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 × S 7 Coincident M2-branes on ∂(AdS4),
Pl/ ∼ N −3/2
• AdS4 has instantons that are exact solutions of interacting theories and
allow 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 × S 7
Consider a conformally coupled scalar with quartic interaction:
1 √ −R + 2Λ 1
S= d4 x g + (∂φ)2 + Rφ2 + λ φ4
2 8πGN 6
8πGN κ2
λ= 6 2
= 6 2
for 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:
1 √
4 −R + 2Λ 6 2
S= d x G + (∂Φ)2 − 2 2 cosh κΦ
2 8πGN κ 3
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:
2
2
ds = dr2 + gij (r, x)dxidxj
r2
gij (r, x) = g(0)ij (x) + r 2g(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 fix
φ(1) (Neumann). Neumann boundary conditions correspond to a boundary
theory with an operator of dimension ∆ = 1, φ(0) . The effective action
becomes 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) = − α φ2
(0) (3)
They interpolate between Neumann (α = 0) and Dirichlet (α = ∞).
The boundary condition can be enforced by adding a boundary term:
3α
√
Sbdy = − d3 x g(0) φ3 (x)
(0) (4)
3
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]:
4 √ −1 1 √ √
Γeff[φ(0) , g(0) ] = √ d x g(0) φ(0) ∂i φ(0) ∂ φ(0) + R[g(0) ]φ(0) + 2 λ( λ − α) φ3
3 i
(0)
3 λ 2
(5)
Redefining φ(0) =ϕ 2 , we get
4 3 √ 1 1 √ √
Γeff[ϕ, g(0) ] = √ d x g(0) (∂ϕ) + R[g(0) ] ϕ + 2 λ( λ − α) ϕ6
2 2
3 λ 2 2
(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 = 8
SCFT describing the IR fixed point of N M2-branes where an operator of
dimension 1 is turned on.
This action is the matter sector of the U (1) N = 2 Chern-Simons action. It
has 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.
1
φ− Rφ − 2λφ3 = 0 (7)
6
Seek solutions (“instantons”) with
2
Tµν = 0 ⇒ ds2 = r2
dr2 + dx2
Unique solution:
2 br
φ= (8)
|λ| −sgn(λ)b2 + (r + a)2 + (x − x0 )2
• 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) + r 2 φ(1) (x) + . . . (9)
1 b
φ(0) (x) = (10)
|λ|1/2 −sgn(λ)b2 + a2 + (x − x0 )2
φ(1) (x) = −α φ(0) (x)2 , α = |λ|a/b > 0 (11)
(12)
7

• Holographic analysis implies α is a deformation parameter of dual CFT
x0 , a2 − b√ parametrize 3d “instanton solution”
2
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 boundary conditions discussed
above
Similar conclusions were reached by Hertog & Horowitz (although only nu-
merically).
Taking the boundary to be S 3 we plot the effective potential to be:

VΛ,Α
Φ
√ √
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 squashing
of an S 2 in the corresponding 11d geometry. This signals a breakdown of
the 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 is
AdS4 (the false vacuum).
Inside, the metric is currently unknown (the true vacuum). One needs to
go beyond sugra to find the true vacuum metric.
The tunneling probability can be computed and equals
 
−Γeff 4π 2 2 1
P ∼e , Γeff = − 1 (13)
 
κ2
 2
κ
1− 6 2 α2
Deformation parameter drives the theory from regime of marginal instability
√
α = 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 following
3-dimensional classical action; with φ(0) = ϕ2 ,
1 √ 1
Sclass[ϕ] = d3x g(0) [(∂ϕ)2 + R[g(0) ] ϕ2 + µ ϕ6 ]
α 2
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 linear
boundary 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 a
solution of the full non-perturbative boundary effective action, including all
higher 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 of
the scalar field. One would like to find similar effects for gravity. The
Euclidean instanton solution itself had an interesting holographic inter-
pretation. It is therefore natural to look for similar solutions involving
the 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 different
self-duality condition. It turns out that self-duality of the Weyl tensor
is compatible with non-zero cosmological constant:
1 γδ
Cµναβ = µν Cγδαβ
2
• This equation can be solved asymptotically. In the Fefferman-Graham
coordinate system:
2
2
ds = dr2 + gij (r, x)
r2
where
gij (r, x) = g(0)ij (x) + r 2g(2)ij (x) + r 3g(3)ij (x) + . . .
11

We find
1
g(2)ij = −Rij [g(0) ] + g(0)ij R[g(0) ]
4
2 kl 2
g(3)ij = − (0)i (0)k g(2)jl = C(0)ij
3 3
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
2
3
• The holographic stress tensor is Tij = 16πGN g(3)ij [SdH,Skenderis, Solo-
dukhin 0002230]. We find that for any g(0)ij the holographic stress
tensor is given by the Cotton tensor:
2
Tij = C(0)ij
8πGN
• Therefore, we can integrate the stress-tensor to obtain the boundary
generating functional:
2 δW
Tij g(0) = √ ij
g δg(0)

• The Cotton tensor is the variation of the gravitational Chern-Simons
action:
k 2
SCS = ωab ∧ dωbb + ωab ∧ ωb c ∧c a
4π 3
k
δSCS = δωαβ ∧ Rβ α
4π
k 1
= δg ij j
kl
l Rik − gil lR . (14)
4π 4
Therefore, the boundary generating functional is the Chern-Simons
gravity term and we find
2 (2N )3/2
k= =
8GN 24

Generalization
Consider the bulk gravity action
1 4
√ 1 √
SEH = − 2 d x G (R[G] − 2Λ) − d3x γ 2K
2κ 2κ2
1 √ 4
+ 2 d3 x γ + R[γ] (15)
2κ
where κ2 = 8πGN , Λ = − 3 . γ is the induced metric on the boundary and
2
K=γ ij K .
ij
We can add to it following bulk term:
2
R∧R = ω ∧ dω + ω∧ω∧ω
M ∂M 3
• It doesn’t change the boundary conditions and gives no contribution to
the equations of motion
• It contributes to the holographic stress energy tensor, precisely by the
Cotton tensor:
3 2
Tij g(0) = g(3)ij + Cij [g(0) ]
16πGN
This now affects any state. Again the dual generating functional contains
the gravitational Chern-Simons term.
12

U (1) gauge fields in AdS
1 2 θ
S = Sgrav + d4 x − Fµν + µνλσ
Fµν Fλσ
4g 2 32π 2
Solve eom:
2
2
ds = dr2 + gij (r, x)dxidxj
r2
gij (r, x) = g(0)ij + r 3g(3)ij + r 4g(4)ij + . . .
Ar (r, x) = A(0) (x) + rA(1) (x) + r 2A(2) (x) + . . .
r r r
Ai(r, x) = A(0) (x) + rA(1) (x) + r 2 A(2) (x) + . . .
i i i
Solve Einstein’s eqs: g(n) and A(n) are determined in terms of lower order co-
i
efficients g(0) , g(3) , A (0) , A(1) [SdH,K. Skenderis,S. Solodukhin hep-th/0002230]
g(0) = bdy (conformal) metric
16πGN
g(3)ij = 3 2
Tij stress-energy tensor
A(0) (x) ≡ Ai(x) = Ji
i
A(1) (x) ≡ Ei (x) = O∆=2,i (x) electric field Fri
i
13

Eom can be exactly solved in this case:
1
AT(r, p) = AT(p) cosh(|p|r) +
i i fi (p) sinh(|p|r)
|p|
Impose regularity at r = ∞:
1 1 1 1
AT =
i AT(p) +
i fi(p) e|p|r + AT(p) −
i fi(p) e−|p|r
2 |p| 2 |p|
⇒ AT(p) +
i
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 terms
perturb the dual CFT and can be computed from the bulk
θ
E≡E− 4π 2
F , Fi ≡ ijk ∂j Ak
S-duality acts as follows:
E = F
F = −E
θ i
τ = −1/τ , τ = + 2 (16)
4π 2 g
The action transforms as follows:
S[A , E ] = S[A, E] + d 3 x 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 = E
New current: O2 A = −F
15

This agrees with Witten’s CFT definition of S-duality [hep-th/0307041]. A
is now regarded as a dynamical field that is integrated over. F is regarded
as a current (conserved by Bianchi). The dual theory has a new background
gauge 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 from
the bulk and is as expected:
g4θ
g2 1/2 4π 2
O 2 O2 A = g θ
Π− ∗d
+ (4πθ2)2
g4 2
4 2
1+ (4π 2 )2
1

IR/UV flow
Take massive b.c. with θ-angle: f = f − θF
1
A+ f=J
m
Solve for A:
m 1/2
A= [(m − )J + θ ∗ dJ]
Γ
1/2 2
Γ = (m − ) + θ2
On-shell action:
m 1/2 1/2
1/2
Son-shell = J (m − )J + θ dJ ∧ J
2 Γ Γ
Includes terms such as 1 Fij [A] −1/2 Fij [A]
2
See QED coupled to fermions [Kapustin,Strassler]
as well as Chern-Simons terms
16

Two-point function:
m|p|
O 2 O2 = ((m − |p|)Πij + θi ijk pk )
2Γ
IR: |p| m, Γ ∼ m2
usual 2-point function for conserved current of dim 2
UV: |p| m, Γ ∼ (1 + θ 2)p2
m θi ijk pk
O2i O2j = Πij − +
1 + θ2 |p|
m2 2θi ijk pk
+ (1 − θ 2 )Πij −
|p|(1 + θ 2 )2 |p|
See also [Leigh,Petkou hep-th/0309177;Kapustin]
Such behavior has also been found in quantum Hall systems [Burgess and
Dolan, 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 holographic
realization of tunneling
• A first step would be to have a holographic description of a single bubble
[Freivogel, Sekino, Susskind, Yeh 0606204]. However, the model was
not embedded in string theory. As a result, the CFT was not unitary
• We will attempt a similar scenario with a model that is straightforwardly
embedded in M-theory
The model
1 d+1
√ d(d − 1) d
√ 2(1 − d)
S[G] = − 2 d x G R[G] + 2
+ d x h 2K[h] + 2
2κ M ∂M
(17)
h = induced metric on boundary
K = extrinsic curvature
d+1=4
18

The solution
In the Lorentzian:
2
2 r
2 dr2 − dt2 + dx2 ⊥ inside
ds = 4 2 b2
(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 requires
r > 0 for the outside solution as well.
• It is a solution of Einstein’s equations with a negative cosmological
constant. The Euclidean solution is completely regular and has no
boundary. In the Lorentzian solution, the boundary is at (r +a)2 +x2 =
⊥
b 2 + t2 and forms only at time t = a2 − b2 . The boundary takes a very
large amount of time to form if a/b 1.
21

In fact, for a/b 1 the space looks locally like de Sitter space:
2 4L4 λ4 2
dsde Sitter = dyµ = 4 λ − 2 2 xµ + . . . dx2
2 2
µ
(L2 + yµ )2
2 L
2 b2 2a2 2 b2
2
dsALAdS = 4 1+ 2 −2 x2 + . . . dx2
µ µ
µ4 µ µ6
2 b2 2a2 b 2a2
2
λ = 1+ 2 , L= 1+ 2 ∼ 3 b/a
µ4 µ µ µ
This correspondence is valid as long as x2 /µ2
µ 1. In particular it is not
valid at very late times.
22

Decay rate
The AdS action vanishes. We compute the on-shell Euclidean action eval-
uated on the instanton solution:
b
ds2 = ψ 2 dr2 + dx2 , ψ=
−b2 + (r + a)2 + x2
1 4
√ 6 3
Sbulk = − 2 d x G R+ 2 = 2 κ2
d4 x ψ 4
2κ
3 2
π2 2 −1 + 2
x
= 1+ .
4κ2 (1 − x2 )3/2
x = b/a
Total result:
15
π2 2 −1 + 2
x2 + 2x3
S[GALAdS ] = 1+
4κ2 (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 coupled
to 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 was
computed.
• The boundary effective action was computed and it agrees with related
proposals: 3d conformally coupled scalar with ϕ6 interaction. In a double
scaling limit the effective action itself has instanton solutions that match
the 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 a
Pontryagin term
• Question: can we get this term in some 3d CFT?
24

• Possible answer: Chern-Simons theory has a 1-loop gravitational anomaly
that couples it to a gravitational Chern-Simons term. This anomaly
shows up in the large N effective action and constitutes one of the
tests of open/closed string duality in the topological A-model [Gopaku-
mar, Vafa 1998]. This gives additional evidence that Chern-Simons
theory may play a role in the large N dual of AdS4
• Pontryagin is likely to be related to corresponding term in 11d sugra
action (in progress)
Gauge fields
• EM duality gives a 1-parameter family of b.c. that interpolate between
D & N. On the boundary it interchanges the source and the conserved
current
• Massive deformations generate a flow of the two-point function. One
finds 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 QED
coupled to fermions at large Nf