higher-spin geometry and string theory

Higher-Spin Higher-Spin GeometryGeometry

and and String TheoryString Theory

Augusto SAGNOTTIAugusto SAGNOTTI

Universita’ di Roma “Tor Vergata”Universita’ di Roma “Tor Vergata”

QG05 – Cala Gonone, September, 2005

Based on:Based on:• Francia, AS, hep-th/0207002,,0212185, 0507144• AS, Tsulaia, hep-th/0311257• AS, Sezgin, Sundell, hep-th/0501156•Also: D. Francia, Ph.D. Thesis, to appear

Based on:Based on:• Francia, AS, hep-th/0207002,,0212185, 0507144• AS, Tsulaia, hep-th/0311257• AS, Sezgin, Sundell, hep-th/0501156•Also: D. Francia, Ph.D. Thesis, to appear

QG05 - Cala Gonone, Sept. 2005QG05 - Cala Gonone, Sept. 2005 22

PlanPlan The (Fang-) Fronsdal equationsThe (Fang-) Fronsdal equations Non-local geometric equationsNon-local geometric equations Local compensator formsLocal compensator forms Off-shell extensionsOff-shell extensions Role in the Vasiliev equationsRole in the Vasiliev equations


The Fronsdal The Fronsdal equationsequations

(Fronsdal, 1978)

1 1 2 1 1

... 1 1 2 1 2 31

... ... ...

... ... ...

...

( . ...) ( ' ...) 0s s s s

s s ssF

1 1 2 1 1

... 1 1 2 1 2 31

... ... ...

... ... ...

...

( . ...) ( ' ...) 0s s s s

s s ssF

Originally from massive Singh-Hagen equations(Singh and Hagen, 1974)

Unusual constraints:Unusual constraints: ' 0, '' 0 ' 0, '' 0

Gauge invariance for massless symmetric tensors:


Bianchi Bianchi identitiesidentities

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

F 1 1 2 3 4... ...3 ( ' ... )s s

F

2.2. Gauge invariance of the LagrangianGauge invariance of the Lagrangian• As in the spin-2 case, F not integrableAs in the spin-2 case, F not integrable• Bianchi identity:Bianchi identity:

2 3 42 2 3 5... ... ...

1' ...

3'' ...

2 2s s sF F

2 3 42 2 3 5... ... ...

1' ...

3'' ...

2 2s s sF F


Constrained Constrained gauge invariancegauge invariance

1 1 1 2 3... ... ...

1' ...

2s s sL F F

1 1 1 2 3... ... ...

1' ...

2s s sL F F

If in the variation of L one inserts:1 1 2... ... ...

s s 1 1 2... ... ...s s

2 2 2 3 2 3 4... ... ... ...

'Bianchiidentity: "

1 1' ... ' ..

2 2s s s sL s F F F

2 2 2 3 2 3 4... ... ... ...

'Bianchiidentity: "

1 1' ... ' ..

2 2s s s sL s F F F

Are these constraints really necessary?Are these constraints really necessary?Are these constraints really necessary?Are these constraints really necessary?


The spin-3 The spin-3 casecase

1 2 3

1 2 3 1 2 3 1 2 323 ' ' 3 'F F

1 2 3

1 2 3 1 2 3 1 2 323 ' ' 3 'F F

A fully gauge invariant (non-local) equation:

1 2 30F 1 2 30F

Reduces to local Fronsdal form upon partial gauge fixing

1 2 3

1 2 3

2' 0F F

1 2 3

1 2 3

2' 0F F

(Francia and AS, 2002)


Spin 3: other Spin 3: other non-local eqsnon-local eqs

Other equivalent forms:

1 2 3 2 3 13 3 1 21 2

1' ' 0'

3FF F F

1 2 3 2 3 13 3 1 21 2

1' ' 0'

3FF F F

Lesson: full gauge invariance with non-local terms

1 2 3 2 31 3 12 3 1 2

01

3F FF F

1 2 3 2 31 3 12 3 1 20

1

3F FF F


Kinetic operatorsKinetic operators

Index-free notation:

(1) 2 ' 0

( )!. .

! !p q p q

F

p qe g

p q

(1) 2 ' 0

( )!. .

! !p q p q

F

p qe g

p q

Now define:

2( 1) ( ) ( ) ( )1 1

'( 1)(2 1) 1

n n n nF F F Fn n n

2( 1) ( ) ( ) ( )1 1

'( 1)(2 1) 1

n n n nF F F Fn n n

2 1( ) [ ]

1(2 1)

nn n

nF n

2 1( ) [ ]

1(2 1)

nn n

nF n

Then:


Kinetic operatorsKinetic operators

2(1)1 1

1 ( ) 0( 1)(2 1) 1k

Fk k k

2(1)1 1

1 ( ) 0( 1)(2 1) 1k

Fk k k

• generic kinetic operator for higher spins• when combined with traces:

Defining:

33 ( ')F H H 33 ( ')F H H

1

1...

1( , ) ...

!s

sx

s

1

1...

1( , ) ...

!s

sx

s


Kinetic operatorsKinetic operators• Are gauge invariant for n > [(s-1)/2]• Satisfy the Bianchi identities

2 1[ 1]

1( ) ( ) 1

12

1'

2n n

nn

nnF F

n

2 1[ 1]

1( ) ( ) 1

12

1'

2n n

nn

nnF F

n

• For n> [(s-1)/2] allow Einstein-like operators

The F(n):

1( ) ( )[ ]

0

( 1) ( )!

2 !

pnn p n p

pp

n pG F

n

1( ) ( )[ ]

0

( 1) ( )!

2 !

pnn p n p

pp

n pG F

n


Geometric Geometric equationsequations

Christoffel connection:Christoffel connection:

h h

Generalizes to all symmetric tensors(De Wit and Freedman, 1980)

1 2 1 1 1

1 2 1 2 1 1

; ... ; ...

; ... ; ...

s s

s sR R

1 2 1 1 1

1 2 1 2 1 1

; ... ; ...

; ... ; ...

s s

s sR R


Geometric Geometric equationsequations

1.1.Odd spins (s=2n+1):Odd spins (s=2n+1):

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

0F 0F 1[ ] ; ...10sn v v

nR

1[ ] ; ...1

0sn v vn

R

0R 0R 1[ ] ; ...1

10sn v v

nR

1[ ] ; ...

1

10sn v v

nR



Bosonic string: Bosonic string: BRSTBRST The starting point is the Virasoro algebra:

In the tensionless limit, one is left with:

Virasoro contracts (no c. charge):


String Field String Field equationequation

Higher-spin massive modes:• massless for 1/’ 0• Free dynamics can be

encoded in:20 ( 0)Q Q

Q

20 ( 0)Q Q

Q

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

NONO trace constraints on or LNONO trace constraints on or L


Low-tension Low-tension limitlimit Similar simplifications hold for the BRST charge:

With zero-modes manifest:


Symmetric Symmetric tripletstriplets(A. Bengtsson, 1986)

(Henneaux,Teitelboim, 1987)(Pashnev, Tsulaia, 1998)(Francia, AS, 2002)(AS, Tdulaia, 2003)

Emerge from

The triplets are:


(A)dS symmetric (A)dS symmetric tripletstriplets Directly, deforming flat-space triplets, or via BRST

(no Aragone-Deser problem) Directly:Directly: insist on relation between C and others BRST: BRST: gauge non-linear constraint algebra Basic commutator:


Compensator Compensator EquationsEquations

In the triplet:

spin-(s-3) compensatorcompensator:

The second becomes:

The first becomes:

Combining them:

Finally (also Bianchi):


(A)dS (A)dS Compensator Compensator

EqsEqs Flat-space compensator equations can be extended to (A)dS: (no Aragone-Deser problem)

Gauge invariant under First can be turned into second via (A)dS Bianchi


Off-Shell Off-Shell Compensator Compensator

EquationsEquations Lagrangian form of compensator: BRST techniques Formulation due to Pashnev and Tsulaia (1997)Formulation due to Pashnev and Tsulaia (1997) Formulation involves a large number of fields (O(s)) Interesting BRST subtleties For spin 3 the fields are:

(AS and Tsulaia, 2003)

Gauge fixing Gauge fixing


Off-Shell Off-Shell Compensator Compensator

EquationsEquations “Minimal” Lagrangians can be built directly for all spins Only two extra fields, Only two extra fields, (spin-(s-3)) and (spin-(s-3)) and (spin-(s-4))(spin-(s-4))


Equation for compensator equation Equation for Equation for current conservation current conservation Lagrange multiplier Lagrange multiplier ::


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

IntegrableIntegrable curvature constraintscurvature constraints on one-forms and zero-forms Cartan integrable systems Key new addition of Vasiliev:Key new addition of Vasiliev: twisted-adjoint representation

(D’Auria,Fre’, 1983)

Minimal caseMinimal case (only symmetric (only symmetric tensors of even rank)tensors of even rank), Sp(2,R)Sp(2,R) zero-form zero-form : : Weyl curvatures one-form A :one-form A : gauge fields


The Vasiliev The Vasiliev equationsequations

Curvature constraints:Curvature constraints:

[extra [extra non comm.non comm. Coords] Coords]

Gauge symmetry:Gauge symmetry:


The Vasiliev The Vasiliev equationsequations “Off-shell”: Riemann-like curvatures

“On-shell”: (Riemann-like = Weyl-like l) Ricci-like = 0Ricci-like = 0 What is the role of Sp(2,R) in this transition?

(AS,Sezgin,Sundell, 2005)

Sp(2,R) generators:

Key on-shell constraint: gauge fields NOT constrainedNOT constrained

Strong constraint:Strong constraint: proper scalar massesproper scalar masses emerge At the interaction level must regulate projectorregulate projector Gauge fields: extended (unconstrained) gauge symmetryGauge fields: extended (unconstrained) gauge symmetry Alternatively: Alternatively: weak constraint, no extra symmetry (Vasiliev) weak constraint, no extra symmetry (Vasiliev)

(Dubois-Violette, Henneaux, 1999) (Bekaert, Boulanger, 2003)


The spin-3 The spin-3 compensatorcompensator(AS,Sezgin,Sundell, 2005)

• In the 0 limit the linearized Vasiliev equations become:

• Can be solved recursively for the W’s in terms of :

• Since C is traceless, the k=2 equation implies:

• Explicitly:

• This implies:

Last term (compensator): “exact” in sense of Dubois-Violette and HenneauxLast term (compensator): “exact” in sense of Dubois-Violette and Henneaux


The Vasiliev The Vasiliev equationsequations Non-linear corrections:Non-linear corrections: from dependence on internal Z- coordinates

• Does the projection that “leaves” the compensators produce singular interactions?• Vasiliev:Vasiliev: works with traceless conditions all over and feels it does• My feeling:My feeling: eventually not, and we are seeing a glimpse of the off-shell form

More work will tell us….More work will tell us….


The EndThe End


FermionsFermions

Notice: 1

2

1

2s

S

S

1

2s

i

sF

1

2

1

2s

S

S

1

2s

i

sF

Example: spin 3/2 (Rarita-Schwinger)

1

2S

S

i v

1

2S

S

i v

0 0 S i 0S i 0



FermionsFermionsOne can again iterate:

The relation to bosons generalizes to:

The Bianchi identity generalizes to:


Fermionic Fermionic TripletsTriplets (Francia and AS, 2003)

Counterparts of bosonic triplets

GSO:GSO: not in 10D susy strings

Yes:Yes: mixed sym generalizations

Directly in type-0 modelstype-0 models

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


Fermionic Fermionic CompensatorsCompensators

Recall:

Spin-(s-2) compensator:

Gauge transformations:

First compensator equation second via BianchiFirst compensator equation second via Bianchi

(recently, also off shell Buchbinder,Krykhtin,Pashnev, 2004)


Fermionic Fermionic CompensatorsCompensators We couldcould extend the fermionic compensator eqs to (A)dS

We could notcould not extend the fermionic triplets BRST:BRST: operator extension does not define a closed algebra

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

(AS and Tsulaia, 2003)


Compensator Compensator Equations (s=3)Equations (s=3)

Gauge transformations:

Field equations:

Gauge fixing: Other extra fields: zero by field equations

higher-spin geometry and string theory

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