Conformal Chern-Simons Gravity ESI Workshop on Higher Spin Hamid R. - - PowerPoint PPT Presentation

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Conformal Chern-Simons Gravity

ESI Workshop on Higher Spin Hamid R. Afshar

Vienna University of Technology

17 April 2012

Branislav Cvetkovi´ c, Sabine Ertl, Daniel Grumiller and Niklas Johansson hep-th/1106.6299, hep-th/1110.5644

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Outline

• Motivation
• Conformal gravity in 3d
• Conserved charges
• Boundary CFT
• Summary

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Partial masslessness

A massive spin-2 field in (A)dS background obeys the linearized equation, Gµν − 1 2m2 (hµν − ¯ gµνh) = 0 Taking the double divergence and the trace of this equation, one

• btains,

¯ ∇µ ¯ ∇νhµν − ¯ ∇2h = 0, [ Λ − D − 1 2 m2 ] h = 0 In the massive case, hµν does not have to be traceless at the partially massless point (Deser et al. 1983, 2001) for which the mass is tuned as, m2 = 2 D − 1Λ

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . 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 − 1Gρρ = 0 which reduces one degree of freedom. Non-linear realiziation of this symmetry, gµν → e2Ω(x)gµν.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Conformal gravity in 3d

Conformal gravity (Weyl 1918) is a gravity theory that a Weyl transformation of the metric, gµν → e2Ω(x)gµν is an exact symmetry of the equations of motion. Conformal gravity in three dimensions (Deser et al. 1982) S = k 4π ∫

M

d3x ǫµνλ Γρµσ ( ∂νΓσλρ + 2

3ΓσντΓτ λρ

) + k 4π ∫

∂M

d2x√−γ ( K αβKαβ − 1 2K 2) . is a topological theory with the equations of motion, C µν ≡ 1 2 ( εµαβ∇αRν

β + εναβ∇αRµ β

) = 0.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Boundary conditions in CSG

• Boundary degrees of freedom emerge under suitable boundary

conditions by the nonvanishing gauge symmetries acting on the boundary.

• In conformal gravities like CSG we can afford a Weyl factor

that in principle can change the boundary condition drastically but 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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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Brown-York Stress tensor

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

functions, δS = 1 2 ∫

∂M

d2x √ −γ(0) ( T αβδγ(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,

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Boundary conditions on the Weyl factor

The appearance of an additional symmetry (Weyl) classifies the boundary 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 = 0.

In gravity theories where we don’t have Weyl symmetry we are always in the first case. These boundary conditions lead to the following correlators between the response functions in case 1, J(z, ¯ z)J(0, 0) = 2k¯ z z3 , T R(z)T R(0) = 6k z4 .

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Asymptotic symmetry group

In a diffeomorphism×Weyl invariant theory the asymptotic symmetry group is generated by a combination of diffeomorphisms generated by a vector field ξ and Weyl rescalings generated by a scalar field Ω: Lξgµν + 2Ωgµν = δgµν . Here δg refers to the transformations that preserve the boundary

• conditions. In the rest to remove gravitational anomaly we require,

f (x+, x−) = f+(x+) + f−(x−), where f± = f0 2 + pf 2 (t ± ϕ) + ∑

n=0

f (n)

± e−in(t±ϕ) .

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Asymptotic symmetry group

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

• I. Trivial Weyl rescaling Ω = O(y2).
• II. Fixed Weyl rescaling Ω = − b

2 ∂ · 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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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

frame formalism

This theory has a first order format S = k 2π ∫

M

Tr ( ω ∧ dω + 2 3 ω ∧ 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) gauge

• field. Once the torsion vanishes, k is quantized for topological
• reasons. The spin-connection 1-form ω defines the curvature

2-form, R = dω + ω ∧ ω.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Gauge theory formulation

A 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 that the Dreibein must be invertible — for a specific partial gauge fixing (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 ∫

M

Tr ( e−1 de )3 − k 4π ∫

∂M

Tr ( ω dee−1) . Which leads to gravitational anomaly (Kraus–Larsen 2005) ∇αT αβ = γαβ

(0)εδγ∂δ∂λΓλ αγ[γ(0)].

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Gauge transformations

Local Poincar´ e transformations take form, δPei µ = −ǫi jkej µθk − (∂µξν)ei ν − ξν∂νei µ δPωi µ = −Dµθi − (∂µξν)ωi ν − ξν∂νωi µ δPλi µ = −ǫi jkλj µθk − (∂µξν)λi ν − ξν∂νλi µ . Weyl transformation, δW ei µ = Ω ei µ δW ωi µ = ǫijkejµekν∂νΩ δW λi µ = −2 Dµ(eiν∂νΩ) − Ω λi µ . We find the gauge generators GP, GW that generate these transformations.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Conserved charges in CSG

Varying the generators and integrating over spacelike hypersurface with boundary leads to a regular term and a boundary term, to

• btain differentiable charges Q we must add a boundary piece to

the generators, ˜ G = G + Γ, which corresponds to the charge, δQP[ξρ] =

2π

∫ dϕ δΓP = − k 2π

2π

∫ dϕ [ ξρ( ei ρ δλiϕ + λi ρ δeiϕ +2ωi ρ δωiϕ ) + 2θi δωiϕ ] . δQW [Ω] =

2π

∫ dϕ δΓW = k π

2π

∫ dϕ (eiµ∂µΩ) δeiϕ . Here QP and QW are the diffeomorphism and Weyl charges.

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Dirac brackets in CSG

Defining the generators of asymptotic symmetries as Tn = ˜ Gξ[ε+ = einx+, ε− = 0] + ˜ GW [ε+] ¯ Tn = ˜ Gξ[ε+ = 0, ε− = −e−inx−] + ˜ GW [ε−] Jn = ˜ GW [Ω = −einx+] we find the corresponding Dirac brackets i{Tn, Tm}∗ = (n − m) Tn+m − k n3 δn+m,0 , i{ ¯ Tn, ¯ Tm}∗ = (n − m) ¯ Tn+m + k n3 δn+m,0 , i{Jn, Jm}∗ = 2k n δn+m,0 .

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Boundary affine algebra in CSG

Shifting from cylinder to the plane and converting the Poisson brackets into commutators by i{q, p} = [ˆ q, ˆ p], the dual CFT of CSG 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 combined diffeomorphisms 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) .

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

The generators of pure diffeomorphism

In the case where f − = 0, Ω = −b 2 ∂+ε+ − ε+ ∂+f , and the compensating Weyl charge can be written as, δQW = k π ∫ 2π dϕ δf ∂ϕ(b 2∂+einx+ + einx+∂+f ) = kδ (∑

m∈Z

m(n − m)f (m)

+

f (n−m)

+

) − ib 2 n∂+f (n)

+

showing that for generating a pure diffeomorphism by ε+ we should shift the left generator as Ln = Ln + 1 4k ∑

m∈Z

: JmJn−m : − b

2 (n + 1) Jn .

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

The generators of pure diffeomorphism

The resulting algebra contains a U(1) current algebra, [Ln, Lm] = (n − m) Ln+m + cL 12 (n3 − n) δn+m,0 , [¯ Ln, ¯ Lm] = (n − m) ¯ Ln+m + cR 12 (n3 − n) δn+m,0 , [Jn, Jm] = 2k n δn+m,0 , [Ln, Jm] = −m Jn+m − bk n(n + 1) δn+m,0 . with cL = −12k + 1 + 6kb2 and cR = 12k. When b = 0, the transformations parametrized by ε− don’t need compensating rescaling so the ¯ L-algebra generates pure diffeomorphisms as well.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Scalar field with background charge

We note that a scalar field with background charge Q with action SQ = 1 4πℓ2

s

∫ d2x√g gαβ∂αX∂βX + Q 4πℓs ∫ d2x√g XR leads to the same algebraic structure. The holomorphic part of the stress tensor is T = − 1 ℓ2

s

: ∂X∂X : +Q ℓs ∂2X whose Fourier-components coincide with the last two terms in Ln upon identifying Q = √ kb and ℓ2

s = 4k. The central charge of the

scalar field is shifted by c = 1 + 6Q2 = 1 + 6kb2

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Other CFT’s

The conservation of the Weyl charge is equivalent to requiring f (n)

+ Ω(n) − = f (n) − Ω(n) +

∀n = 0 . A functional relation between f+ and f− f (n)

+

= Cnf (n)

−

for some constants Cn. The f− = 0 choice satisfies this. For general Cn the allowed Weyl rescalings have Fourier modes of the form Ω = −e−inx− − C−neinx+. Computing the corresponding charge 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 corresponding generators commute with each other.

. . . . . .

. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . 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 order

formalism.

• Our holographic results are very based on the boundary

conditions where the asymptotic line-element is conformally AAdS3.

• The holomorphic Weyl factor in the theory emerged as a free

chiral boson in the theory shifting cL → cL + 1 + 6kb2 where the shift by one is a quantum contribution.

• The dynamics of this scalar field in the CFT is determined

solely by boundary and consistency conditions like removing the gravitational anomaly.

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. Outline . . Motivation . . . . . . . . . . . Conformal Chern-Simons Gravity . . . . . Boundary CFT . . Conclusion

Discussion

• CSG allows for geometries that are not diffeomorphic to each
• ther, but nevertheless are gauge-equivalent.
• Similar features to those described here are likely to occur in

any theory of gravity with additional gauge symmetries.