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• 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
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,
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,
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,
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)
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.
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
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.
• 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.