Instantons and Chern-Simons Terms in

AdS4/CFT3

Sebastian de Haro

King’s College, London

Kolymbari, July 3, 2007

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

Motivation

• M-theory/sugra in AdS4 × S7 ' Coincident M2-branes on ∂(AdS4),`Pl/` ∼ N−3/2

• AdS4 has instantons that are exact solutions of interacting theories andallow us to probe the CFT away from the conformal vacuum

• Instantons can mediate non-perturbative instabilities

• Holographic description of gravitational tunneling effects

• Conformal holography [hep-th/0606276,0611315]: one expects a re-fined version of holography (integrability) for:

1) exact instanton solutions

2) Weyl invariant theories

Typically, the dual effective action is a “topological” action

1

This talk:

1. Conformally coupled scalar:

Bulk: scalar instantons – instability

Bdy: effective action – 2+1 conformally coupled scalar w/ ϕ6

2. Gravity: SD solutions – gravitational Chern-Simons

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

2

Conformally Coupled Scalars and AdS4 × S7

Consider a conformally coupled scalar with quartic interaction:

S =1

2

∫

d4x√g

[−R+ 2Λ

8πGN+ (∂φ)2 +

1

6Rφ2 + λφ4

]

λ = 8πGN

6`2= κ2

6`2for M-theory embedding

A field redefinition makes the scalar field minimally coupled:

κφ/√

6 = tanh(κΦ/√

6) ⇒ −√

6/κ < φ <√

6/κ → −∞ < Φ <∞

gµν = cosh2(κΦ/√

6)Gµν

The action becomes:

S =1

2

∫

d4x√G

[

−R+ 2Λ

8πGN+ (∂Φ)2 − 6

κ2`2cosh

(

√

2

3κΦ

)]

This is a consistent truncation of the N = 8 4d sugra action [Duff,Liu 1999]

3

Boundary effective action

The bulk metric is asymptotically Euclidean ALAdS:

ds2 =`2

r2

(

dr2 + gij(r, x)dxidxj

)

gij(r, x) = g(0)ij(x) + r2g(2)ij(x) + . . . (1)

The scalar field has the following expansion:

φ(r, x) = r∆− φ(0)(x) + r∆+ φ(1)(x) + . . .

∆− = 1

∆+ = 2 (2)

There is a choice of boundary conditions: to fix φ(0) (Dirichlet) or to fixφ(1) (Neumann). Neumann boundary conditions correspond to a boundarytheory with an operator of dimension ∆ = 1, φ(0). The effective actionbecomes a function of Γ[φ(0)] in the fixed background g(0)ij.

4

Consider a 1-parameter family of boundary conditions that preserve AdS4

symmetries:

φ(1) = −`αφ2(0) (3)

They interpolate between Neumann (α = 0) and Dirichlet (α = ∞).

The boundary condition can be enforced by adding a boundary term:

Sbdy = −`3α

3

∫

d3x√g(0) φ

3(0)(x) (4)

This corresponds to adding a triple trace deformation in the boundary theory.

For generic potentials, numerical solutions of the eom with the above bound-ary conditions were studied by [Hertog and Horowitz 2004]

5

For the quartic potential, keeping up to two derivative terms gives [IP hep-th/0703152]:

Γeff[φ(0), g(0)] =4

3√λ

∫

d3x√g(0)

(

φ−1(0)∂iφ(0)∂

iφ(0) +1

2R[g(0)]φ(0) + 2

√λ(

√λ− α)φ3

(0)

)

(5)Redefining φ(0) = ϕ2, we get

Γeff[ϕ, g(0)] =4

3√λ

∫

d3x√g(0)

(

1

2(∂ϕ)2 +

1

2R[g(0)]ϕ

2 + 2√λ(

√λ− α)ϕ6

)

(6)⇒ 3d conformally coupled scalar field with ϕ6 interaction.

[SdH,AP 0606276; SdH, AP, IP 0611315][Hertog, Horowitz hep-th/0503071]

This should arise in the large N limit of the strongly coupled 3d N = 8SCFT describing the IR fixed point of N M2-branes where an operator ofdimension 1 is turned on.

This action is the matter sector of the U(1) N = 2 Chern-Simons action. Ithas recently been proposed to be dual to AdS4 [Schwarz, hep-th/0411077;Gaiotto, Yin, arXiv:0704.3740]

6

Instanton solutions

The bulk equations of motion admit exact instanton solutions.

�φ− 1

6Rφ− 2λφ3 = 0 (7)

Seek solutions (“instantons”) with

Tµν = 0 ⇒ ds2 = `2

r2

(

dr2 + d~x2)

Unique solution:

φ =2

`√

|λ|br

−sgn(λ)b2 + (r+ a)2 + (~x− ~x0)2(8)

• λ < 0 ⇒ solution is regular everywhere• λ > 0 ⇒ solution is regular everywhere provided

a > b ≥ 0

• a/b labels different boundary conditions:

φ(r, x) = r φ(0)(x) + r2 φ(1)(x) + . . . (9)

φ(0)(x) =1

|λ|1/2b

−sgn(λ)b2 + a2 + (~x− ~x0)2(10)

φ(1)(x) = −αφ(0)(x)2 , α =

√

|λ|a/b > 0 (11)

(12)

7

• Holographic analysis implies α is a deformation parameter of dual CFT~x0, a2 − b2 parametrize 3d “instanton solution”For α >

√λ (a > b) the effective potential becomes unbounded from below.

This is the holographic image of the vacuum instability of AdS4 towardsdressing by a non-zero scalar field with mixed boundary conditions discussedabove

Similar conclusions were reached by Hertog & Horowitz (although only nu-merically).

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.

8

The instability region is φ→√

6/κ which corresponds to the total squashingof an S2 in the corresponding 11d geometry. This signals a breakdown ofthe supergravity description in this limit.

In the Lorentzian Coleman-De Luccia picture, our solution describes an ex-panding bubble centered at the boundary. Outisde the bubble, the metric isAdS4 (the false vacuum).

Inside, the metric is currently unknown (the true vacuum). One needs togo beyond sugra to find the true vacuum metric.

The tunneling probability can be computed and equals

P ∼ e−Γeff , Γeff =4π2`2

κ2

1√

1 − κ2

6`2α2

− 1

(13)

Deformation parameter drives the theory from regime of marginal instabilityα = κ/

√6` to total instability at α→ ∞.

9

Discussion and open points

• The boundary value of the scalar field is itself a solution of the following3-dimensional classical action; with φ(0) = ϕ2,

Sclass[ϕ] =1

α

∫

d3x√g(0)[(∂ϕ)

2 +1

2R[g(0)]ϕ

2 + µϕ6]

µ = λ(1 − a2/b2)

This agrees with the boundary effective action computed earlier when a/b→1 i.e. µ� 1 (marginal instability). a/b = 1 is also the supersymmetric linearboundary condition.

• They also agree at finite µ, in the double scaling limit a/b → 1, G/` → ∞.In this regime there is no good reason to trust the supergravity computation.

• A priori there is no reason why these actions should agree. ϕ(x) is asolution of the full non-perturbative boundary effective action, including allhigher derivatives. We have truncated to the two-derivative terms.

• The agreement may or may not be a coincidence. However, in the near-extremal situation a/b→ 1 there is possibly a non-renormalization theorem.

10

Gravitational instantons

• The previous solution described tunneling between two local minima ofthe scalar field. One would like to find similar effects for gravity. TheEuclidean instanton solution itself had an interesting holographic inter-pretation. It is therefore natural to look for similar solutions involvingthe gravitational field.

• Instanton solutions with Λ = 0 have self-dual Riemann tensor. However,self-duality of the Riemann tensor implies Rµν = 0

• In spaces with a cosmological constant we have to choose a differentself-duality condition. It turns out that self-duality of the Weyl tensoris compatible with non-zero cosmological constant:

Cµναβ =1

2εµν

γδCγδαβ

• This equation can be solved asymptotically. In the Fefferman-Grahamcoordinate system:

ds2 =`2

r2

(

dr2 + gij(r, x))

where

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

11

We find

g(2)ij = −Rij[g(0)] +1

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

g(3)ij = −2

3ε(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〉 = 3`2

16πGNg(3)ij [SdH,Skenderis, Solo-

dukhin 0002230]. We find that for any g(0)ij the holographic stresstensor is given by the Cotton tensor:

〈Tij〉 =`2

8πGNC(0)ij

• Therefore, we can integrate the stress-tensor to obtain the boundarygenerating functional:

〈Tij〉g(0)=

2√g

δW

δgij(0)

• The Cotton tensor is the variation of the gravitational Chern-Simonsaction:

SCS =k

4π

∫ (

ωab ∧ dωb

b +2

3ωa

b ∧ ωbc ∧c a)

δSCS =k

4π

∫

δωαβ ∧Rβα

=k

4π

∫

δgijεjkl

[

∇lRik −1

4gil∇lR

]

. (14)

Therefore, the boundary generating functional is the Chern-Simonsgravity term and we find

k =`2

8GN=

(2N)3/2

24

Generalization

Consider the bulk gravity action

SEH = − 1

2κ2

∫

d4x√G (R[G] − 2Λ) − 1

2κ2

∫

d3x√γ 2K

+1

2κ2

∫

d3x√γ

(

4

`+ `R[γ]

)

(15)

where κ2 = 8πGN , Λ = − 3`2. γ is the induced metric on the boundary and

K = γijKij.

We can add to it following bulk term:∫

M

R ∧R =

∫

∂M

(

ω ∧ dω+2

3ω ∧ ω ∧ ω

)

• It doesn’t change the boundary conditions and gives no contribution tothe equations of motion• It contributes to the holographic stress energy tensor, precisely by theCotton tensor:

〈Tij〉g(0)=

3`2

16πGN

(

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

This now affects any state. Again the dual generating functional containsthe gravitational Chern-Simons term.

12

U(1) gauge fields in AdS

S = Sgrav +

∫

d4x

(

− 1

4g2F 2µν +

θ

32π2εµνλσFµνFλσ

)

Solve eom:

ds2 =`2

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) + . . .

Solve Einstein’s eqs: g(n) and A(n)i are determined in terms of lower order co-

efficients g(0), g(3), A(0), A(1) [SdH,K. Skenderis,S. Solodukhin hep-th/0002230]

g(0) = bdy (conformal) metric

g(3)ij = 16πGN

3`2〈Tij〉 stress-energy tensor

A(0)i (x) ≡ Ai(x) = Ji

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

13

Eom can be exactly solved in this case:

ATi (r, p) = AT

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

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

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

We will write this as

f = −√

|�|ΠA , Πij = δij − pipj/p2

14

S-duality in AdS4

• The bulk eom are invariant under e.m. transformations

• Bulk action is invariant up to boundary terms. These boundary termsperturb the dual CFT and can be computed from the bulk

E ≡ E − θ4π2 F , Fi ≡ εijk∂jAk

S-duality acts as follows:

E′ = F

F ′ = −E

τ ′ = −1/τ , τ =θ

4π2+

i

g2(16)

The action transforms as follows:

S[A′, E′] = S[A,E] +

∫

d3x EA

In the old theory, A (F) is the source and E is the conserved current. S-duality interchanges the roles of the current and the source.New source: J ′ = ENew current: 〈O′

2〉A′ = −F

15

This agrees with Witten’s CFT definition of S-duality [hep-th/0307041]. Ais now regarded as a dynamical field that is integrated over. F is regardedas a current (conserved by Bianchi). The dual theory has a new backgroundgauge field v, E = ∗dv. The action changes precisely by the above Chern-Simons term.

The action of S-duality on the two-point function can be computed fromthe bulk and is as expected:

〈O′2O′

2〉A′ =g2

1 + g4θ2

(4π2)2

�1/2 Π −

g4θ4π2

1 + g4θ2

(4π2)2

∗ d

IR/UV flow

Take massive b.c. with θ-angle: f = f − θF

A+1

mf = J

Solve for A:

A =m

Γ[(m− �

1/2)J + θ ∗ dJ]

Γ = (m− �1/2)2 + θ2�

On-shell action:

Son-shell =m

2

∫

J�1/2

Γ(m− �

1/2)J + θ dJ ∧ �1/2

ΓJ

Includes terms such as 12

∫

Fij[A]�−1/2Fij[A]

See QED coupled to fermions [Kapustin,Strassler]as well as Chern-Simons terms

16

Two-point function:

〈O2O2〉 =m|p|2Γ

((m− |p|)Πij + θiεijkpk)

IR: |p| � m, Γ ∼ m2

usual 2-point function for conserved current of dim 2UV: |p| � m, Γ ∼ (1 + θ2)p2

〈O2iO2j〉 =m

1 + θ2

(

Πij −θiεijkpk

|p|

)

+

+m2

|p|(1 + θ2)2

(

(1 − θ2)Πij −2θiεijkpk

|p|

)

See also [Leigh,Petkou hep-th/0309177;Kapustin]Such behavior has also been found in quantum Hall systems [Burgess andDolan, hep-th/0010246]

17

Instanton Decay into AdS (in progress)

• Coleman-de Luccia (1980): tunneling between two stable ground states(true vacuum and false vacuum) including gravity effects

• It is very hard to give a detailed QG/stringy description of eternal infla-tion, where tunneling continues forever. One would like a holographicrealization of tunneling

• A first step would be to have a holographic description of a single bubble[Freivogel, Sekino, Susskind, Yeh 0606204]. However, the model wasnot embedded in string theory. As a result, the CFT was not unitary

• We will attempt a similar scenario with a model that is straightforwardlyembedded in M-theory

The model

S[G] = − 1

2κ2

[∫

M

dd+1x√G

(

R[G] +d(d− 1)

`2

)

+

∫

∂M

ddx√h

(

2K[h] +2(1 − d)

`2

)]

(17)

h = induced metric on boundaryK = extrinsic curvatured+ 1 = 4

18

The solution

In the Lorentzian:

ds2 =

{

`2

r2

(

dr2 − dt2 + dx2⊥)

inside4`2b2

(b2−(r+a)2−x2⊥+t2)2

(

dr2 − dt2 + dx2⊥)

outside

inside: (r+ a)2 + x2⊥ < t2 + b2 + 2br

outside: (r+ a)2 + x2⊥ ≥ t2 + b2 + 2br

Regularity requires a > b ≥ 0

19

• The region inside the bubble has lowest action and is pure AdS.

• Locally, both spaces are pure AdS. Once we glue them, the global ge-ometry changes. Regularity of the Euclidean instanton solution requiresr > 0 for the outside solution as well.

• It is a solution of Einstein’s equations with a negative cosmologicalconstant. The Euclidean solution is completely regular and has no

boundary. In the Lorentzian solution, the boundary is at (r+a)2+x2⊥ =

b2 + t2 and forms only at time t = a2 − b2. The boundary takes a verylarge amount of time to form if a/b� 1.

21

In fact, for a/b� 1 the space looks locally like de Sitter space:

ds2de Sitter =4L4

(L2 + y2µ)

2dy2

µ = 4

[

λ2 − 2λ4

L2x2µ + . . .

]

dx2µ

ds2ALAdS = 4

[

`2b2

µ4

(

1 +2a2

µ2

)

− 2`2b2

µ6x2µ + . . .

]

dx2µ

λ2 =`2b2

µ4

(

1 +2a2

µ2

)

, L =`b

µ

(

1 +2a2

µ2

)

∼ 3`b/a

This correspondence is valid as long as x2µ/µ

2 � 1. In particular it is notvalid at very late times.

22

Decay rate

The AdS action vanishes. We compute the on-shell Euclidean action eval-uated on the instanton solution:

ds2 = ψ2(

dr2 + d~x2)

, ψ =b`

−b2 + (r+ a)2 + ~x2

Sbulk = − 1

2κ2

∫

d4x√G

(

R+6

`2

)

=3

`2κ2

∫

d4xψ4

=π2`2

4κ2

(

1 +−1 + 3

2x2

(1 − x2)3/2

)

.

x = b/a

Total result:

S[GALAdS] =π2`2

4κ2

(

1 +−1 + 15

2x2 + 2x3

(1 − x2)3/2

)

P ∼ e−S[GALAdS]

• Probability vanishes when x→ 1 (a = b): instantons disappear

• x→ 0 region of total instability.

23

Discussion and summary

• Self-dual configurations in the bulk of AdS4 are dual to CFT’s coupledto sources whose effective action is given by some relative of the Chern-Simons action.

Scalar fields

• Generalized boundary conditions that correspond to multiple trace oper-ators destabilize AdS nonperturbatively. The instanton decay rate wascomputed.

• The boundary effective action was computed and it agrees with relatedproposals: 3d conformally coupled scalar with ϕ6 interaction. In a doublescaling limit the effective action itself has instanton solutions that matchthe bulk instantons

Gravity

• Gravitational instantons correspond to theories whose stress-energy ten-sor is given by the Cotton tensor. The generating functional is a Chern-Simons gravity term. This result generalizes to any theory containing aPontryagin term

• Question: can we get this term in some 3d CFT?

24

• Possible answer: Chern-Simons theory has a 1-loop gravitational anomalythat couples it to a gravitational Chern-Simons term. This anomalyshows up in the large N effective action and constitutes one of thetests of open/closed string duality in the topological A-model [Gopaku-mar, Vafa 1998]. This gives additional evidence that Chern-Simonstheory may play a role in the large N dual of AdS4

• Pontryagin is likely to be related to corresponding term in 11d sugraaction (in progress)

Gauge fields

• EM duality gives a 1-parameter family of b.c. that interpolate betweenD & N. On the boundary it interchanges the source and the conservedcurrent

• Massive deformations generate a flow of the two-point function. Onefinds the conserved current in the IR, but the S-dual gauge field in UV

• In UV, we find the effective action discussed by Kapustin for 3d QEDcoupled to fermions at large Nf