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, G µν − 1 2 m 2 ( h µν − ¯ g µν h ) = 0 Taking the double divergence and the trace of this equation, one obtains, [ ] Λ − D − 1 ∇ ν h µν − ¯ ∇ µ ¯ ¯ ∇ 2 h = 0 , m 2 h = 0 2 In the massive case, h µν does not have to be traceless at the partially massless point (Deser et al. 1983, 2001) for which the mass is tuned as, 2 m 2 = D − 1Λ . . . . . .

Outline Motivation Conformal Chern-Simons Gravity Boundary CFT Conclusion . . . . . . . . . . . . . . . . . . . . . Gauge enhancement The following scalar gauge invariance appears, ( 2Λ ) ∇ ( µ ¯ ¯ h µν → h µν + ∇ ν ) + ( D − 1)( D − 2) ¯ g µν ζ This new gauge symmetry induces a Bianchi identity, Λ ∇ ν G µν + D − 1 G ρρ = 0 ∇ µ ¯ ¯ which reduces one degree of freedom. Non-linear realiziation of this symmetry, g µν → e 2Ω( x ) g µν . . . . . . .

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, g µν → e 2Ω( x ) g µν is an exact symmetry of the equations of motion. Conformal gravity in three dimensions (Deser et al. 1982) S = k ∫ d 3 x ǫ µνλ Γ ρµσ ∂ ν Γ σλρ + 2 3 Γ σντ Γ τ λρ ( ) 4 π M d 2 x √− γ K αβ K αβ − 1 + k ∫ ( 2 K 2 ) . 4 π ∂ M is a topological theory with the equations of motion, C µν ≡ 1 ( ) β + ε ναβ ∇ α R µ ε µαβ ∇ α R ν = 0 . β 2 . . . . . .

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 g µν = e 2 φ ( x + , x − , y ) ( dx + dx − + dy 2 ) + h µν . y 2 with h + − = h ± y = O (1) , h ++ h −− = O (1 / y ) and φ = b ln y + f ( x + , x − ) + O ( y 2 ) . . . . . . .

Outline Motivation Conformal Chern-Simons Gravity Boundary CFT Conclusion . . . . . . . . . . . . . . . . . . . . . Brown-York Stress tensor • Using the AdS/CFT dictionary, we calculating the response functions, δ S = 1 ∫ √ ( ) T αβ δγ (0) αβ + J αβ δγ (1) d 2 x − γ (0) αβ 2 ∂ M in Gaussian normal coordinate, ( e ρ ∝ 1 / y ) ds 2 = d ρ 2 + αβ e 2 ρ + γ (1) αβ e ρ + γ (2) γ (0) dx α dx β ( ) αβ + . . . 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 ( y 2 ) , 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, z ) J (0 , 0) � = 2 k ¯ z � T R ( z ) T R (0) � = 6 k � J ( z , ¯ z 3 , z 4 . . . . . . .

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: L ξ g µν + 2Ω g µν = δ g µν . Here δ g 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 ± = f 0 2 + p f ± e − in ( t ± ϕ ) . f ( n ) ∑ 2 ( t ± ϕ ) + n � =0 . . . . . .

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 ( y 2 ). II. Fixed Weyl rescaling Ω = − b 2 ∂ · ε − ε · ∂ f + O ( y 2 ). III. Free Weyl rescaling Ω = Ω( x + ) + Ω( x − ) + O ( y 2 ). With the diffeomorphisms ξ ± = ε ± ( x ± ) − y 2 2 ∂ ∓ ∂ · ε + O ( y 3 ) , ξ y = y 2 ∂ · ε + O ( y 3 ) , in all three cases above. . . . . . .

Outline Motivation Conformal Chern-Simons Gravity Boundary CFT Conclusion . . . . . . . . . . . . . . . . . . . . . frame formalism This theory has a first order format ( ) S = k ∫ ω ∧ d ω + 2 3 ω ∧ ω ∧ ω + λ ∧ T Tr 2 π M where ω is the spin connection one-form and T = d e + 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 = d ω + ω ∧ ω. . . . . . .

Outline Motivation Conformal Chern-Simons Gravity Boundary CFT Conclusion . . . . . . . . . . . . . . . . . . . . . Gauge theory formulation A Chern–Simons gauge theory with SO (3 , 2) gauge group, S CS = k ∫ A ∧ d A + 2 ( ) 3 A ∧ A ∧ A , Tr 4 π M 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 k ∫ ) 3 − k ∫ e − 1 d e ω d ee − 1 ) ( ( ∆ S = . Tr Tr 12 π 4 π M ∂ M Which leads to gravitational anomaly (Kraus–Larsen 2005) ∇ α T αβ = γ αβ (0) ε δγ ∂ δ ∂ λ Γ λ αγ [ γ (0) ] . . . . . . .

Outline Motivation Conformal Chern-Simons Gravity Boundary CFT Conclusion . . . . . . . . . . . . . . . . . . . . . Gauge transformations Local Poincar´ e transformations take form, δ P e i µ = − ǫ i jk e j µ θ k − ( ∂ µ ξ ν ) e i ν − ξ ν ∂ ν e i µ δ P ω i µ = −D µ θ i − ( ∂ µ ξ ν ) ω i ν − ξ ν ∂ ν ω i µ δ P λ i µ = − ǫ i jk λ j µ θ k − ( ∂ µ ξ ν ) λ i ν − ξ ν ∂ ν λ i µ . Weyl transformation, δ W e i µ = Ω e i µ δ W ω i µ = ǫ ijk e j µ e k ν ∂ ν Ω δ W λ i µ = − 2 D µ ( e i ν ∂ ν Ω) − Ω λ i µ . We find the gauge generators G P , G W that generate these transformations. . . . . . .

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 obtain differentiable charges Q we must add a boundary piece to the generators, ˜ G = G + Γ, which corresponds to the charge, 2 π 2 π ∫ d ϕ δ Γ P = − k ∫ δ Q P [ ξ ρ ] ξ ρ ( e i ρ δλ i ϕ + λ i ρ δ e i ϕ [ = d ϕ 2 π 0 0 + 2 θ i δω i ϕ +2 ω i ρ δω i ϕ ) ] . 2 π 2 π ∫ d ϕ δ Γ W = k ∫ d ϕ ( e i µ ∂ µ Ω) δ e i ϕ . δ Q W [Ω] = π 0 0 Here Q P and Q W 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 G ξ [ ε + = e inx + , ε − = 0] + ˜ ˜ G W [ ε + ] = T n G ξ [ ε + = 0 , ε − = − e − inx − ] + ˜ ¯ ˜ G W [ ε − ] T n = G W [Ω = − e inx + ] ˜ J n = we find the corresponding Dirac brackets i { T n , T m } ∗ = ( n − m ) T n + m − k n 3 δ n + m , 0 , T m } ∗ = ( n − m ) ¯ T n + m + k n 3 δ n + m , 0 , i { ¯ T n , ¯ i {J n , J m } ∗ = 2 k n δ n + m , 0 . . . . . . .