Basics of Higher Spin Gravity

Ergin Sezgin

Texas A & M University

Istanbul 2011 Conference, Koc University, August 3

Background

I HS gravity was formulated by Vasiliev in 1992. In its simplestform it is an extension of ordinary gravity that includes amassless scalar and massless fields with spins s = 3, 4, . . . ,∞each occurring once.

I Cosmological constant is essential. Its inverse powers occur inhigher derivative interactions.

I For a brief history of the subject, see:E.S. and P. Sundell, hep-th/0105001For a review, see:M. Vasiliev, arXiv:hep-th/9910096

I Connection with tensionless string on AdS5 × S5 and freeYang-Mills theory in the large N limit:

Haggi-Mani and Sundborg 2000, Sundborg 2001

In this limit the α′

R2 = 1g2YMN

→∞

Developed further by:

E.S. & Sundell, 2002Beisert, Bianchi, Morales, Samtleben, 2003...

See also: E. Witten, talk given at J.H. Schwarz 60th BirthdayConference, 2001.

Holography conjecturesE.S. and Sundell, hep-th/0205131:

Vasilev’s HS gravity (Type A with parity even bulk scalar) is dualto free SU(N) valued scalar field theory in 1/N2 expansion where

L ∼ tr(∂φ)2 , SHS =N2

R2AdS

∫d4xLHS

Klebanov and Polyakov, hep-th/0210114:

Vasiliev’s HS gravity is dual to either free O(N) vector model in1/N expansion

L ∼ 12(∂~φ) · (∂~φ) (UV fixed point) SHS = N

R2AdS

∫d4xLHS

or strongly coupled IR fixed point reached by double-tracedeformation

∆L ∼ λ2N (~φ · ~φ)2

(in the λ→∞ limit), depending on whether the bulk scalar isquantized with regular or irregular boundary condition (have AdSenergies E0 = 1 or 2, respectively).

Variant conjecture: E.S. and P. Sundell, 2003

Type B HS gravity which has with odd-parity bulk scalar is dual tofree fermion field at IR fixed point, or a strongly coupled UV fixedpoint reached by double trace deformation

∆L ∼ λ2N (ψψ)2 (Gross-Neveu model)

Evidence for the conjecture so far in theories in the case of bothboundary conditions, and to leading order in 1/N has been found.

Cubic scalar field couplings:

Petkou, 2003E.S. and P. Sundell, 2003

Arbitrary three point functions:

X.Yin and S. Giombi, 2009 & 2010

Structure and Properties of HS Gravity

Spectrum?

Consider the free O(N) vector model:

S = 12

∫d3x

∑Na=1(∂µφ

a)2

O(N) singlet conserved currents:

J(µ1···µs) = φa←→∂ (µ1 · · ·

←→∂ µs)φ

a + · · · s = 0+, 2, 4, ...

This is the spectrum of massless HS fields in minimal bosonic HSgravity in 4D.

Type A Model: φaφa ↔ bulk scalarType B Model: ψψ ↔ bulk pseudo-scalar

Introduce commuting spinors (yα, yα), and package all the gaugefields into a master field Aµ(x, y, y):

Aµ(x, y, y) = eααµ yαyα+ ωαβµ yαyβ + ωαβµ yαyβ + · · ·

+Aα1...αs−1α1···αs−1µ yα1 · · · yas−1 yα1 · · · yαs−1 + · · ·

Impose constraints on A(x, y, y) such that only s = 2, 4, 6, ... areindependent fields.

What about the scalar field φ?

Combine φ and spin s Weyl tensors φα1···αs+2 into a master 0-form:

Φ(x, y, y) = φ(x) + φαα(x)yαyα + · · ·+ φα1···α4(x)yα1 · · · yα4

+φααα1···α4(x)yαyαyα1 · · · yα4 + · · ·

Impose constraint on Φ(x, y, y) such that

φαα ∼ ∂ααφ , φααα1···α4 ∼ ∂ααφα1···α4 , etc

To construct the HS algebra, introduce the oscillator algebra

yα ? yβ = yαyβ + iεαβ , yα ? yβ = yαyβ + iεαβ

where the yα ? yβ denotes the operator product and yαyβ = yβyαdenotes the Weyl ordered product.

HS extension of SO(3, 2) is obtained from the algebra ofpolynomials P (y, y) modulo the projection and reality conditions:

τ(P (y, y)) ≡ P (iy, iy) = −P , P † = −P .

These conditions define the Lie algebra hs(4) with respect to thebracket [P,Q] = P ? Q−Q ? P , since τ and the † have theproperties:

τ(P ? Q) = τ(Q) ? τ(P ) , (P ? Q)† = Q† ? P † .

The algebra hs(4) is a direct sum of spaces of monomials in y andy of degree 2, 6, 10, ....

SO(3, 2) generators: yαyβ , yαyβ , yαyα

hs(4) generators: yα1yα2 · · · yαn yα1 yα2 · · · yαm with

n+m = 4`+ 2 , ` = 0, 1, 2, ...

The hs(4) gauge theory is constructed by introducing a one-formgauge field A = dxµAµ(x, y, y) and a zero-form Φ(x, y, y)satisfying the conditions

τ(A) = −A , A† = −A τ(Φ) = π(Φ) , Φ† = π(Φ)

where

τ(f(y, y)) = f(iy, iy)

π(f(y, y)) = f(−y, y)

π(f(y, y)) = f(y,−y) .

The idea is to construct hs(4) invariant gauge theory such thattreating Φ and HS fields as weak fields, the y-expansion to lowestorder in weak fields would yield:

Gravity

Rαβ,γδ = Φαβγδ (Einstein eq.)

Rαβ,γδ = 0

Rαβ,γδ = 0

Higher spins

{F

(1)αβ,γ1...γ2s−2

= Φαβγ1...γ2s−2

F(1)αβ,γ1...γkγk+1...γ2s−2

= 0

Scalar

∇ααΦβ1...βm

β1...βn = iΦαβ1...βmαβ1...βn

−imnεα(β1εα(β1Φβ2...βm)

β2...βn)

The stated task turns out to be extremely difficult, as it requiresan unmanageable Weyl curvature expansion, with great difficultydoable just up to cubic order (Fradkin and Vasiliev). Breakthroughcame with Vasiliev’s work in 1992, which solved this problem asfollows.Introduce commuting internal spinorial coordinates (zα, zα) andthe associative product rules

yα ? yβ = yαyβ + iεαβ , zα ? zβ = zαzβ − i εαβ ,

yα ? zβ = yαzβ − i εαβ , zα ? yβ = zαyβ + i εαβ

Next, introduce the Grassmann even master fields

A = dxµAµ + dzαAα + dzαAα , Φ

obeying

τ(A) = −A , A† = −A τ(Φ) = π(Φ) , Φ† = π(Φ)

The τ , π and π maps are defined by

τ(y, y, z, z) = (iy, iy,−iz,−iz)

π(y, y, z, z) = (−y, y,−z,z)

π(y, y, z, z) = (y,−y, z,−z)

Next, define ?-product as

f ∗ g = fexp

[i

( ←−∂

∂zα+

←−∂

∂yα

)( −→∂

∂zα−−→∂

∂yα

)

+i

( ←−∂

∂zα−←−∂

∂yα

)(−→∂∂zα

+

−→∂

∂yα

)]g

Next, define curvatures as

F = dA+ A ? A ,

DΦ = dΦ + A ? Φ− Φ ? π(A) .

Vasiliev HS field equations are then expressed as constraints onthese curvatures:

F = ic1 dzα ∧ dzα Φ ? κ+ ic2 dz

α ∧ dzα Φ ? κ

DΦ = 0 , κ = eiyαzα , κ = e−iy

αzα

Invariant under HS gauge transformations:

δA = dε+ [A, ε]? , δΦ = Φ ? π(ε)− ε ? Φ

Signature (3, 1): c∗1 = c2

Signatures: (4, 0) and (2, 2): c∗1 = c1, c∗2 = c2

Chiral Model: c2 = 0 for (4,0), (2,2)

with suitable reality conditions on y, y, z, z understood in (4, 0)and (2, 2) signatures.C. Iazeolla, E. Sezgin and P. Sundell, arXiv:0706.2983

Expansion in CurvaturesStart from the initial conditions

Aµ|Z=0 = Aµ(x, y, y) , Φ|Z=0 = Φ(x, y, y)

Integrate the constraints

DαΦ = 0 , Fαβ = iεαβΦ ? κ , Fαµ = 0

Assuming Φ weak, we obtain perturbative solution

Φ(Φ) , Aµ(Φ, A) , Aα(Φ)

Constraint Fµν = 0 and DµΦ = 0 can then be shown to beequivalent to:

Fµν |Z=0 = 0 , DµΦ|Z=0 = 0

With the solutions for (A, Φ) substituted, these are the full HSfield equations.

Lorentz Transformations

Vierbein and Lorentz connection in Eµ = eµ + ωµ where

eµ = 12ie

ααµ yαyα , ωµ = 1

4iωαβµ yαyβ − h.c.

Writing Aµ = Eµ +W ′µ, one finds that W ′µ do not transform asLorentz tensors. The remedy is to define

Aµ = Eµ +Wµ +Kµ

Kµ ≡ iωαβµ

(Aα ? Aβ

)Z=0− h.c.

The component field in Wµ defined in this way do transformcorrectly as Lorentz tensors.

HS equations in weak field expansion

Treating gravity as strong field, Φ and HS fields as weak fields, theresulting HS field equations are of the form

eαβ ∧Rβα = eα

β ∧ Jβα ,

e(α1

β ∧ F (1)

α2...αs−1)βα1...αs−1= e(α1

β ∧ Jα2...αs−1)βα1...αs−1

(∇2 + 2)φ = ∇µPµ − i2(σµ)ααPµ,αα .

Sources J and P are highly complicated functions of Weylcurvatures and their derivatives. Undetermined so far beyond thelowest (quadratic) level.

Working in Z-space

Very advantageous in finding exact solutions and in amplitudecomputations. In this approach, the constraints

Fµν = 0 , Fµα = 0 , DµΦ = 0

are integrated in simply connected spacetime regions, with

Aµ = L−1 ? ∂µL

Aα = L−1 ? (A′α + ∂α)L

Φ = L−1 ? Φ′ ? π(L)

where L = L(x; y, y, z, z) is a gauge function, and A′α and Φ′ arex-independent: ∂µA

′α = 0 , ∂µΦ′ = 0

Solve the remaining constraints in Z-space:

2∂[αA′β] + [A′α, A

′β]? = − i

2εαβΦ′ ? κ

∂αA′β− ∂βA

′α + [A′α, A

′β]? = 0

∂αΦ′ + A′α ? Φ′ + Φ′ ? π(A′α) = 0

with initial condition C ′(y, y) = Φ′|Z=0.

Let us restrict our attention to gauge functions of the form:L = L(x; y, y). The gauge fields can then be obtained from

eµ + ωµ +Wµ = L−1∂µL−Kµ

with

Kµ = iωµαβ (L−1 ? A′α ? A

′β ? L)

∣∣∣Z=0

Gauge fields, including the metric, are thus obtained by algebraicmeans!

Prokushkin and Vasiliev, 1999Iazeolla, E.S. and Sundell, 2007: SO(3, 1) invariant and otherexact solutionsDidenko and Vasiliev, 2009: Static BPS black holesGiombi and Yin, 2010: Great simplifications in amplitudecomputations

Vasiliev equations in terms of deformed oscillators:

Sα = zα − 2iAα , Sα = zα − 2iAα

obeying π(Sα) = −π(Sα) and π(Sα) = −π(Sα). Then Vasilievequations become:

S[α ? Sβ] = −iεαβ(1− Φ ? κ)

Sα ? Sα = Sα ? Sα , Sα ? Φ + Φ ? π(Sα) = 0

dW ′ + W ′ ? W ′ = 0

dΦ + [W ′, Φ]π = 0 , dS + [W ′, S]? = 0

Interaction Ambiguity

In the deformed oscillator algebra

S[α ? Sβ] = −iεαβ(1− Φ ? κ)

we still have the freedom to have instead

S[α ? Sβ] = −iεαβ(1− Φ ? eiΘ ? κ)

where Θ is an arbitrary function of Φ ? π(Φ). (Vasiliev, 1990).

Assigning the master scalar intrinsic parity ±1, and imposing paritysymmetry reduces the interaction ambiguity to:

Type A model : Θ = 0 , Type B model : Θ = π/2 .

(E.S. and Sundell, 2003). They contain the scalars in the D(1, 0)and D(2, 0) irreps of AdS4 group, respectively. Relevant forholography.

Interaction ambiguity is possibly relevant to the study of mattercoupled CS theories in 3D as holographic duals of bulk HS gravity.This remains to be seen.

Giombi and Yin, Simons Workshop, 2011.

S. Giombi, S. Minwalla, S. Prakash, S. Trivedi, X. Yin, to appear

O. Aharony, G. Gur-Ari, and R. Yacoby, in progress

Higher Spin Supergravities

Straightforward. Introduce anti-commuting oscillators. Considerfor example N = 8 HS supergravity. Introduce anticommutingoscillators:

θi ? θj = δij + θiθj , i = 1, ..., 8

Master fields A, Φ now depend on θi as well. Extend the definitionof the τ and π maps for these fields. Impose

Φ† = π (Φ ) ? Γ , Γ = θ1 · · · θ8

In the HS field equations κ→ κΓ Resulting with HS extension ofde Wit-Nicolai N = 8, D = 4 supergravity. E.S. and Sundell, 1997

E. Bergshoeff, A. Salam, E.S. and Y. Tanii, 1988 conjectured thatthe spectrum of massless states of M2 brane on AdS4 × S7, aregiven by the symmetric product of OSp(8|4) singletons:

`\s 0 12 1 3

2 2 52 3 7

2 4 92

5 112

6 · · ·

0 70 56 28 8 11 1+1 8 28 56 70 56 28 8 12 1 8 28 56 70 56 28 8 13 1 8 28 56 70 · · ·4 1 · · ·...

Higher Dimensions

I Straightforward generalization of Vasilev equations in spinorformalism not possible, since special properties in D = 4 notavailable in D > 4.

I Only linearized equations have been constructed in 5D and7D so far. E.S. and P. Sundell, 2001 and 2002

I Introducing vector oscillators Y iA and ZiA, where

A = 0, 1, . . . , D − 1, D is an SO(D − 1, 2) vector index and iis an Sp(2, R) doublet index, obeying suitable commutationrules, Vasiliev proposed HS equations in D dimensions.

A = dxµAµ(x, Y, Z) + dZiAAiA(x, Y, Z) , Φ = Φ(x, Y, Z) ,

I Sp(1, R) invariance needs to be imposed. There are unsettledproblems. Also the SUSY extension is not known.A. Sagnotti, E.S. and P. Sundell, 2005

Action?

Multiplying the constraints by Lagrange multipliers and integratingcannot provide the answer.Recent development:N. Boulanger and P. Sundell, 2011 introduced auxiliary extramaster forms and Lagrange multiplier master forms to write downan action.Structure and consequences not well understood yet.

Global formulation, geometry and observables?E.S. and P. Sundell, 2011

I proposed an infinite dimensional structure group thatgeneralizes the Lorentz group.

I proposed a formula for minimal area

I constructed on-shell closed abelian forms of positive evendegrees.

I introduced tensorial coset coordinates and demonstrate howsingle derivatives with respect coordinates of higher ranksfactorize into multiple derivatives with respect to lower ranks.e.g.

∂µνρφ ∼ ∂µ∂ν∂ρφ

Open problems

I Develop the action formalism

I Study quantum behaviour (e.g. how powerful is HS symmetryin controlling the effective action)

I Study spontaneous breaking of HS symmetry. Grouptheoretical analysis shows that the candidate Goldstone modesare composites of a scalar with spin s− 2 field.L. Girardello, M. Porrati and A. Zaffaroni, 2002

I Explore holographic duals and study the interactionambiguities in HS gravity

Open problems (con’ed)

I Introduce internal symmetry

I Construct HS supergravities in D > 4

I Find the relation between HS gravity and string field theory intensionless limit.Augusto Sagnotti’s talk.

I Explore solutions of HS gravity and investigate theirproperties.

I Investigate the cosmological implications of HS gravity(tensionless string phase followed by spontaneous symmetrybraking. Any consequences for the CMB data?)