holograms of conformal chern simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/uppsala2011.pdf ·...
TRANSCRIPT
![Page 1: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/1.jpg)
Holograms of conformal Chern–Simons gravity
Niklas Johansson
Vienna University of Technology
Uppsala, June 30, 2011
Work done with H. Afshar, B. Cvetkovic, S. Ertl and D. Grumiller
![Page 2: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/2.jpg)
We study...
SCS =k
4π
∫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–Yorkresponses, correlators... [arXiv:1107.xxxx]
• This talk: What does the Weyl symmetry give rise to at theboundary?
![Page 3: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/3.jpg)
We study...
SCS =k
4π
∫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–Yorkresponses, correlators... [arXiv:1107.xxxx]
• This talk: What does the Weyl symmetry give rise to at theboundary?
![Page 4: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/4.jpg)
We study...
SCS =k
4π
∫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–Yorkresponses, correlators... [arXiv:1107.xxxx]
• This talk: What does the Weyl symmetry give rise to at theboundary?
![Page 5: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/5.jpg)
We study...
SCS =k
4π
∫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–Yorkresponses, correlators... [arXiv:1107.xxxx]
• This talk: What does the Weyl symmetry give rise to at theboundary?
![Page 6: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/6.jpg)
We study...
SCS =k
4π
∫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–Yorkresponses, correlators... [arXiv:1107.xxxx]
• This talk: What does the Weyl symmetry give rise to at theboundary?
![Page 7: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/7.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 8: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/8.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 9: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/9.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 10: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/10.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!
• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 11: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/11.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!
• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 12: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/12.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 13: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/13.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 14: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/14.jpg)
Warm-up: (2+1)-dimensional EH gravity
SEH =∫M3
√−g(R − 2Λ) Gauge symmetry: δgµν = ∇(µξν)
Boundary conditions: [Brown, Henneaux ’86]
gµν = gAdSµν + hµν = gAdS
µν +
O(1) O(1) O(y)O(1) O(y)
O(1)
µν
Diffeos that preserve the BCs: ξ± = ε±(x±) +O(y 2)
• Canonical realization =⇒ boundary charge!• Central extension of the algebra!• Gauge symmetry → global symmetry!
iLn, Lm = (n −m)Ln+m + c12 (n3 − n)δn+m
y
6
x±
∂M3M3
![Page 15: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/15.jpg)
Conformal Chern-Simons gravity
Gauge sym: δgµν = ∇(µξν), δgµν = 2Ω(x)gµν
Boundary conditions: [arXiv:1107.xxxx].
gµν = eφ(x ,y)(
gAdSµν + hµν
)Gauge trafo’s that preserve the BCs: depends on BC on φ.
![Page 16: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/16.jpg)
Conformal Chern-Simons gravity
Gauge sym: δgµν = ∇(µξν), δgµν = 2Ω(x)gµν
Boundary conditions: [arXiv:1107.xxxx].
gµν = eφ(x ,y)(
gAdSµν + hµν
)
Gauge trafo’s that preserve the BCs: depends on BC on φ.
![Page 17: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/17.jpg)
Conformal Chern-Simons gravity
Gauge sym: δgµν = ∇(µξν), δgµν = 2Ω(x)gµν
Boundary conditions: [arXiv:1107.xxxx].
gµν = eφ(x ,y)(
gAdSµν + hµν
)Gauge trafo’s that preserve the BCs: depends on BC on φ.
![Page 18: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/18.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)
φ ≡ 0 : diffeo → Vir 2, c = −c.
Weyl → trivial
φ ≡ ffixed(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → trivial
φ = ffree(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → nontrivial charge
Conservation of the Weyl charge =⇒ consistency conditions.
![Page 19: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/19.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)φ ≡ 0 : diffeo → Vir 2, c = −c.
Weyl → trivial
φ ≡ ffixed(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → trivial
φ = ffree(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → nontrivial charge
Conservation of the Weyl charge =⇒ consistency conditions.
![Page 20: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/20.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)φ ≡ 0 : diffeo → Vir 2, c = −c.
Weyl → trivial
φ ≡ ffixed(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → trivial
φ = ffree(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → nontrivial charge
Conservation of the Weyl charge =⇒ consistency conditions.
![Page 21: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/21.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)φ ≡ 0 : diffeo → Vir 2, c = −c.
Weyl → trivial
φ ≡ ffixed(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → trivial
φ = ffree(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → nontrivial charge
Conservation of the Weyl charge =⇒ consistency conditions.
![Page 22: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/22.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)φ ≡ 0 : diffeo → Vir 2, c = −c.
Weyl → trivial
φ ≡ ffixed(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → trivial
φ = ffree(x) + . . . : diffeo + Weyl → Vir 2, c = −c
Weyl → nontrivial charge
Conservation of the Weyl charge =⇒ consistency conditions.
![Page 23: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/23.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)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, Lm] = (n −m) Ln+m +c
12(n3 − n) δn+m
[Jn, Jm] = 2k n δn+m
[Ln, Jm] = −mJn+m
Pure diffeos =⇒ Sugawara shift Ln → Ln +∑
k JkJn−k .
Thank you!
![Page 24: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/24.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)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, Lm] = (n −m) Ln+m +c
12(n3 − n) δn+m
[Jn, Jm] = 2k n δn+m
[Ln, Jm] = −mJn+m
Pure diffeos =⇒ Sugawara shift Ln → Ln +∑
k JkJn−k .
Thank you!
![Page 25: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/25.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)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, Lm] = (n −m) Ln+m +c
12(n3 − n) δn+m
[Jn, Jm] = 2k n δn+m
[Ln, Jm] = −mJn+m
Pure diffeos =⇒ Sugawara shift Ln → Ln +∑
k JkJn−k .
Thank you!
![Page 26: Holograms of conformal Chern Simons gravityquark.itp.tuwien.ac.at/~grumil/pdf/Uppsala2011.pdf · Holograms of conformal Chern{Simons gravity Niklas Johansson Vienna University of](https://reader034.vdocuments.site/reader034/viewer/2022050308/5f7042e005383f1a4d763283/html5/thumbnails/26.jpg)
Conformal Chern-Simons gravity
gµν = eφ(x ,y)(gAdSµν + hµν
)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, Lm] = (n −m) Ln+m +c
12(n3 − n) δn+m
[Jn, Jm] = 2k n δn+m
[Ln, Jm] = −mJn+m
Pure diffeos =⇒ Sugawara shift Ln → Ln +∑
k JkJn−k .
Thank you!