
(1) 2 ' 0
( )!. .! !
p q p q
F
p qe gp q
ϕ ϕ ϕ
+
≡ − ∂ ∂ ⋅ + ∂ =
⎛ ⎞+∂ ∂ ≡ ∂⎜ ⎟
⎝ ⎠Now define:Now define:
2( 1) ( ) ( ) ( )1 1'
( 1)(2 1) 1n n n nF F F F
n n n+ ∂ ∂
= + − ∂ ⋅+ + +
2 1( ) [ ]
1(2 1)n
n nnF nδ
+
−
∂= + ΛThen:Then:
Budapest, Sept. 2007Budapest, Sept. 2007 3636
Kinetic operators for integer spinKinetic operators for integer spinKinetic operators for integer spin
2(1)1 11 ( ) 0
( 1)(2 1) 1k
Fk k kξ ξ ξ
⎡ ⎤∂ ∂+ ∂ ⋅ ∂ − ∂ ⋅ ∂ Φ =⎢ ⎥+ + +⎣ ⎦
∏
•• is the generic kinetic operator for higher spinsis the generic kinetic operator for higher spins•• when combined with traces can be reduced towhen combined with traces can be reduced to
Defining:Defining:
3 ( 3 ')F H Hδ= ∂ = Λ
1
1 ...1( , ) ...!
s
sx
sμμ
μ μ
ξ
ξ ξ ξ ϕ
ξ
Φ =
∂∂ =
∂
Budapest, Sept. 2007Budapest, Sept. 2007 3737
Kinetic operators for integer spinKinetic operators for integer spinKinetic operators for integer spin
•• Are gauge invariant for n > [(sAre gauge invariant for n > [(s--1)/2]1)/2]•• Satisfy the Bianchi identitiesSatisfy the Bianchi identities
FermionsFermionsFermionsOne can again iterate:One can again iterate:
The relation to bosons generalizes to:The relation to bosons generalizes to:
Bianchi identities:Bianchi identities:
Gauge transformations: Gauge transformations:
Budapest, Sept. 2007Budapest, Sept. 2007 4545
Lecture IIILecture IIILecture III
Relation with String TheoryRelation with String Theory
BRST formulation for the bosonic stringBRST formulation for the bosonic stringLowLow--tension limit and tripletstension limit and triplets
Compensator equations for Higher SpinsCompensator equations for Higher SpinsCompensator equations and minimal LagrangiansCompensator equations and minimal LagrangiansAn alternative offAn alternative off--shell formulationshell formulation
Budapest, Sept. 2007Budapest, Sept. 2007 4646
Bosonic string: BRSTBosonic string: BRSTBosonic string: BRSTThe starting point is the Virasoro algebra: The starting point is the Virasoro algebra:
In the tensionless limit, one is left with:In the tensionless limit, one is left with:
Virasoro contracts (no c. charge):Virasoro contracts (no c. charge):
Budapest, Sept. 2007Budapest, Sept. 2007 4747
Low-tension limitLowLow--tension limittension limitSimilar simplifications for BRST charge:Similar simplifications for BRST charge:
Making zeroMaking zero--modes manifest: modes manifest:
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String Field equationString Field equationString Field equationHigherHigher--spin massive modes:spin massive modes:
•• massless for 1/massless for 1/αα’’ 00•• Free dynamics can be encoded in:Free dynamics can be encoded in:
20 ( 0)Q Q
Q
ψ
δ ψ
= =
= Λ
(Kato and Ogawa, 1982)(Kato and Ogawa, 1982)(Witten, 1985)(Witten, 1985)(Neveu, West et al, 1985)(Neveu, West et al, 1985)
NONO trace constraints on j or Ltrace constraints on j or L
Physical state conditions:Physical state conditions:Propagate spins s,sPropagate spins s,s--2, 2, ……, 0 or 1, 0 or 1
Budapest, Sept. 2007Budapest, Sept. 2007 5151
(A)dS symmetric triplets(A)dS symmetric triplets(A)dS symmetric tripletsCan build directly, deforming flatCan build directly, deforming flat--space triplets, or via BRSTspace triplets, or via BRST
Directly:Directly: insist on relation between C and othersinsist on relation between C and othersBRST: BRST: gauge nongauge non--linear constraint algebralinear constraint algebra
Basic commutator: Basic commutator:
Budapest, Sept. 2007Budapest, Sept. 2007 5252
(A)dS symmetric triplets(A)dS symmetric triplets(A)dS symmetric tripletsCan also deform directly the equations without C:Can also deform directly the equations without C:
It is convenient to define It is convenient to define
Bianchi identity and first equation then become:Bianchi identity and first equation then become:
Budapest, Sept. 2007Budapest, Sept. 2007 5353
(A)dS symmetric triplets(A)dS symmetric triplets(A)dS symmetric tripletsThe deformed equations can be derived from The deformed equations can be derived from
Alternatively: Alternatively: modify momenta in contracted Virasoromodify momenta in contracted Virasorogauge algebra gauge algebra non linearnon linear (but M,N diagonal on triplets)(but M,N diagonal on triplets)
Describe a spin-s gauge field with:NO trace constraints on the gauge parameterNO trace constraints on the gauge fieldFirst can be reduced to minimal non-local form
BUT:NOT Lagrangian equations
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)dS Compensator Eqs(A)dS Compensator Eqs(A)dS Compensator EqsFlatFlat--space compensator equations can be extended to (A)dS:space compensator equations can be extended to (A)dS:
Gauge invariant underGauge invariant under
The first can be turned into the second via (A)dS BianchiThe first can be turned into the second via (A)dS Bianchi
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Compensator EquationsCompensator EquationsCompensator EquationsIt is possible to obtain a Lagrangian form of the compensatorIt is possible to obtain a Lagrangian form of the compensator
equations, using BRST techniquesequations, using BRST techniquesEssentially in Pashnev and Tsulaia (1997)Essentially in Pashnev and Tsulaia (1997)Formulation involves number of fields ~ sFormulation involves number of fields ~ sInteresting BRST subtletiesInteresting BRST subtletiesHere we discuss explicitly spin s=3Here we discuss explicitly spin s=3
First compensator equation second via BianchiFirst compensator equation second via Bianchi
Budapest, Sept. 2007Budapest, Sept. 2007 6161
Fermionic Compensators Fermionic Compensators Fermionic Compensators We We couldcould extend the fermionic compensator eqs to (A)dS backgrounds extend the fermionic compensator eqs to (A)dS backgrounds [We [We could notcould not extend the fermionic triplets] extend the fermionic triplets]
First compensator equation First compensator equation second via (A)dS Bianchi identity:second via (A)dS Bianchi identity:
Budapest, Sept. 2007Budapest, Sept. 2007 6262
Off-Shell truncation of tripletsOffOff--Shell truncation of tripletsShell truncation of triplets( B( Buchbinduchbinder, er, KrykhtinKrykhtin, , ReshetnyakReshetnyak 2007 )2007 )
•• start from a start from a triplettriplet (s,s(s,s--2,2,……) )
•• add add two (gauge invariant) Lagrange two (gauge invariant) Lagrange multipliersmultipliers
•• Lagrangian : Lagrangian :
λ λ and and μμ : set to zero by the field equations: set to zero by the field equations
OffOff--shell reduction of triplets :shell reduction of triplets :
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Minimal local LagrangiansMinimal local LagrangiansMinimal local Lagrangians
““MinimalMinimal”” local Lagrangians with local Lagrangians with unconstrained unconstrained gauge symmetry:gauge symmetry:
The Lagrangians are:The Lagrangians are:
(Francia, AS, 2005; Francia, Mourad and AS, 2007)
Can be nicely extended to Can be nicely extended to (A)dS backgrounds(A)dS backgrounds
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Lecture IVLecture IVLecture IV
External currentsExternal currents
Lessons for the nonLessons for the non--local formulationlocal formulationVDVZ discontinuityVDVZ discontinuity
The Vasiliev construction (and the compensator)The Vasiliev construction (and the compensator)Problems with higherProblems with higher--spin interactionsspin interactionsThe Vasiliev constructions and freeThe Vasiliev constructions and free--differential algebrasdifferential algebrasStrong vs weak projection: recovery of the compensatorStrong vs weak projection: recovery of the compensator
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Minimal local LagrangiansMinimal local LagrangiansMinimal local Lagrangians
““MinimalMinimal”” local Lagrangians with local Lagrangians with unconstrained unconstrained gauge symmetry:gauge symmetry:
The Lagrangians are:The Lagrangians are:
(Francia, AS, 2005; Francia, Mourad and AS, 2007)
Can be nicely extended to Can be nicely extended to (A)dS backgrounds(A)dS backgrounds
•• Residues of current exchanges reflect the Residues of current exchanges reflect the degrees of freedomdegrees of freedom
•• For s=1 : For s=1 :
•• For all s : For all s :
Budapest, Sept. 2007Budapest, Sept. 2007 6767
External currents : local caseExternal currents : local caseExternal currents : local case
K K ““doubly tracelessdoubly traceless”” using double trace constraintusing double trace constraintB: determines multiplier B: determines multiplier β β for double trace constraintfor double trace constraint
The exchange involves, correctly, a The exchange involves, correctly, a traceless conserved currenttraceless conserved current
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External currents : non-local caseExternal currents : nonExternal currents : non--local caselocal caseHow about the nonHow about the non--local version of the theory? local version of the theory?
Apparently:Apparently: different choices for the field equation, EQUIVALENT without cudifferent choices for the field equation, EQUIVALENT without currentsrrents
S=3 :S=3 :
Bianchi identityBianchi identity: changes after every iteration: changes after every iteration
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Naively: Naively:
Solution:Solution: modify the nonmodify the non--local Lagrangian equationlocal Lagrangian equation
Incorrect current exchange ! Incorrect current exchange !
Budapest, Sept. 2007Budapest, Sept. 2007 7070
VD-V-Z Discontinuity for HSVDVD--VV--Z Discontinuity for HSZ Discontinuity for HS(van Dam, Veltman; Zakharov, 1970)(van Dam, Veltman; Zakharov, 1970)
For all s and D, m=0 : For all s and D, m=0 :
VDVZ VDVZ discontinuitydiscontinuity follows in general comparing D and (D+1) massless exchanges follows in general comparing D and (D+1) massless exchanges First present for s=2 via DFirst present for s=2 via D--dependence of dependence of ρρ(D,s) (D,s) For all s:For all s: can describe irreducibly a massive field acan describe irreducibly a massive field a’’ la Scherkla Scherk--Schwarz from (D+1) dimensions : Schwarz from (D+1) dimensions :
[ e.g. for s=2 : [ e.g. for s=2 : hhMNMN (h(hμνμν cos(my) , Acos(my) , Aμμ sin(my), sin(my), ϕϕ cos(my) )cos(my) ) ] ]
(A)dS extension, first discussed, for s=2, by Higuchi and Porrat(A)dS extension, first discussed, for s=2, by Higuchi and Porrati i Discontinuity Discontinuity smooth interpolation in (msmooth interpolation in (mLL) ) 22
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HS InteractionsHS InteractionsHS Interactions
Problems with interacting higher spins :Problems with interacting higher spins :
Inconsistent equations (derivatives imply further conditions)Inconsistent equations (derivatives imply further conditions)Coupling with gravity leaves Coupling with gravity leaves ““nakednaked”” Weyl tensors Weyl tensors (Aragone, Deser,1979)
Coleman Coleman -- MandulaMandula
(Berends, Burgers, van Dam, 1982)(Bengtsson2, Brink, 1983)
(Fradkin and Vasiliev, 1980’s)
(Vasiliev, 1990, 2003)(Sezgin, Sundell, 2001)
Way out:Way out:
Infinitely many interacting fields Infinitely many interacting fields NonNon--vanishing vanishing ““cosmological constantcosmological constant”” ΛΛVasiliev equationsVasiliev equations:: paradigmatic exampleparadigmatic example
Budapest, Sept. 2007Budapest, Sept. 2007 7272
Aragone-Deser problemAragoneAragone--Deser problemDeser problem
Flat space Lagrangians: usually gauge invariant as a result ofFlat space Lagrangians: usually gauge invariant as a result ofcancellations between terms that differ by commutatorscancellations between terms that differ by commutators
Moving to curved backgrounds introduces in general nakedMoving to curved backgrounds introduces in general nakedRiemann tensorsRiemann tensors
Miracle of supergravity: such terms combine into Einstein Miracle of supergravity: such terms combine into Einstein tensors, which allows cancellationstensors, which allows cancellations
Independent higherIndependent higher--spin propagation possible in spin propagation possible in conformally flat conformally flat backgroundsbackgrounds
(A)dS: (A)dS: massmass--like termslike terms, but , but ““freefree””. . Central role in Vasiliev equationsCentral role in Vasiliev equations
(Aragone, Deser, 1979)(Aragone, Deser, 1979)
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HS InteractionsHS InteractionsHS InteractionsVasilievVasiliev’’ s setting:s setting:
1.1. Extend Extend the frame formulation of gravity the frame formulation of gravity ::
For spinFor spin--ss: :
2.2. ∞∞ -- dim. HSdim. HS--algebra via algebra via oscillatorsoscillators (coordinates and momenta):(coordinates and momenta):
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HS InteractionsHS InteractionsHS Interactions3.3. [Weyl ordered (symmetric) polynomials in (x,p) or [Weyl ordered (symmetric) polynomials in (x,p) or ∗∗-- products ] products ]
4.4. A oneA one--form Aform A in adjoint of HS algebra :in adjoint of HS algebra :(all (all ωω’’s : HS vielbeins and connections) s : HS vielbeins and connections)
5.5. A zeroA zero--form form ΦΦ in the in the ““twisted adjointtwisted adjoint””::
One writes the spinOne writes the spin--2 equation in the form: 2 equation in the form: ““ Riemann = Weyl Riemann = Weyl ””whose trace gives the familiar equation whose trace gives the familiar equation ““ Ricci=0Ricci=0””““ Weyl Weyl ”” (+ derivatives) :(+ derivatives) : fields with Youngfields with Young--tableau structure tableau structure
Scalar : ϕ
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HS InteractionsHS InteractionsHS InteractionsTWO forms of Vasilev equations :TWO forms of Vasilev equations :
1.1. 44--dim spinors dim spinors (2(2--component formalism) component formalism)
2.2. DD--dim vectorsdim vectors
Let us try to see what these equations mean Let us try to see what these equations mean (focusing on the case with vector oscillators)(focusing on the case with vector oscillators)
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HS InteractionsHS InteractionsHS Interactions
Background independent (nonBackground independent (non--Lagrangian) !Lagrangian) !A:A: s fields for every (even) rank s s fields for every (even) rank s ((““adjointadjoint”” of HS algebra) of HS algebra)
(Generalized vielbeins and connections) (Generalized vielbeins and connections) [Chan[Chan--Paton extension to all (even and odd) ranks] Paton extension to all (even and odd) ranks]
ΦΦ:: ∞∞ fields for every rank s fields for every rank s ((““twisted adjointtwisted adjoint”” of HS algebra)of HS algebra)(Generalized Weyl and their covariant derivatives) (Generalized Weyl and their covariant derivatives)
ΦΦ∗∗κκ:: converts converts ““twisted adjointtwisted adjoint”” ΦΦ to adjointto adjointConsistent (almost by inspection) : Bianchi for F implies seconConsistent (almost by inspection) : Bianchi for F implies second eq ! d eq !
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HS InteractionsHS InteractionsHS Interactions
Gauge field A in (x,z) space:Gauge field A in (x,z) space:
Internal equations:Internal equations: power series in power series in ΦΦ by successive iterations by successive iterations
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HS InteractionsHS InteractionsHS Interactions
Lowest order in Lowest order in ΦΦ ““Riemann = WeylRiemann = Weyl”” + + ……
with
““Internal Internal ”” equations: equations:
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HS InteractionsHS InteractionsHS Interactions
k is a crucial ingredient :k is a crucial ingredient :
Uniform description of HS interactions Uniform description of HS interactions
Budapest, Sept. 2007Budapest, Sept. 2007 8080
HS InteractionsHS InteractionsHS Interactions
Cartan Integrable System :Cartan Integrable System :
e.g. Cherne.g. Chern--Simons theory Simons theory manifestly consistent eqsmanifestly consistent eqsgauge covariancegauge covariancemanifest diff covariancemanifest diff covariancenonnon--LagrangianLagrangian
NEW INGREDIENT :NEW INGREDIENT : 00--form form ΦΦ
(Sullivan, 1977; D’Auria, Fre’, 1982)
Budapest, Sept. 2007Budapest, Sept. 2007 8181
HS InteractionsHS InteractionsHS Interactions
Some missing ingredientsSome missing ingredients : : Y Y iAiA, Z , Z iAiA to build HS algebra extending SO(2,D) to build HS algebra extending SO(2,D) must select Sp(2,R) singletsmust select Sp(2,R) singletsK K i j i j : Sp(2,R) generators (bilinears in Y, Z) : Sp(2,R) generators (bilinears in Y, Z)
REMOVE TRACES to obtain dynamical equationsREMOVE TRACES to obtain dynamical equations
Weak projection :Weak projection : remove traces symmetrically from A and remove traces symmetrically from A and ΦΦ (Vasiliev, 2003)(Vasiliev, 2003)
Strong :Strong : leave traces in A leave traces in A (AS,Sezgin,Sundell, 2004)(AS,Sezgin,Sundell, 2004)
s = 3 :s = 3 :a first equationa first equation, analogous of the vielbein postulate giving , analogous of the vielbein postulate giving ωω(e)(e)a second equationa second equation, defining a second, defining a second--order kinetic operator order kinetic operator
a third equationa third equation giving the constraint giving the constraint
Field TheoryField TheoryString Theory (an instance with String Theory (an instance with ““spontaneous breakingspontaneous breaking””))
In these lectures:In these lectures:Free (unconstrained) HigherFree (unconstrained) Higher--Spin fieldsSpin fieldsRelation with VasilievRelation with Vasiliev’’ constructionconstruction
Some reviews:
N. Bouatta, G. Compere, A.S., hep-th/0609068 D. Francia and A.S., hep-th/0601199
A.S., Sezgin, Sundell, hep-th/0501156 X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep-th/0503128A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 2008 (?)
Some reviews:Some reviews:
N. Bouatta, G. Compere, A.S., hepN. Bouatta, G. Compere, A.S., hep--th/0609068 th/0609068 D. Francia and A.S., hepD. Francia and A.S., hep--th/0601199 th/0601199
A.S., Sezgin, Sundell, hepA.S., Sezgin, Sundell, hep--th/0501156 th/0501156 X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hepX. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, hep--th/0503128th/0503128A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 200A.S., P. Sundell, D. Sorokin, M.A. Vasiliev, Phys. Reports, 2008 (?) 8 (?)