The Holographic Cotton Tensor

Recent Advances in Topological Quantum Field Theory University of Lisbon, September 14, 2012 Sebastian de Haro (ITFA and AUC, University of Amsterdam)

AdS/CFT: generalities and holographic renormalization

Boundary graviton: duality and CFT coupled to gravity

Instanton solutions

Holographic Cotton Tensor. SdH, UvA

Holographic duality that relates:

Gravity (string theory, M-theory) in (𝑑 + 1)-dimensional AdS space to:

A CFT on the (conformal) boundary of this space (𝑑-dimensional).

The duality works for both Euclidean and Lorentzian signatures.

Euclidean case much better understood.

Interested in instanton solutions.


Quadric in ℝ1,𝑑+1:

One can choose coordinates:

d𝑠2 = d𝑦2 + sinh2 𝑦 dΩ𝑑2

There is a boundary 𝐒𝑑 at 𝑦 = ∞.

We will use local coordinates (half space 𝑟 > 0):

d𝑠2 =ℓ2

𝑟2d𝑟2 + d𝑥 𝑑

2

Induces (conformally) flat metric on the boundary.

The CFT lives on this conformal boundary.


− 𝑋𝑑+1 2+ 𝑋𝑖

2

𝑑

𝑖=0

= −ℓ2

Setting up QFT in AdS space, fields may contain a classical and a quantum part:

Φ = Φcl + 𝛿Φ

Both satisfy equation of motion in the bulk.

𝛿Φ is part of Hilbert space → normalizable, 𝜙 1

▪ Holographically: vev of operator ⟨𝒪⟩

Φcl need not be: background → 𝜙 0

▪ Holographically: source 𝐽


□ − 𝑚2 Φ = 0

𝑍bulk 𝜙 0 = 𝒟Φ 𝑒−𝑆 Φ

𝜙 0

= 𝑒𝑊CFT 𝐽 = 𝒟𝜙 𝑒−𝑆CFT 𝜙 −𝑆int 𝐽,𝒪

𝑆int 𝐽, 𝒪 = d𝑑𝑥 𝐽 𝑥 𝒪 𝜙 𝑥

If 𝐽 is independent of 𝑥, it is the coupling constant for that operator.


Φcl = 𝜙 0 = 𝐽

Semi-classical approximation in the bulk: Boundary theory is strongly coupled in that case.

Typically: large 𝑁 and large 𝜆 = 𝑔YM2 𝑁~

ℓAdS

ℓ𝑠

4

(for AdS5): strong ‘t Hooft coupling. Expectation values of operators:

𝜙+ bulk solution for Φ (normalizable mode 𝛿Φ)


−𝑆on−shell 𝜙 0 = 𝑊CFT[𝐽]

𝒟Φ 𝑒−𝑆 Φ

𝜙 0

= 𝑒𝑊CFT 𝐽

𝒪(∆+) 𝑥𝐽

= −𝛿𝑊CFT 𝐽

𝛿𝐽 𝑥= − Δ+ − Δ− 𝜙+ 𝑥

𝑑 = 3, conformally coupled scalar field in bulk:

𝜙 0 (𝑥) is boundary condition at 𝑟 = 0.

Regularity at 𝑟 = ∞ imposes:


Φ 𝑟, 𝑥 = 𝜙 0 𝑥 + 𝑟 𝜙 1 𝑥 + ⋯

Φ 𝑟, 𝑥 =1

𝜋2 d3𝑥′

𝑟

𝑟2 + 𝑥 − 𝑥 ′ 2 2 𝜙 0 𝑥′ = 𝜙 0 𝑥 + 𝑟

1

𝜋2 d3𝑥′

𝜙 0 𝑥′

𝑥 − 𝑥 ′ 4 + ⋯

𝒪(∆+) 𝑥𝐽

= −𝛿𝑊CFT 𝐽

𝛿𝐽 𝑥= − Δ+ − Δ− 𝜙+ 𝑥

𝒪 2 𝑥 𝒪 2 𝑥′𝐽=0

=𝛿2𝑊CFT 𝐽

𝛿𝐽 𝑥 𝛿𝐽 𝑥′ 𝐽=0

=1

𝜋2 𝑥 − 𝑥 ′ 4 ⇒

When computing correlation functions from bulk, we encounter divergences as 𝑟 → 0.

Need formalism for generic boundary metric not just flat. Allows computation correlation functions of stress-

energy tensor.

Allows computation CFT in any background. Take into account back-reaction. Holographic renormalization systematic method

to do this.


Bulk metric 𝐺𝜇𝜈 (𝑑 = 3):

d𝑠2 = 𝐺𝜇𝜈 𝑟, 𝑥 d𝑥𝜇d𝑥𝜈 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖 d𝑥𝑗

𝑔𝑖𝑗 𝑟, 𝑥 = 𝑔 0 𝑖𝑗 𝑥 + 𝑟2 𝑔 2 𝑖𝑗 𝑥 + 𝑟3𝑔 3 𝑖𝑗 𝑥 + ⋯

Solve Einstein’s equations perturbatively in 𝑟 for given boundary values of metric:

𝜌 2𝑔′′ − 2𝑔′𝑔−1𝑔′ + Tr 𝑔−1𝑔′ 𝑔′ − Ric 𝑔 − 𝑔′ − Tr 𝑔−1𝑔′ 𝑔= 0

𝛻𝑗𝑔𝑖𝑗′ = 𝛻𝑖Tr 𝑔−1𝑔′

Tr 𝑔−1𝑔′′ =1

2Tr 𝑔−1𝑔′ 2


𝜇 = 𝑟, 𝑖 𝑖 = 1,2,3

𝜌 = 𝑟2

𝑅𝜇𝜈 = −3

ℓ2 𝐺𝜇𝜈

Solve Einstein’s equations perturbatively in 𝜌:

𝑔 0 , 𝑔 3 undetermined (= b.c.)

Higher 𝑔 𝑛 ’s:


𝑔 𝑛 = 𝑔 𝑛 𝑔 0 , 𝑔 3

𝑔 2 𝑖𝑗 𝑥 = −𝑅𝑖𝑗 𝑔 0 +1

4𝑔 0 𝑖𝑗𝑅 𝑔 0

𝑔 4 𝑖𝑗 𝑥 = quartic in derivatives

Holographic recipe:

Regularize the bulk action at 𝑟 = 𝜖

Add counterterms that do not modify eom

Send 𝜖 → 0, obtain finite result.


𝑇𝑖𝑗 𝑥𝑔 0

=2

𝑔 0

𝛿𝑆on−shell 𝑔 0

𝛿𝑔0

𝑖𝑗𝑥

𝑆 = 𝑆bulk + 𝑆GH + 𝑆ct

= −1

2𝜅2 d4𝑥 𝑔 𝑅 𝐺 +6

ℓ2 −1

2𝜅2 d3𝑥 𝛾 2𝐾 −4

ℓ− ℓ𝑅 𝛾

𝜕𝑀𝜖𝑀𝜖

𝜅2 = 8𝜋𝐺N

𝐾𝑖𝑗 =1

2𝜕𝑛𝛾𝑖𝑗 𝛾𝑖𝑗 =

ℓ2

𝑟2 𝑔𝑖𝑗= induced metric 𝐾 = 𝛾𝑖𝑗𝐾𝑖𝑗

To compute 𝑆on−shell 𝑔 0 , need to solve

eom all the way to interior.

Only need its variation


=2

𝑔 0

𝛿𝑆on−shell 𝑔 0

𝛿𝑔0

𝑖𝑗𝑥

Compute the regularized action with counterterm subtraction, vary and take limit.

It is enough to know the divergent terms.


The expectation value:


= lim𝜖→0

1

det 𝑔 𝜖, 𝑥

𝛿𝑆sub 𝑥, 𝜖

𝛿𝑔𝑖𝑗 𝑥, 𝜖

Result:



=3ℓ2

16𝜋𝐺N 𝑔 3 𝑖𝑗 𝑥


Goal: understand holography of graviton and whether CFT can be coupled to dynamical gravity. What does this give in bulk?

Standard normalizability analysis: 𝑔 3 normalizable, depends on a choice of

solution: vev of stress-energy.

𝑔 0 is non-normalizable: b.c., corresponds to source on boundary. Or is it?

Ishibashi-Wald (2004): both modes are normalizable.


This means that either 𝑔 0 or 𝑔 3 can be

interpreted as boundary gravitons.

Correspond to different CFT’s.

Since both modes can be normalized in the bulk, they need not be a priori fixed.

We may set up dynamical equation that selects certain solutions.

May couple CFT to gravity.


Investigate the dynamics of boundary graviton in simple case: self-dual Weyl.

Boundary graviton does not solve full Einstein equations, but more restrictive one.

Physically, self-dual solutions are ‘gravitational instantons’ that signal instability of bulk under deformation of b.c. Decay towards new vacuum.


In spaces without cosmological constant, natural condition is self-duality of Riemann.

It automatically implies 𝑅𝜇𝜈 = 0.

If cosmological constant non-zero, need to choose a different condition. Self-duality of Weyl tensor is compatible with cosmological constant and Euclidean signature:


𝐶𝜇𝜈𝛼𝛽 =1

2 휀𝜇𝜈

𝛾𝛿 𝐶𝛾𝛿𝛼𝛽

Simplest case: vanishing Weyl tensor.

Bulk metric is conformally flat.

On-shell, the Weyl tensor reduces to:

Together with Einstein’s equations:


𝐶𝜇𝜈𝛼𝛽 = 𝑅𝜇𝜈𝛼𝛽 +1

ℓ2 𝐺𝜇𝛼𝐺𝜈𝛽 − 𝐺𝜇𝛽𝐺𝜈𝛼

ℓ2𝑅𝑖𝑗𝑘𝑙 𝑔 = 𝑔𝑖𝑘𝑔𝑗𝑙′ + 𝑔𝑗𝑙𝑔𝑖𝑘

′ − 𝑔𝑖𝑙𝑔𝑗𝑘′ − 𝑔𝑗𝑘𝑔𝑖𝑙

′ − 𝜌 𝑔𝑖𝑘′ 𝑔𝑗𝑙

′ − 𝑔𝑖𝑙′ 𝑔𝑗𝑘

′

𝛻𝑖𝑔𝑗𝑘′ − 𝛻𝑗𝑔𝑖𝑘

′ = 0

𝑔′′ −1

2𝑔′𝑔−1𝑔′ = 0

𝜌 = 𝑟2

(1)

(2)

(3)

Equation 3 implies: Series terminates at order 𝜌2: 𝑔 4 is last non-

vanishing coefficient. (3) can be integrated:

Equation (2) implies that the Cotton tensor of

𝑔 0 vanishes: The boundary metric is conformally flat.

Bulk metric conformally flat iff boundary metric is conformally flat.


𝑔′′′ = 0

𝑔 = 1 +𝜌

2𝑔 2 𝑔 0

−1 𝑔 0 1 +𝜌

2𝑔 0

−1𝑔 2

𝐶𝑖𝑗 =1

2 𝜖𝑖

𝑘𝑙𝛻𝑘 𝑅𝑗𝑙 −1

4𝑔𝑗𝑙𝑅

Next case: non-zero, self-dual Weyl tensor

𝐶𝜇𝜈𝛼𝛽 =1

2 휀𝜇𝜈

𝛾𝛿 𝐶𝛾𝛿𝛼𝛽

Solve coupled equations asymptotically:

d𝑠2 = 𝐺𝜇𝜈 𝑟, 𝑥 d𝑥𝜇d𝑥𝜈 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖 d𝑥𝑗


𝑔 2 𝑖𝑗 𝑥 = −𝑅𝑖𝑗 𝑔 0 +1

4𝑔 0 𝑖𝑗𝑅 𝑔 0


𝑅𝜇𝜈 = −3

ℓ2 𝐺𝜇𝜈

Result:

𝑔 3 𝑖𝑗 𝑥 = −2

3 𝜖 0 𝑖

𝑘𝑙𝛻 0 𝑘𝑔 2 𝑗𝑙 =2

3 𝐶𝑖𝑗 𝑔 0

𝐶𝑖𝑗 =1

2𝜖𝑖

𝑘𝑙𝛻𝑘 𝑅𝑗𝑙 −1

4𝑔𝑗𝑙𝑅

Combine with holographic interpretation of 𝑔 3 as 1-point function of stress-energy:

Integrate stress-tensor to obtain boundary generating functional:

𝑇𝑖𝑗 𝑔 0=

2

𝑔 0

𝛿𝑊

𝛿𝑔0

𝑖𝑗


Bulk Einstein


ℓ2

8𝜋𝐺N 𝐶𝑖𝑗 𝑔 0

This can be integrated to yield the boundary generating function:

𝛿𝑆CS = −1

2 Tr 𝛿Γ ∧ 𝑅 = −

1

2 𝜖𝑖𝑗𝑘𝑅𝑖𝑗𝑙

𝑚 𝛿Γ𝑘𝑚𝑙 =: − 𝐶𝑖𝑗𝛿𝑔𝑖𝑗

𝑆CS = −1

4 Tr Γ ∧ dΓ +

2

3Γ ∧ Γ ∧ Γ

We get:


𝑊 = −ℓ2

64𝜋𝐺N Tr Γ ∧ dΓ +

2

3Γ ∧ Γ ∧ Γ

At next level 𝑔 0 , we get a compatibility

condition for the curvature:

𝑔 4 𝑖𝑗 =1

4𝑔 2 𝑔 0

−1𝑔 2 𝑖𝑗−

3

4𝜖𝑖

𝑘𝑙𝛻𝑘𝑔 3 𝑗𝑙

Also: 𝑔 4 𝑖𝑗 =1

4□𝑅𝑖𝑗 −

1

16𝛻𝑖𝛻𝑗𝑅 −

1

24𝑔𝑖𝑗□𝑅 +

1

2𝑅𝑘𝑙𝑅𝑖𝑘𝑗𝑙 +

1

2𝑅𝑖

𝑘𝑅𝑗𝑘 −1

2𝑅𝑅𝑖𝑗 −

1

4𝑔𝑖𝑗𝑅

𝑘𝑙𝑅𝑘𝑙 +9

16∙4𝑔𝑖𝑗𝑅

2

Non-linear gravity theory in 3d. Needs to be studied further.


Holographic dictionary: 𝑔 0 boundary graviton

Stress-energy tensor: Both modes are normalizable (linearized

fluctuations). 𝑔 3 is related to boundary graviton of CFT2 via

Cotton tensor. Solutions with zero bulk Weyl tensor have zero

boundary Cotton tensor. Bulk solution with BTZ black hole on boundary.



=3ℓ2

16𝜋𝐺N 𝑔 3 𝑖𝑗 𝑥

Solutions with self-dual Weyl tensor:

Generating functional can be integrated and gives the gravitational Chern-Simons action.

At next level, we get a non-linear consistency condition for the curvature.



ℓ2

8𝜋𝐺N 𝐶𝑖𝑗 𝑔 0

Maldacena (1997) Witten (1998) Skenderis, Solodukhin (2000) SdH, Skenderis, Solodukhin (2000) SdH, Petkou (2008, 2012)


Thank you!


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The Holographic Cotton Tensor