Ergin Sezgin- Aspects of Higher Spin Gravity

8/3/2019 Ergin Sezgin- Aspects of Higher Spin Gravity

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Aspects of Higher Spin Gravity

Ergin Sezgin

Texas A & M University

Milos Conference, June 24, 2011
http://find/http://goback/


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Topics

Some background

Structure and properties of Vasilievs HS gravity

Recent results on geometry and observables in HS gravitybased on: E.S. and P. Sundell, arXiv:1103.2360


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Some Background

See, e.g., introduction in E.S. and P. Sundell, hep-th/0105001

Fierz & Pauli 1939: Free massive HS eqs in Minkowski4 Weinberg 1964, Weinberg & Witten 1980: No-go theorems

for HS interactions Fronsdal 1978: Free massless limit in Minkowski4 Fronsdal 1978: Free massless HS eqs in AdS4 Fradkin & Vasiliev 1987: HS algebras, cubic gravitational

interactions in AdS4 Vasiliev 1987-1992: Full HS gravity eqs of motion in AdS4


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Connection with M-theory:

E. Bergshoeff, A. Salam, E.S. and Y. Tanii, 1988

\s 012 1

32 2

52 3

72 4

9

25

11

26

0 70 56 28 8 1

1 1+1 8 28 56 70 56 28 8 1

2 1 8 28 56 70 56 28 8 1

3 1 8 28 56 70

4 1

.

..

The conjectured spectrum of massless states of M2 brane onAdS4 S

7, given by symmetric product of OSp(8|4) singletons.


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Connection with tensionless string on AdS5 S5 and free

Yang-Mills theory in the large N limit:

Haggi-Mani and Sundborg 2000, Sundborg 2001

In this limit the

R2= 1

g2YM

N

Developed further by:

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

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


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HST4/CFT3 correspondenceE.S. and Sundell conjecture, hep-th/0205131:

Vasilevs 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()2 , SHS =N2

R2AdS

d4xLHS

Klebanov and Polyakov conjecture, hep-th/0210114:

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

L 12() () (UV fixed point) SHS = NR2AdS

d4xLHSor strongly coupled IR fixed point reached by double-tracedeformation

L 2N( )2


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

Type B HS gravity which has with odd-parity bulk scalar is dual to

free fermion field at IR fixed point, or a strongly coupled UV fixedpoint reached by double trace deformation

L 2N()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, 2003

E.S. and P. Sundell, 2003

Arbitrary three point functions:

X.Yin and S. Giombi, 2009 & 2010


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What about the matter coupled CS gauge theories in 3D? Are

they dual to any HS gravity in 4D bulk?

Results to appear by:

Giombi, Minwalla, Prakash, Trivedi and Yin

Will come back to this later.

Visit also the web page for:

Simons Center Workshop on higher spin theories and holography,March 14-18, 2011


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Structure and Properties of HS Gravity

Free O(N) vector model:

S = 12

d3x

Na=1(

a)2

O(N) singlet conserved currents:

J(1s) = a(1 s)

a + s = 0+, 2, 4,...

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

Type A Model: aa bulk scalarType B Model: bulk pseudo-scalar


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Introduce commuting spinors (y, y), and package all the gauge

fields into a master field A(x,y, y):

A(x,y, y) = e yy +

yy +

yy +

+A1...s11s1 y1 yas1 y1 ys1 +

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

independent fields.


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What about the scalar field ?

Combine and spin s Weyl tensors 1s+2 into a master 0-form:

(x,y, y) = (x) + (x)yy + + 14(x)y

1 y4

+14(x)y

y

y1 y

4 +

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

, 14 14 , etc

T h HS l b ll i h ( ) i d i h h HS


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To construct the HS algebra, call it hs(4), associated with the HSgauge fields, let (y, y

) obey the oscillator algebra

y y = yy + i , y y = yy + i

where the y y denotes the operator product and yy = yydenotes the Weyl ordered product.

SO(3, 2) generators: yy , yy , yyhs(4) generators: y1y2 yn y1 y2 ym with

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

The hs(4) algebra A has an anti-involution and involutions and acting on Weyl ordered function f hs(4) as

(f(y, y)) = f(iy,iy)

(f(y, y)) = f(y, y) , (f(y, y)) = f(y,y) .


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

(A) = A , A = A () = () , = ()


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

Gravity

R, = (Einstein eq.)

R, = 0

R,

= 0

Higher spins

F(1)

,1...2s2= 1...2s2

F(1)

,1...kk+1...2s2= 0

Scalar

1...m1...n = i1...m

1...n

imn(1(12...m)

2...n)

Th t t d t k t t t b t l diffi lt it i


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The stated task turns out to be extremely difficult, as it requiresan unmanageable Weyl curvature expansion, manageable just upto cubic order (Fradkin and Vasiliev). Breakthrough came withVasilievs work in 1992, which solved this problem.

Introduce commuting internal spinorial coordinates (z, z) andthe associative product rules

y y = yy + i , z z = zz i ,

y z = yz i , z y = zy + i

Next, introduce the Grassmann even master fields

A = dx A + dz A + dz A , obeying

(A) = A , A = A () = () , = ()


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The , and maps are defined by

(y, y,z, z) = (iy,iy,iz,iz)

(y, y,z, z) = (y, y,z,z)

(y, y,z, z) = (y,y, z,z)

Next, define -product as

fg = fexpi

z

+

y

z

y

+i

z

y

z

+

y

g


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Next, define curvatures as

F = dA + A A ,D = d + A (A) .Vasiliev HS field equations are then expressed as constraints onthese curvatures:

F = ic1 dz dz + ic2 dz dz D = 0 , = eiyz , = eiyz


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Invariant under HS gauge transformations:

A = d + [A,] , = () Signature (3, 1): c1 = c2

Signatures: (4, 0) and (2, 2): c1

= c1

, c2

= 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

E i i C


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Expansion in CurvaturesStart from the initial conditions

A|Z=0 = A(x,y, y) , |Z=0 = (x,y, y)Integrating the constraints

D

= 0 ,

F = i

,

F = 0

Assuming weak, we obtain perturbative solution() , A(, A) , A()Constraint F = 0 and D = 0 can then be shown to beequivalent to:F|Z=0 = 0 , D|Z=0 = 0Substituting the solutions for (A,), these are the full HS fieldequations.

L T f i


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Lorentz Transformations

Vierbein and Lorentz connection in E = e + where

e = 12i e yy , = 14i

yy h.c.

Writing A = E + W

, one finds that W

do not transform asLorentz tensors. The remedy is to define

A = E + W + K

K i

A AZ=0 h.c.The component field in W defined in this way do transformcorrectly as Lorentz tensors.

W ki i Z


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Working in Z-space

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

F = 0 , F = 0 , D = 0are integrated in simply connected spacetime regions, with

A = L1 L

A =

L1 (

A + )

L

= L1 (L)where

L =

L(x; y, y,z, z) is a gauge function, and

A and

are

x-independent: A = 0 , = 0


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Solve the remaining constraints in Z-space:

2[

A] + [

A,

A ] =

i

2

A A + [A, A] = 0 + A + (A) = 0

with initial condition C

(y, y) = |Z=0.

Let us restrict our attention to gauge functions of the form:


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

1LK

with

K = i (L1 A

A

L)Z=0

Gauge fields, including the metric, are thus obtained algebraically,without any other integration in spacetime!

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 amplitude

computations

Vasiliev equations in terms of deformed oscillators:


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Vasiliev equations in terms of deformed oscillators:

S = z 2iA , S = z 2iAobeying (

S) = (

S) and (

S) = (

S). Then Vasiliev

equations become:

S[ S] = i(1 )

S

S =

S

S ,

S

+

(

S) = 0

dW +W W = 0d + [W,] = 0 , dS+ [W, S] = 0

Interaction Ambiguity


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Interaction Ambiguity

In the deformed oscillator algebra

S[ S] = i(1 )we still have the freedom to have instead

S[

S] = i(1

ei )

where is an arbitrary function of (). (Vasiliev, 1990).Assigning the master scalar intrinsic parity 1, and imposing paritysymmetry reduces the interaction ambiguity to:

Type A model : = 0 , Type B model : = /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 for

holography.

Further Generalization


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Further Generalization

Proposed by colorvioletE.S. nd P. Sundell, in arXiv:1103.2360.Consider the system

F + Fr(

)

Jr = 0 ,

D

= 0 ,

d

Jr = 0

[f , Jr] = 0 , f A , (Jr) = Jr , r = 1,...,nfor some n, and where

Fr() = n=0 f2n+1, r() () n

d f2n+1,r = 0 , f2n+1,r, f

= 0


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Interaction ambiguity is possibly relevant to the study of matter

coupled 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

Aharony et al, private communication

Wadia et al: Talk in this conference on W symmetry of CS gaugetheory coupled to fermions.

Global Formulation and Structure Groups


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Global Formulation and Structure Groups

Recall that we have gauge fields on M (the x-space), valued inhigher spin Lie algebra

hs(4) =

P(Y, Z) : (

P) = (

P) =

P

ad P1(P2) = P1, P2

The structure group

t

hs(4)sl(2,C)

Decompose M into coordinate charts MI labelled by I, i.e.M =

IMI.

Th l i l d li h i f bi f l ll


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The classical moduli space then consists of gauge orbits of locallydefined field configurations

WI,I,I, S,Iglued together by transition functions GII = exp(tII ) with tII tdefined on MIMI . The classical moduli space thus encodes aprincipalt-bundle with connection = t(W )where t denotes the projection tot, and an associatedt-bundlewith section (E,, S) where

E = (1t)(

W )


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Observables: O WI,I,I, S,I have the invariance properties: IO 0 off-shell for locally defined

I = t(I I)

IO = 0 on-shell for parameters I = (1t)(I) thatform sections of at-bundle associated to the principal bundle.

Thus, the observables are left invariant on-shell by thediffeomorphisms ofM.

Four natural phases (choices of structure group) in minimal


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bosonic higher spin gravity:

an unbroken phase with structure algebra hs(4)sl(2,C); a broken phase with -even higher-spin structure algebrahs+(4)sl(2,C)where

hs+(4) =1

2(1 + )hs(4)

(proposed by E.S. and P. Sundell).

a broken phase with chiral higher-spin structure algebraconsisting of all polynomials in hs+(4) that are purelyholomorphic or anti-holomorphic (proposed by Vasiliev).

the broken Lorentz invariant phase with structure algebragiven by the canonical sl(2,C) which is typically assumed in

the literature on HS gravity.


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Using the geometric formulation -even structure group, we have:

proposed a formulae minimal areas of higher spin metrics

constructed on-shell closed abelian forms of positive even

degrees.

we introduced tensorial coset coordinates and demonstratehow single derivatives with respect coordinates of higher ranksfactorize into multiple derivatives with respect to lower ranks.

Ergin Sezgin- Aspects of Higher Spin Gravity

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