Holograms of conformal Chern–Simons gravity

Holograms of conformal Chern–Simons gravity

Niklas Johansson

Vienna University of Technology

Uppsala, June 30, 2011

Work done with H. Afshar, B. Cvetkovi´c, S. Ertl and D. Grumiller

We study...

S_CS= k 4π

Z

M³

(Γ∧dΓ +2 3Γ³)

[Deser, Jackiw & Templeton, ’82],[Horne & Witten, ’89]

• Topological. (Gauge symmetries: diffeos + Weyl.)

• ∂M³6=∅ =⇒ non-trivial dynamics.

• Holographic description: Partially massless gravitons, Brown–York responses, correlators... [arXiv:1107.xxxx]

• This talk: What does the Weyl symmetry give rise to at the boundary?

We study...

S_CS= k 4π

Z

M³

(Γ∧dΓ +2 3Γ³)

[Deser, Jackiw & Templeton, ’82],[Horne & Witten, ’89]

• Topological. (Gauge symmetries: diffeos + Weyl.)

• ∂M³6=∅ =⇒ non-trivial dynamics.

• Holographic description: Partially massless gravitons, Brown–York responses, correlators... [arXiv:1107.xxxx]

• This talk: What does the Weyl symmetry give rise to at the boundary?

We study...

S_CS= k 4π

Z

M³

(Γ∧dΓ +2 3Γ³)

[Deser, Jackiw & Templeton, ’82],[Horne & Witten, ’89]

• Topological. (Gauge symmetries: diffeos + Weyl.)

• ∂M³ 6=∅ =⇒ non-trivial dynamics.

• Holographic description: Partially massless gravitons, Brown–York responses, correlators... [arXiv:1107.xxxx]

• This talk: What does the Weyl symmetry give rise to at the boundary?

We study...

S_CS= k 4π

Z

M³

(Γ∧dΓ +2 3Γ³)

[Deser, Jackiw & Templeton, ’82],[Horne & Witten, ’89]

• Topological. (Gauge symmetries: diffeos + Weyl.)

• ∂M³ 6=∅ =⇒ non-trivial dynamics.

• Holographic description: Partially massless gravitons, Brown–York responses, correlators... [arXiv:1107.xxxx]

• This talk: What does the Weyl symmetry give rise to at the boundary?

We study...

S_CS= k 4π

Z

M³

(Γ∧dΓ +2 3Γ³)

[Deser, Jackiw & Templeton, ’82],[Horne & Witten, ’89]

• Topological. (Gauge symmetries: diffeos + Weyl.)

• ∂M³ 6=∅ =⇒ non-trivial dynamics.

• Holographic description: Partially massless gravitons, Brown–York responses, correlators... [arXiv:1107.xxxx]

• This talk: What does the Weyl symmetry give rise to at the boundary?

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν)

Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry!

i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry! i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m

y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry! i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m

y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry! i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m

y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry! i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m

y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry!

i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m

y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^±=^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry!

i{L_n,L_m}= (n−m)L_n+m+₁₂^c (n³−n)δ_n+m y

6 x^±

∂M³ M³

Warm-up: (2+1)-dimensional EH gravity

S_EH=R

M³

√−g(R−2Λ) Gauge symmetry: δg_µν =∇_(µξ_ν) Boundary conditions: [Brown, Henneaux ’86]

gµν =g_µν^AdS+hµν =g_µν^AdS+





O(1) O(1) O(y) O(1) O(y) O(1)





µν

Diffeos that preserve the BCs: ξ^± =^±(x^±) +O(y²)

• Canonical realization =⇒ boundary charge!

• Central extension of the algebra!

• Gauge symmetry → global symmetry!

i{L¯_n,¯L_m}= (n−m)¯L_n+m+₁₂^¯c (n³−n)δ_n+m y

6 x^±

∂M³ M³

Conformal Chern-Simons gravity

Gauge sym: δg_µν=∇_(µξ_ν),δg_µν = 2Ω(x)g_µν

Boundary conditions: [arXiv:1107.xxxx]. g_µν =e^φ(x,y)

g_µν^AdS+h_µν

Gauge trafo’s that preserve the BCs: depends on BC on φ.

Conformal Chern-Simons gravity

Gauge sym: δg_µν=∇_(µξ_ν),δg_µν = 2Ω(x)g_µν Boundary conditions: [arXiv:1107.xxxx].

g_µν =e^φ(x,y)

g_µν^AdS+h_µν

Gauge trafo’s that preserve the BCs: depends on BC on φ.

Conformal Chern-Simons gravity

Gauge sym: δg_µν=∇_(µξ_ν),δg_µν = 2Ω(x)g_µν Boundary conditions: [arXiv:1107.xxxx].

g_µν =e^φ(x,y)

g_µν^AdS+h_µν

Gauge trafo’s that preserve the BCs: depends on BC onφ.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

φ≡0 : diffeo → Vir²,c =−¯c. Weyl → trivial

φ≡f_fixed(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → trivial

φ=ffree(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → nontrivial charge

Conservation of the Weyl charge =⇒ consistency conditions.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

φ≡0 : diffeo → Vir²,c =−¯c. Weyl → trivial

φ≡f_fixed(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → trivial

φ=ffree(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → nontrivial charge

Conservation of the Weyl charge =⇒ consistency conditions.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

φ≡0 : diffeo → Vir²,c =−¯c. Weyl → trivial

φ≡f_fixed(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → trivial

φ=ffree(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → nontrivial charge

Conservation of the Weyl charge =⇒ consistency conditions.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

φ≡0 : diffeo → Vir²,c =−¯c. Weyl → trivial

φ≡f_fixed(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → trivial

φ=ffree(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → nontrivial charge

Conservation of the Weyl charge =⇒ consistency conditions.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

φ≡0 : diffeo → Vir²,c =−¯c. Weyl → trivial

φ≡f_fixed(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → trivial

φ=ffree(x) +. . .: diffeo+Weyl → Vir²,c =−¯c Weyl → nontrivial charge

Conservation of the Weyl charge =⇒ consistency conditions.

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

Conservation of the Weyl charge =⇒ consistency conditions.

Simplest case: φ=φ(x⁺) and Ω = Ω(x⁺)

[Ln,Lm] = (n−m)Ln+m+ c

+1

12 (n³−n)δn+m

[¯Ln,L¯m] = (n−m) ¯Ln+m+ c¯

12(n³−n)δn+m

[J_n,J_m] = 2k nδ_n+m

[L_n,J_m] =−mJ_n+m

Pure diffeos =⇒ Sugawara shift L_n→L_n+P

kJ_kJ_n−k. Thank you!

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

Conservation of the Weyl charge =⇒ consistency conditions.

Simplest case: φ=φ(x⁺) and Ω = Ω(x⁺) [Ln,Lm] = (n−m)Ln+m+ c

+1

12 (n³−n)δn+m

[¯Ln,L¯m] = (n−m) ¯Ln+m+ c¯

12(n³−n)δn+m

[J_n,J_m] = 2k nδ_n+m

[L_n,J_m] =−mJ_n+m

Pure diffeos =⇒ Sugawara shift L_n→L_n+P

kJ_kJ_n−k. Thank you!

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

Conservation of the Weyl charge =⇒ consistency conditions.

Simplest case: φ=φ(x⁺) and Ω = Ω(x⁺) [Ln,Lm] = (n−m)Ln+m+ c+1

12 (n³−n)δn+m

[¯Ln,L¯m] = (n−m) ¯Ln+m+ c¯

12(n³−n)δn+m

[J_n,J_m] = 2k nδ_n+m [L_n,J_m] =−mJ_n+m

Pure diffeos =⇒ Sugawara shift L_n→L_n+P

kJ_kJ_n−k.

Thank you!

Conformal Chern-Simons gravity

g_µν =e^φ(x,y) g_µν^AdS+h_µν

Conservation of the Weyl charge =⇒ consistency conditions.

Simplest case: φ=φ(x⁺) and Ω = Ω(x⁺) [Ln,Lm] = (n−m)Ln+m+ c+1

12 (n³−n)δn+m

[¯Ln,L¯m] = (n−m) ¯Ln+m+ c¯

12(n³−n)δn+m

[J_n,J_m] = 2k nδ_n+m [L_n,J_m] =−mJ_n+m

Pure diffeos =⇒ Sugawara shift L_n→L_n+P

kJ_kJ_n−k. Thank you!