Instantons and Chern-Simons terms in

AdS4/CFT3: Gravity on the Brane?

Sebastian de Haro

King’s College, London

QGT08, Kolkata, India, 9 January 2008

Based on:

• SdH and A. Petkou, hep-th/0606276, JHEP 12

(2006) 76

• SdH, I. Papadimitriou and A. Petkou, hep-th/0611315,

PRL 98 (2007) 231601

• SdH and Peng Gao, hep-th/0701144,

Phys. Rev. D76(2007) 106008

• SdH and A. Petkou, in progress

Motivation

1. Understanding the theory on the worldvolume of

a large number of membranes.

• It is a 3d SCFT with N = 8 supersymmetry that

arises as IR limit of 3d N = 8 SYM.

• It is dual to supergravity in AdS4.

2. Gravity/gauge duality beyond weak coupling.

1

Evidence for some exotic properties:

• No coupling constant.

• No decoupling of modes.

• Higher-spin fields.

• Electric-magnetic duality properties.

• Gravity?

Strategy

⇒ Use AdS/CFT to learn about this CFT.

I will consider AdS4 × S7 ≃ 3d SCFT on ∂(AdS4)

ℓPl/ℓ ∼ N−3/2

⇒ Problem: it is a weak/strong coupling duality

Use instanton solutions and duality properties

2

Instantons

Instantons describe tunneling across potential barrier

Important quantum effects on gravity

Difficulties:

• Need exact solutions of interacting theories

• String theory/M-theory on these backgrounds not

well understood beyond supergravity approximation

⇒ Explicit exact solutions of M-theory compactified

down to 4 dimensions

⇒ Holographic description

3

• From the point of view of holography, the use of

instantons is that they probe a non-conformal vac-

uum.

• In this vacuum, purely topological degrees of free-

dom can become dynamical.

• This is tightly connected with the presence of du-

ality and is special to AdS4/CFT3.

4

Duality of 3d CFT’s

There is a duality conjecture for the 3d CFT’s of

interest: a generalization of electric-magnetic dual-

ity for higher spins relating IR and UV theories [Leigh

and Petkou, ’03]

5

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spin

dimension

20

1

2

3

1

Deformation

Double−trace Dualization and "double−trace" deformations

Weyl−equivalence of UIR of O(4,1)

= s+1∆Unitarity bound

Duality conjecture relating IR and UV CFT3’s.

• Instantons describe the self-dual point of duality

• Duality plays an essential role in finding their holo-

graphic description

• Typically, the dual effective action is “topological”6

Plan:

1. Conformally coupled scalar:

Bulk: scalar instantons → instability

Boundary: effective action → 3d conformally

coupled scalar field with ϕ6 potential

2. Gravitational instantons and topologically mas-

sive gravity

3. U(1) gauge fields, RG flows and S-duality

7

Conformally Coupled Scalars and AdS4 × S7

The model: a conformally coupled scalar with quartic

interaction:

S =1

2

∫

d4x√

g

−R + 2Λ

8πGN+ (∂φ)2 +

1

6R φ2 + λ φ4

(1)

• This model arises in M-theory compactification

on S7. It is a consistent truncation of the N = 8 4d

sugra action where we only keep the metric and one

scalar field [Duff,Liu 1999].

• The coupling is then given by λ = 8πGN6ℓ2

8

Potential ℓ2V (φ) in units where 8πGN = 1 for a

background with negative cosmological constant.

9

• There are two extrema: φ = 0 and φ =√

6/8πGN .

• Naively we would expect to roll down the hill by

small perturbations. However, in the presence of

gravity such a picture is misleading: one needs to

take into account kinetic terms and boundary terms

[E. Weinberg ’86]

• In fact, the φ = 0 point is known to be stable. It is

the well-known AdS4 vacuum with standard choice

of boundary conditions [Breitenlohner and Freedman,

’82]

10

• But it is unstable for generic choice of boundary

conditions. One has to consider tunneling effects

due to the presence of kinetic terms [E. Weinberg

’82]. Instanton effects can mediate the decay.

• I will construct such instantons explicitly for one

particular choice of boundary conditions and compute

the decay rate.

• There is an interesting holographic dual descrip-

tion of the decay that I will also analyze

Instantons

Instanton solutions: exact solutions of the Euclidean

equations of motion with finite action.

’t Hooft instantons have zero stress-energy tensor:

T00 ∼ E2 − B2 = 0 (2)

We will likewise look for solutions with

Tµν = 0 (3)

11

• They are “ground states” of the Euclidean theory

• The problem of solving the eom is simplified be-

cause there is no back-reaction on the metric:

ds2 =ℓ2

r2

(

dr2 + d~x2)

, r > 0 (4)

We need to solve the Klein-Gordon equation in an

AdS4 background:

�φ − 1

6R φ − 2λ φ3 = 0 (5)

Unique solution with vanishing stress-energy tensor:

φ =2

ℓ√

|λ|br

−sgn(λ)b2 + (r + a)2 + (~x − ~x0)2

(6)

• λ < 0 ⇒ solution is regular everywhere

• λ > 0 ⇒ solution is regular everywhere provided

a > b ≥ 0

• α = a/b labels different boundary conditions:

φ(r, x) = r φ(0)(x) + r2 φ(1)(x) + . . . (7)

It gives the relation between φ(1) and φ(0):

φ(1)(x) = −ℓα φ2(0)(x) (8)

12

Holographic analysis

1) α is a deformation parameter of the dual CFT

2) ~x0, a2 − b2 parametrize 3d instanton vacuum

• For α >√

λ (a > b) the effective potential becomes

unbounded from below. This is the holographic

image of the vacuum instability of AdS4 towards

dressing by a non-zero scalar field with mixed bound-

ary conditions discussed above.

Similar conclusions were reached by Hertog & Horowitz

[2005] (although only numerically).

13

Taking the boundary to be S3 we plot the effective

potential to be:

Φ-

VΛ,Α

The global minimum φ(0) = 0 for α <√

λ becomes

local for α >√

λ. There is a potential barrier and

the vacuum decays via tunneling of the field.

14

• The instability region is φ →√

6/8πGN , which cor-

responds to the total squashing of an S2 in the cor-

responding 11d geometry. This signals a breakdown

of the supergravity description in this limit.

• In the Lorentzian Coleman-De Luccia picture, our

solution describes an expanding bubble centered at

the boundary. Outisde the bubble, the metric is AdS4

(the false vacuum).

Inside, the metric is currently unknown (the true vac-

uum). One needs to go beyond sugra to find the true

vacuum metric.15

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ζ

z1

z2

Ro(t)

Expansion of the bubble towards the bulk in the

Lorentzian.

R0(t): radius of the bubble

z1, z2: boundary coordinates

ζ: radial AdS coordinate16

The tunneling probability can be computed and

equals

P ∼ e−Γeff , Γeff =4π2ℓ2

κ2

1√

1 − κ2

6ℓ2α2

− 1

(9)

The deformation parameter α drives the theory from

regime of marginal instability α = κ/√

6ℓ (P → 0) to

total instability at α → ∞ (P → 1).

17

Boundary description of the instability

• We have seen that the bulk instability is mirrored

by the unboundedness of the CFT effective potential

• According to the usual AdS/CFT recipe, the bound-

ary generator of correlation functions W [φ(0)] at large

N is obtained from the bulk on-shell sugra action:

φ(r, x) = r∆− φ(0)(x) + . . .

eSbulkon-shell[φ] = eW [φ(0)] ≡ 〈e

∫

d3x φ(0)(x)O(x)〉CFT (10)

18

• The Wilsonian effective action Γ[〈O(x)〉] can be

obtained from W [φ(0)] but this is in practice a compli-

cated procedure. In the current example, dim O = 2

• Bulk analysis implies there is a duality between

φ(0) ↔ φ(1), hence O ↔ O

• Therefore we can use duality to obtain the effective

action where operator O of dimension 1 is turned on:

(φ(0), φ(1)) = (J, 〈O〉) = (〈O〉, J)

W [J] = Γ[〈O〉]Γ[〈O〉] = W [J] (11)

19

The effective action can be computed:

Γeff =1√λ

∫

d3x√

g(0)

(

φ−1(0)

∂iφ(0)∂iφ(0) +

1

2R[g(0)]φ(0)

+2√

λ(√

λ − α)φ3(0)

)

(12)

Redefining φ(0) = ϕ2, we get

Γeff[ϕ, g(0)] =1√λ

∫

d3x√

g(0)

(

1

2(∂ϕ)2 +

1

2R[g(0)]ϕ2

+2√

λ(√

λ − α)ϕ6)

(13)

⇒ 3d conformally coupled scalar field with ϕ6

interaction.

[SdH,AP 0606276; SdH, AP, IP 0611315]

[Hertog, Horowitz hep-th/0503071]

20

This describes the large N limit of the strongly cou-

pled 3d N = 8 SCFT where an operator of dimension

1 is turned on. This CFT describes N coincident

M2-branes for large N away from the conformal fixed

point.

This action is the matter sector of the U(1) N = 2

Chern-Simons action. It has recently been pro-

posed to be dual to AdS4 [Schwarz, hep-th/0411077;

Gaiotto, Yin, arXiv:0704.3740]. See also Bagger and

Lambert arXiv:0712.3738 [hep-th]

Warming up: U(1) instantons

• In the usual AdS/CFT interpretation, the boundary

values of fields are identified with sources in the CFT,

whereas canonical momenta give one-point functions:

Ai(x) := Ai(0, x) = Ji(x)

Ei(x) := ∂rAi(r, x)|r=0 = 〈Oi(x)〉J=0 (14)

Solving the bulk equations of motion amounts to im-

posing the regularity condition:

Ei(p) = −|p| Ai(p)

⇒ 〈Oi(x)〉J=0 = 0 . (15)

21

This is the usual conformal vacuum. Now consider a

more general boundary condition:

mAi(x) + Ei(x) − θBi(x) = Ji(x) . (16)

We can separate:

Ai(x) = ai(x) + Ai[J(x)] . (17)

ai(x) satisfies a dynamical equation:

ai(x) +1

Mǫijk∂jaj(x) = 0 . (18)

〈Oi(x)〉J=0 = ai(x) . (19)

〈Oi〉 satisfies the same dynamical equation as ai and it

can be integrated to an effective action. In this case,

this action turns out to be the massive topological

gauge theory of Deser and Jackiw in 3d.

Gravitational instantons

• The scalar field described tunneling between two

local minima of the scalar field. One would like to

find similar effects for gravity. Should reveal extra

holographic degrees of freedom.

• Instanton solutions with Λ = 0 have self-dual Rie-

mann tensor. However, self-duality of the Riemann

tensor implies Rµν = 0.

• In spaces with a cosmological constant we need

22

to choose a different self-duality condition. It turns

out that self-duality of the Weyl tensor is compatible

with a non-zero cosmological constant:

Cµναβ =1

2ǫµν

γδCγδαβ

•This equation can be solved asymptotically. In the

Fefferman-Graham coordinate system:

ds2 =ℓ2

r2

(

dr2 + gij(r, x) dxidxj)

where

gij(r, x) = g(0)ij(x) + r2g(2)ij(x) + r3g(3)ij(x) + . . .

We find

g(2)ij = −Rij[g(0)] +1

4g(0)ij R[g(0)]

g(3)ij = −2

3ǫ(0)i

kl∇(0)kg(2)jl =2

3C(0)ij

Cij is the 3d Cotton tensor.

• In general d, conformal flatness is measured by the

Weyl tensor [see e.g. Skenderis and Solodukhin, hep-

th/9910023]

• In d = 3, the Weyl tensor identically vanishes and

conformal flatness ⇔ Cij = 0

• The holographic stress tensor is 〈Tij〉 = 3ℓ2

16πGNg(3)ij

[SdH,Skenderis, Solodukhin 0002230]. We find that

for any g(0)ij the holographic stress tensor is given

by the Cotton tensor:

〈Tij〉 =ℓ2

8πGNC(0)ij

• Therefore, we can integrate the stress-tensor to

obtain the boundary generating functional:

〈Tij〉g(0)=

2√

g

δW

δgij(0)

• The Cotton tensor is the variation of the gravi-

tational Chern-Simons action:

SCS =k

4π

∫

(

ωab ∧ dωb

b +2

3ωa

b ∧ ωbc ∧c

a)

δSCS =k

4π

∫

δωαβ ∧ Rβ

α

=k

4π

∫

δgijǫjkl[

∇lRik − 1

4gil∇lR

]

. (20)

Therefore, the boundary generating functional is the

Chern-Simons gravity term.

Generalization

Consider the bulk gravity action

SEH = − 1

2κ2

∫

d4x√

G (R[G] − 2Λ) − 1

2κ2

∫

d3x√

γ 2K

+1

2κ2

∫

d3x√

γ

(

4

ℓ+ ℓ R[γ]

)

(21)

where κ2 = 8πGN , Λ = − 3ℓ2

. γ is the induced metric

on the boundary and K = γijKij.

We can add to it following bulk term:

∫

MR ∧ R =

∫

∂M

(

ω ∧ dω +2

3ω ∧ ω ∧ ω

)

23

• It doesn’t change the boundary conditions and gives

no contribution to the equations of motion

• It contributes to the holographic stress energy ten-

sor, precisely by the Cotton tensor:

〈Tij〉g(0)=

3ℓ2

16πGN

(

g(3)ij + Cij[g(0)])

This now affects any state. Again the dual gen-

erating functional contains the gravitational Chern-

Simons term. One can show that such terms arise

from R6 terms in M-theory [in progress].

Regularity of solutions

We have solved a perturbative series for the metric:

gij(r, x) = g(0)ij(x) + r2g(2)ij(x) + r3g(3)ij(x) + . . .

We want regular the solutions in the interior r = ∞.

We solve Einstein’s equations perturbatively: gij =

δij + hij. The solution is

hij(r, x) =∫ d3p

(2π)3eip·x f(r, p)

f(0, p)Πijkl h(0)kl(p)

(

Πijkl

)2= Πijkl (22)

24

Regular solutions have:

f(r, p) =

√

π

2(1 + |p|r) e−|p|r .

Expanding in powers of r, we get

hij(0, x) = h(0)ij(x)

h(3)ij(x) =1

3�

3/2h(0)ij (23)

This is a Rubin-type boundary condition. Combine

with regularity condition:

�h(0)ij = α ǫikl∂k�1/2h(0)jl + (i ↔ j) (24)

This can be rewritten as

δGij = − α

�1/2δCij (25)

This is the linearized version of the 3d topologically

massive gravity theory of Deser, Jackiw and Temple-

ton!

• Rubin-type boundary conditions for gravity are only

possible in AdS4 [Ishibashi, Wald].

• The fact that we have deformed the boundary con-

ditions by instantons is crucial. It leads to a dynam-

ical equation for the graviton on the boundary.

U(1) gauge fields in AdS

S = Sgrav +∫

d4x

− 1

4g2F2

µν +θ

32π2ǫµνλσFµνFλσ

Solve eom:

ds2 =ℓ2

r2

(

dr2 + gij(r, x)dxidxj)

gij(r, x) = g(0)ij + r3g(3)ij + r4g(4)ij + . . .

Ar(r, x) = A(0)r (x) + rA(1)

r (x) + r2A(2)r (x) + . . .

Ai(r, x) = A(0)i (x) + rA

(1)i (x) + r2A

(2)i (x) + . . .

26

Solve Einstein’s eqs: g(n) and A(n)i are determined in

terms of lower order coefficients g(0), g(3), A(0), A(1)

[SdH,K. Skenderis,S. Solodukhin hep-th/0002230]

A(0)i (x) ≡ Ai(x) = Ji

A(1)i (x) ≡ Ei(x) = 〈O∆=2,i(x)〉 electric field Fri

Eom can be exactly solved in this case:

ATi (r, p) = AT

i (p) cosh(|p|r) +1

|p| fi(p) sinh(|p|r)

27

Impose regularity at r = ∞:

ATi =

1

2

ATi (p) +

1

|p| fi(p)

e|p|r+1

2

ATi (p) −

1

|p| fi(p)

e−|p|r

⇒ ATi (p) + 1

|p| fi(p) = 0 regularity

S-duality acts as follows:

E′ = B

B′ = −E

τ ′ = −1/τ , τ =θ

4π2+

i

g2(26)

The action transforms as:

S[A′, E′] = S[A, E] +∫

d3x (E − θ B)A

28

With usual boundary conditions, B is fixed at the

boundary and corresponds to a source in the CFT.

This source couples to a conserved current (oper-

ator). E corresponds this conserved current. S-

duality interchanges the roles of the current and the

source.

New source: J ′ = E

New current: 〈O′〉A′ = −B

⇒ (J, 〈O〉) ↔ (〈O′〉, J ′)

29

We can check how S-duality acts on two-point func-

tions of the current O:

〈Oi(p)Oj(−p)〉A=0 =1

g2|p|Πij +

θ

(4π2)2i ǫijk pk

Πij = δij −pipj

|p|2 (27)

〈O′i(p)O′

j(−p)〉A′ =g2

1 + g4θ2

(4π2)2

|p|Πij −g4θ4π2

1 + g4θ2

(4π2)2

i ǫijk pk

(28)

in other words

τ → −1/τ

RG flow

• So far we have considered the CFT at the confor-

mal fixed point, either IR (Dirichlet) or UV (Neu-

mann). We will now consider how S-duality acts on

RG flows

• Deforming the boundary conditions in a way that

breaks conformal invariance (introducing mass pa-

rameter)

⇔ adding relevant operator that produces flow to-

wards new IR fixed point

30

Massive boundary condition: A + 1m (E − θB) = J

IR: 〈OiOj〉 = |p|Πij + θ i ǫijk pk

2-point function for conserved current of dim 2.

UV: 〈OiOj〉 = m2

|p|2(1+θ2)2(|p|Πij − θ i ǫijk pk)

S-dual current coming from dualizing gauge field.

See also [Leigh,Petkou hep-th/0309177;Kapustin]

Such behavior has also been found in quantum Hall

systems [Burgess and Dolan, hep-th/0010246]

31

Conclusions

• Instanton configurations in the bulk of AdS4 are

dual to CFT’s whose effective action is given by a

topological term, typically a relative of the Chern-

Simons action. They describe tunneling effects in

M-theory.

• Bulk instantons modify the boundary conditions.

Boundary degrees of fredom become dynamical.

32

Scalar fields

• Generalized b.c. that correspond to multiple trace

operators destabilize AdS4 nonperturbatively by

dressing of the scalar field. The Lorentzian picture

is in terms of tunneling to a new vacuum. The

tunneling rate was computed.

• Boundary effective action was computed and it

agrees with related proposals: 3d conformally cou-

pled scalar with ϕ6 interaction. Boundary instantons

match bulk instantons and describe the decay.

33

Gravity

• Bulk instantons correspond to boundary configura-

tions whose stress-energy tensor is the Cotton tensor.

The generating functional is the gravitational Chern-

Simons term.

• Roman boundary conditions for gravity amount to

a linearized graviton that satisfies a dynamical equa-

tion. For the case of the Chern-Simons term we get

topologically massive gravity [Deser, Jackiw, Tem-

pleton].

34

Gauge fields

• Electric-magnetic duality interpolates between D

& N. On the boundary it interchanges the source and

the conserved current (dual CFT3’s).

• Massive deformations generate RG flow of the two-

point function. One finds the conserved current in

the IR, but the S-dual gauge field in UV.

35

Outlook

• Gravitational anomalies [SdH and Petkou, in progress].

• Decoupling of gravity? [special to 4d].

36