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