Manifestly conformal descriptions and highersymmetries of bosonic singletons

Maxim Grigoriev

Based on:

M.G. and Xavier Bekaert, arXiv:arXiv:0907.3195 [hep-th]

Singletons:

Conformal scalar and spinor (Dirac, 1963)More generally, conformal fields originating from unitary Poincareirreps (Siegel, 1989)

Spin-s singletons:these are Lorentz fields in d -dimensions (d even) whose curvaturesare described by

. . . .

s

d2

+ (anti)self-duality in each column.

Scalar singleton underlies the structure of higher spin interactions ind+ 1-dim AdS space. In particular:

• Flato–Fronsdal theorem: tensor product of two singletons equalsa direct sum of AdS HS fields (Flato-Fronsdal, 1978)

• Underling the AdS HS field interactions is a higher spin (HS)algebra (certain enveloping of the conformal algebra) (Vasileiv,1990, 2003)

• In terms of conformal fields, HS algebra is an algebra of highersymmetries of the equations of motion for the conformal scalar.This was identified in Eastwood, 2002 as enveloping of o(d, 2)

• In d = 4 algebras of symmetrises were identified in some casesin (Fushchich, Nikitin 1987) and for all massless fields in d = 4in (Pohjanpelto, Anco 2003, 2008) and (Vasiliev 2002)

• From the 1st-quantized point of view: singleton is a quantizedmassless scalar particle. The respective UIR is its physical sub-space while HS algebra is an algebra of its quantum observ-ables.

It is natural to expect that extending this picture to general singletonswould lead to understanding interactions of mixed-symmetry fieldson AdS

Two aims:

1. To construct explicitly-local and explicitly o(d, 2)-invariant formulation of bosonic singletons.

2. To describe higher symmetries of singleton equa-tions of motion using the constructed formulation

Plan

• To explicitly illustrate the construction in the knowncase of a scalar singleton

• To show how the construction generalizes to the cases > 0

• Rederive the Eastwood classification of higher sym-metries of the scalar singleton.

• To extend the classification to s > 0

• To describe the structure of invariants

Ambient space description of the conformal geometry

Conformal space can be seen as a quotient of the hypercone (Dirac,1936)

Y · Y = 0 , Y ∼ λY λ 6= 0

in Rd,2. Scalar product is defined using the ambient space metric

ηAB = diag−−+ + . . .+

e.g. Y 2 = Y · Y = Y AY BηAB .

In terms of constrained dynamics the phase space corresponding tothe conformal space (cotangent bundle over the conformal space) isdetermined by the following first class constraints

Y · Y = 0 , Y · P = 0 ,

For Y variables:Y ·Y = 0 restricts to the hypercone while Y ·P = 0 implements theidentification X ∼ λX as a gauge symmetry.

For the momenta P :Y · P = 0 and the gauge symmetry generated by Y · Y = 0 restrictthe momentum space accordingly.

Scalar particle on the projective hypercone

In order to describe scalar particle one adds the mass-shell con-straint P 2 = 0. The entire set of constraints:

Y · Y = 0 , 1

2(Y · P + P · Y ) = 0 , P · P = 0

These form sp(2) algebra w.r.t. the Poisson bracket.This constrained system was considered by many authors:Marnelius, Bars, Vasileiv, . . .

After quantization: space of states – functions in Y . Constraints:

Y 2 , Y · ∂

∂Y+d+ 2

2,

∂

∂Y· ∂

∂Y

Representation of sp(2) on functions in Y .In the appropriate functional space these constraints single out

usual conformal scalar in d-dimensions Marnelius, 1979.

Apparent disadvantage: locality is not explicit because d-dimensionalfield theory is described in terms of d+ 2 dimensional fields.

Algebraic remarks: Howe duality

Introducing notation ZAa = (Y A, PA) generators of sp(2) ando(d− 1, 1) can be written as follows:

tab = ZAa ZBb ηAB JAB = εabZAa Z

Bb

These algebras form a dual pair in the sense of Howe duality.

More precisely, coordinates ZAa are naturally considered as those onthe tensor product of 2-dim. symplectic space and d+2-dim. pseu-doEucledean one.

Maintaining locality: ambient space as a fiber

The way out is to consider fields that are o(d − 1, 1) tensors definedon the true conformal space.The well-known examples (Stelle–West,1979) for AdS gravity andVasileiv, 2001 for AdS HS fields takes into account AdS geometrythrough AdS connection and the compensator field i.e., use CartandescriptionAnalogous approach to conformal fields is also known ((Preitschopf,Vasiliev 1998)What we need here is a more refined description that is analogous tothe Vasiliev unfolded description. From the first quantized point ofview it can be seen as an analog of Fedosov quantization.

More precisely, its BRST version adapted to constrained systemsBatalin,Fradkin 86, Fradkin Linetsky 1994, M.G. Lyakhovich 2000,Batalin, M.G., Lyakhovich 2001 and cotangent bundles Bordemannet all 1997, Barnich, M.G. Semikhatov, Tipunin 2004

To maintain locality: puting ambient space as the fiber over the truespace-time

The construction is best illustrated in the first quantized terms andusing BRST formalism.

1st step:Scalar particle BRST operator:

Ω0 = c+P2 + 1

2c0(Y · P + P · Y ) + c−Y

2 + ghosts

2nd step:Introduce new pure gauge variables PA, XB along with new con-straints PA − PA = 0 and new ghost variables ΘA. The equivalentBRST operator

Ω = ΘA(PA − PA) + Ω ,

Ω = c+P2+ 1

2c0((X+Y )·P+P ·(X+Y ))+c−(X+Y )2+ghosts

3rd step: Identifying dXA ≡ ΘA it is easy to see that one can usegeneral coordinates XA and general local frame:

Ω = −ΘAPA + ΘAEBAPB + ωCABYBPC + Ω

where E and W are compatible vielbein and flat connection satisfy-ing

∇2 = 0 , ∇X(X) = E

4th step: One can eliminate extra coordinates X and respective θbecause they are pure gauge. Using coordinate representation for theremaining “intrinsic” coordinates xµ, θµ the equivalent BRST opera-tors takes the form:

Ω = θµ∂

∂xµ− θµeAµ

∂

∂Y A− ωBµA(x)Y A ∂

∂Y B+ Ω

The cartesian coordinate XA expressed through the intrinsic coordi-nates on the conformal space are components conformal compensatorfield V , i.e. V A(x) = XA(x).

Note that V · V = 0 and V is naturally defined up to rescaling

V → λV λ 6= 0

.In the frame where V A = const the BRST operator takes the follow-ing form

Ω = ∇+ Ω , ∇ = d + ωBA (Y A + V A) ∂

∂Y B,

Ω = c+P2 + 1

2c0((Y +V ) ·P +P ·(Y +V ))+c−(Y +V )2 +ghosts

∇ – o(d, 2)- flat covariant derivative, Ω – sp(2) BRST operator.Equations of motion, gauge symmetries, etc.:

ΩΨ(0) = 0 , δΨ(0) = ΩΨ(−1) , . . .

Familiar form of the generating BRST formulation of various fields:Poincare invariant fields: Barnich, M.G., Semikhatov, Tipunin, 2004,Alkalaev, M.G., Tipunin, 2008AdS fields: Barnich, M.G., 2006, Alkalaev, M.G. to appear

More generally, can be constructed taking as a fiber the space equippedwith the action of two Howe dual algebra and an appropriate space-time manifold.

Reduction to Lorentz fields

Adapted frame:

V + = 1, V − = 0, V a = 0 ,η+− = 1 , ηa+ = ηa− = 0 , ηab = diag(−+ + . . .+) ,

The system can be reduced to the Lorentz description: Recall:

Ω = ∇+ Ω = ∇+ c+∂

∂Y· ∂

∂Y

+ c0( ∂

∂Y ++ Y · ∂

∂Y− d− 2

2) + (Y · Y + 2Y −) ∂

∂b−+ ghosts

Two steps of the reduction:1st step: eliminate Y − and b− by reducing to cohomology of

Y −∂

∂b−

2nd step: eliminate Y + and c0 by reducing to cohomology of

c0∂

∂Y +

.Finally:

Ωreduced = ∇+ c+∂

∂Y a

∂

∂Ya

In the Cartesian coordinates where eaµ = δaµ and ω = 0 this is equiv-alent to

Ωstandard = c+∂

∂xa∂

∂xa

leading to a usual massless Klein-Gordon equations of motion.

Explicitly o(d, 2)-invariant and explicitly-local description of the con-formal scalar

spin-s singletons

Representation space:

(functions in Y ) → (functions in Y )⊗ (Grassmann algebra in ϑAi )

ϑAi –usual fermionic oscillators.Constraint algebra:

sp(2)− algebra → osp(2s|2)− superalgebra

Besides sp(2) constraints Y 2, Y · ∂∂Y + d+2

2 , ∂∂Y ·

∂∂Y one has

“algebraic constraints”

ϑi ·∂

∂ϑj− δji

d+ 22

, ϑi · ϑj ,∂

∂ϑi· ∂

∂ϑj

forming o(2s) algebra. Together with 4s fermionic constraints

X · ϑi , X · ∂

∂ϑi,

∂

∂X· ϑi ,

∂

∂X· ∂

∂ϑi

all these form osp(2s|2) superalgebra.

In the appropriate functional space these determine spin-s singleton.(Siegel 1988, Arvidsson, Marnelius 2006).

These constraints originate from extended supersymmetric spin-ning particle model Howe, Penati, Pernici,Townsend 1988

Algebraic structure: Howe dual o(d, 2) and osp(2s|2) algebras.Introducing ZAα = Y A, PA, ϑAi , πiA the generators are

tαβ = ZAαZBβ ηAB JAB = εαβZAαZ

Bβ

Just like in the conformal particle case locality can be maintained bypassing to the generating BRST description

Ω = ∇+ Ω , Ω = CItI + 1

2UKIJC

ICJPK ,

where tI = tαβ – osp(2s|2)-generators. In contrast some auxiliaryoff-shell constraints to eliminate doubling of states.

Explicitly o(d, 2)-invariant and explicitly-local description of spin s-singletons

Higher symmetriesEquations of motion:

TIΨ = 0 , TI − differential operators

Symmetry: operator K that maps solutions to solutions, i.e.

TIKΨ = 0 =⇒ [TI ,K] = UJI TJ

Equivalence:K ∼ K +W JTJ

This is nothing but the Dirac definition of the observable for the con-strained system with 1st class constraints TI

inequivalent symmetries ' first-quantized observables

Using BRST formalism: Ω = CITI+. . ., CI ,PJ – ghost variables

H0([Ω, ·], operators) ' inequivalent symmetries

More precisely, this is true if Ω is proper. Otherwise there can beadditional classes.

Inequivalent symmetries is naturally an associative algebra.The product is the operator product induced in the cohomology. Fromthe 1st-quantized point of view – algebra of quantum observbles.

What we need to compute is the cohomology of 1~ [Ω, ·] at zeroth

ghost number.

For Ω of the form Ω = ∇+ Ω the cohomology is isomorphic to thatof 1

~ [Ω, ·] (if De Rham cohomology do not contribute)

Work in terms of Weyl symbols. Operator multiplication → Weyl∗-product. For

Ω = CITI + 1

2UKIJC

ICJPK

D = TI∂

∂PI+ CI [TI , ·]∗ + ghosts +O(~2) .

Cohomology computation: first cohomology of δ = TI∂∂PI

(Kozuledifferential). If no P-dependent classes then all cohomology can beassumed totally traceless (e.g. in scalar case TI = P 2, Y · P, Y 2,i.e. all possible traces) one ends up with [TI , φ]∗ = 0 i.e. invariantsof osp(2s|2) in this representation.

s = 0

In this case: no P-dependent cohomology classes of δ = TI∂∂PI

(Barnich, M.G., Semikhatov, Tipunin, 2004).

Higher symmetries are then given by totally traceless sp(2)-invariantpolynomials in Y, P . These are generated by Y APB − Y BPA, i.e.form an enveloping algebra of o(d, 2) in this representation (East-wood, 2002)

s > 0

In this case δ = TI∂∂PI

in general can have P-dependent classes.However, these originate from relations between constraints and areto be eliminated through additional terms in the BRST operator

Koszule differential → Koszule–Tate differential

Hence these classes can not contribute to nontrivial symmetries. Oneends up with invariants of osp(2s|2) in

traceless polynomials in Y A, PA and ϑAi , πiA

Structure of invariants

osp(2s|2)-invariants in representations of these type are known in themath literature (Sergeev, 1992,1998)More precisely,

U = U0 ⊕ U1

ρ(g)O = O for O ∈ U0 , ρ(g)O = det(g)O for O ∈ U1 ,

for all g ∈ O(2s) (for a tensor representation the action of o(2s) ⊂osp(2s|2) can be integrated to the group).

U0 – enveloping of o(d, 2)U1 – “chiral” symmetries

Recall, that irreducible singletons are also self-dual ∗kφ = φ∗k-Hodge conjugation in k-th column defined such that ∗k∗k = 1.It turns out that chiral symmetries are not compatible, e.g.

σk(O) = ∗kO∗k , σk ⊂ O(2s) det(σk) = −1

Finally, for an irreducible singleton the algebra of higher symmetriesis an enveloping of o(d, 2).

Conclusions

• Provides a higher analog of the usual higher spin algebra andgives a 1st-quantized interpretation to HS algebras.

• First step towards generalization of the Vasileiv nonlinear equa-tions to mixed-symmetry AdS fields

• Provides a framework to study a spin-s version of the Flato-Fronsdal theorem

• Together with agamous description for AdS (gauge) fields canbe used as an explicitly covariant and gauge invariant frame-work to analyze bulk/boundary correspondence ((partially)gaugefixed versions (Metsaev, 1999,. . . ))