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