Conformal Chern-Simons Gravityquark.itp.tuwien.ac.at/~grumil/ESI2012/slides/Afshar_ESI.pdf ·...

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.Outline

. .Motivation

. . . . . . . . . . .Conformal Chern-Simons Gravity

. . . . .Boundary CFT

. .Conclusion

Conformal Chern-Simons GravityESI Workshop on Higher Spin

Hamid R. Afshar

Vienna University of Technology

17 April 2012

Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller and Niklas Johansson

hep-th/1106.6299, hep-th/1110.5644

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.Outline

. .Motivation



. .Conclusion

Outline

• Motivation

• Conformal gravity in 3d

• Conserved charges

• Boundary CFT

• Summary

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.Outline

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. .Conclusion

Partial masslessnessA massive spin-2 field in (A)dS background obeys the linearizedequation,

Gµν −1

2m2 (hµν − gµνh) = 0

Taking the double divergence and the trace of this equation, oneobtains,

∇µ∇νhµν − ∇2h = 0,

[Λ− D − 1

2m2

]h = 0

In the massive case, hµν does not have to be traceless at thepartially massless point (Deser et al. 1983, 2001) for which themass is tuned as,

m2 =2

D − 1Λ

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. .Conclusion

Gauge enhancement

The following scalar gauge invariance appears,

hµν → hµν +

(∇(µ∇ν) +

2Λ

(D − 1)(D − 2)gµν

)ζ

This new gauge symmetry induces a Bianchi identity,

∇µ∇νGµν +Λ

D − 1Gρ

ρ = 0

which reduces one degree of freedom. Non-linear realiziation ofthis symmetry,

gµν → e2Ω(x)gµν .

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. .Conclusion

Conformal gravity in 3dConformal gravity (Weyl 1918) is a gravity theory that a Weyltransformation of the metric,

gµν → e2Ω(x)gµν

is an exact symmetry of the equations of motion. Conformalgravity in three dimensions (Deser et al. 1982)

S =k

4π

∫Md3x εµνλ Γρµσ

(∂νΓ

σλρ +

23Γ

σντΓ

τλρ

)+

k

4π

∫∂Md2x

√−γ(KαβKαβ − 1

2K 2).

is a topological theory with the equations of motion,

Cµν ≡ 1

2

(εµαβ∇αR

νβ + εναβ∇αR

µβ

)= 0.

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. .Conclusion

Boundary conditions in CSG

• Boundary degrees of freedom emerge under suitable boundaryconditions by the nonvanishing gauge symmetries acting onthe boundary.

• In conformal gravities like CSG we can afford a Weyl factorthat in principle can change the boundary condition drasticallybut doesn’t affect the equations of motion

gµν = e2φ(x+, x−, y)

(dx+dx− + dy2

y2+ hµν

).

with

h+− = h±y = O(1) , h++h−− = O(1/y)

and

φ = b ln y + f (x+, x−) +O(y2).

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. .Conclusion

Brown-York Stress tensor

• Using the AdS/CFT dictionary, we calculating the responsefunctions,

δS =1

2

∫∂M

d2x

√−γ(0)

(Tαβδγ

(0)αβ + Jαβδγ

(1)αβ

)in Gaussian normal coordinate, (eρ ∝ 1/y)

ds2 = dρ2 +(γ(0)αβ e2ρ + γ

(1)αβ eρ + γ

(2)αβ + . . .

)dxαdxβ

where γ(1) describes new additional Weyl graviton mode,

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. .Conclusion

Boundary conditions on the Weyl factor

The appearance of an additional symmetry (Weyl) classifies theboundary conditions on the Weyl factor

φ = b ln y + f (x+, x−) +O(y2),

to three cases:

I. Trivial Weyl factor φ = 0.

II. Fixed Weyl factor δφ = 0.

III. Free Weyl factor δφ 6= 0.

In gravity theories where we don’t have Weyl symmetry we arealways in the first case. These boundary conditions lead to thefollowing correlators between the response functions in case 1,

〈J(z , z)J(0, 0)〉 = 2kz

z3, 〈TR(z)TR(0)〉 = 6k

z4.

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.Outline

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. .Conclusion

Asymptotic symmetry group

In a diffeomorphism×Weyl invariant theory the asymptoticsymmetry group is generated by a combination of diffeomorphismsgenerated by a vector field ξ and Weyl rescalings generated by ascalar field Ω:

Lξgµν + 2Ωgµν = δgµν .

Here δg refers to the transformations that preserve the boundaryconditions. In the rest to remove gravitational anomaly we require,

f (x+, x−) = f+(x+) + f−(x

−),

where

f± =f02+

pf2(t ± ϕ) +

∑n 6=0

f(n)± e−in(t±ϕ) .

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.Outline

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. .Conclusion

Asymptotic symmetry group

The classification of the boundary conditions on the Weyl rescalingto three different cases will induce a same classification on ASG;

I. Trivial Weyl rescaling Ω = O(y2).

II. Fixed Weyl rescaling Ω = −b2 ∂ · ε− ε · ∂f +O(y2).

III. Free Weyl rescaling Ω = Ω(x+) + Ω(x−) +O(y2).

With the diffeomorphisms

ξ± = ε±(x±)− y2

2∂∓∂ · ε+O(y3) ,

ξy =y

2∂ · ε+O(y3) ,

in all three cases above.

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. .Conclusion

frame formalism

This theory has a first order format

S =k

2π

∫MTr

(ω ∧ dω +

2

3ω ∧ ω ∧ ω + λ ∧ T

)where ω is the spin connection one-form and

T = de + ω ∧ e

is the torsion tow-form. We can think of ω as an SO(2, 1) gaugefield. Once the torsion vanishes, k is quantized for topologicalreasons. The spin-connection 1-form ω defines the curvature2-form,

R = dω + ω ∧ ω.

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. .Conclusion

Gauge theory formulationA Chern–Simons gauge theory with SO(3, 2) gauge group,

SCS =k

4π

∫M

Tr(A ∧ dA+ 2

3 A ∧ A ∧ A),

recovers the first order action — as well as the requirement thatthe Dreibein must be invertible — for a specific partial gaugefixing (Horne–Witten 1989), breaking

SO(3, 2) → SL(2,R)L × SL(2,R)R × U(1)Weyl.

The first order action differs from 2nd order (metric) action by

∆S =k

12π

∫MTr(e−1 de

)3 − k

4π

∫∂M

Tr(ω dee−1

).

Which leads to gravitational anomaly (Kraus–Larsen 2005)

∇αTαβ = γαβ(0)ε

δγ∂δ∂λΓλαγ [γ

(0)].

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. .Conclusion

Gauge transformations

Local Poincare transformations take form,

δPeiµ = −εi jke

jµθ

k − (∂µξν)e i ν − ξν∂νe

iµ

δPωiµ = −Dµθ

i − (∂µξν)ωi

ν − ξν∂νωiµ

δPλiµ = −εi jkλ

jµθ

k − (∂µξν)λi

ν − ξν∂νλiµ .

Weyl transformation,

δW e iµ = Ω e iµ

δWωiµ = εijkejµek

ν∂νΩ

δWλiµ = −2Dµ(e

iν∂νΩ)− Ωλiµ .

We find the gauge generators GP , GW that generate thesetransformations.

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. .Conclusion

Conserved charges in CSG

Varying the generators and integrating over spacelike hypersurfacewith boundary leads to a regular term and a boundary term, toobtain differentiable charges Q we must add a boundary piece tothe generators, G = G + Γ, which corresponds to the charge,

δQP [ξρ] =

2π∫0

dϕ δΓP = − k

2π

2π∫0

dϕ[ξρ(e iρ δλiϕ + λi

ρ δeiϕ

+2ωiρ δωiϕ

)+ 2θi δωiϕ

].

δQW [Ω] =

2π∫0

dϕ δΓW =k

π

2π∫0

dϕ (e iµ∂µΩ) δeiϕ .

Here QP and QW are the diffeomorphism and Weyl charges.

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. .Conclusion

Dirac brackets in CSG

Defining the generators of asymptotic symmetries as

Tn = Gξ[ε+ = e inx

+, ε− = 0] + GW [ε+]

Tn = Gξ[ε+ = 0, ε− = −e−inx− ] + GW [ε−]

Jn = GW [Ω = −e inx+]

we find the corresponding Dirac brackets

iTn, Tm∗ = (n −m)Tn+m − k n3 δn+m,0 ,

iTn, Tm∗ = (n −m) Tn+m + k n3 δn+m,0 ,

iJn, Jm∗ = 2k n δn+m,0 .

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. .Conclusion

Boundary affine algebra in CSGShifting from cylinder to the plane and converting the Poissonbrackets into commutators by iq, p = [q, p], the dual CFT ofCSG with the aforementioned boundary conditions takes the form

[Ln, Lm] = (n −m) Ln+m − k (n3 − n) δn+m,0 ,

[Ln, Lm] = (n −m) Ln+m + k (n3 − n) δn+m,0 ,

[Jn, Jm] = 2k n δn+m,0 .

The Virasoro generators are defined as generators of the combineddiffeomorphisms and Weyl rescalings acting on the boundary

ξ± = ε±(x±)− y2

2∂∓∂ · ε+O(y3) ,

ξy =y

2∂ · ε+O(y3) ,

Ω = −b

2∂ · ε− ε · ∂f +O(y2) .

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. .Conclusion

The generators of pure diffeomorphismIn the case where f − = 0,

Ω = −b

2∂+ε

+ − ε+ ∂+f ,

and the compensating Weyl charge can be written as,

δQW =k

π

∫ 2π

0dϕ δf ∂ϕ(

b

2∂+e

inx+ + e inx+∂+f )

= kδ

(∑m∈Z

m(n −m)f(m)+ f

(n−m)+

)− ib

2n∂+f

(n)+

showing that for generating a pure diffeomorphism by ε+ weshould shift the left generator as

Ln = Ln +1

4k

∑m∈Z

: JmJn−m : −b2 (n + 1)Jn .

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.Outline

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. .Conclusion

The generators of pure diffeomorphism

The resulting algebra contains a U(1) current algebra,

[Ln, Lm] = (n −m)Ln+m +cL12

(n3 − n) δn+m,0 ,

[Ln, Lm] = (n −m) Ln+m +cR12

(n3 − n) δn+m,0 ,

[Jn, Jm] = 2k n δn+m,0 ,

[Ln, Jm] = −mJn+m − bk n(n + 1) δn+m,0 .

with cL = −12k + 1 + 6kb2 and cR = 12k . When b = 0, thetransformations parametrized by ε− don’t need compensatingrescaling so the L-algebra generates pure diffeomorphisms as well.

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. .Conclusion

Scalar field with background charge

We note that a scalar field with background charge Q with action

SQ =1

4π`2s

∫d2x

√g gαβ∂αX∂βX +

Q

4π`s

∫d2x

√g XR

leads to the same algebraic structure. The holomorphic part of thestress tensor is

T = − 1

`2s: ∂X∂X : +

Q

`s∂2X

whose Fourier-components coincide with the last two terms in Ln

upon identifying Q =√kb and `2s = 4k. The central charge of the

scalar field is shifted by

c = 1 + 6Q2 = 1 + 6kb2

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. .Conclusion

Other CFT’s

The conservation of the Weyl charge is equivalent to requiring

f(n)+ Ω

(n)− = f

(n)− Ω

(n)+ ∀n 6= 0 .

A functional relation between f+ and f−

f(n)+ = Cnf

(n)−

for some constants Cn. The f− = 0 choice satisfies this. Forgeneral Cn the allowed Weyl rescalings have Fourier modes of theform Ω = −e−inx− − C−ne

inx+ . Computing the correspondingcharge is straightforward and yields

Q[Ω = −e−inx− − C−neinx+ ] = 2kin (|Cn|2 − 1)f

(n)−

Note that the charge vanishes for |Cn| = 1 and the correspondinggenerators commute with each other.

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. .Conclusion

Summary

• In this talk we focused on conformal Chern–Simons gravity(CSG) and performed a holographic analysis of it.

• We did this by doing canonical analysis in its first orderformalism.

• Our holographic results are very based on the boundaryconditions where the asymptotic line-element is conformallyAAdS3.

• The holomorphic Weyl factor in the theory emerged as a freechiral boson in the theory shifting cL → cL + 1 + 6kb2 wherethe shift by one is a quantum contribution.

• The dynamics of this scalar field in the CFT is determinedsolely by boundary and consistency conditions like removingthe gravitational anomaly.

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. .Conclusion

Discussion

• CSG allows for geometries that are not diffeomorphic to eachother, but nevertheless are gauge-equivalent.

• Similar features to those described here are likely to occur inany theory of gravity with additional gauge symmetries.

Conformal Chern-Simons Gravityquark.itp.tuwien.ac.at/~grumil/ESI2012/slides/Afshar_ESI.pdf ·...

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