Instantons and Chern-Simons Terms in AdS4/CFT3: Gravity on the Brane?
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
Motivation1. Understanding the theory on the worldvolume ofa large number of membranes.• It is a 3d SCFT with N = 8 supersymmetry thatarises 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 × S 7 ≃ 3d SCFT on ∂(AdS4) ℓPl/ℓ ∼ N −3/2⇒ Problem: it is a weak/strong coupling dualityUse instanton solutions and duality properties 2
InstantonsInstantons describe tunneling across potential barrierImportant quantum eﬀects on gravityDiﬃculties:• Need exact solutions of interacting theories• String theory/M-theory on these backgrounds notwell understood beyond supergravity approximation⇒ Explicit exact solutions of M-theory compactiﬁeddown to 4 dimensions⇒ Holographic description 3
• From the point of view of holography, the use ofinstantons 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’sThere is a duality conjecture for the 3d CFT’s ofinterest: a generalization of electric-magnetic dual-ity for higher spins relating IR and UV theories [Leighand Petkou, ’03] 5
dimension Weyl−equivalence of UIR of O(4,1) 1 0 Unitarity bound 1 0 ∆ = s+1 1 0 3 1 0 1 0 Double−trace 2 1 0 Dualization and "double−trace" deformations 1 0 Deformation 10 1 0 1 2 spin 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 → instabilityBoundary: eﬀective action → 3d conformallycoupled scalar ﬁeld with ϕ6 potential2. Gravitational instantons and topologically mas-sive gravity3. U (1) gauge ﬁelds, RG ﬂows and S-duality 7
Conformally Coupled Scalars and AdS4 × S 7The model: a conformally coupled scalar with quarticinteraction: 1 4 √ −R + 2Λ 1 S= d x g + (∂φ) + R φ2 + λ φ4 2 2 8πGN 6 (1)• This model arises in M-theory compactiﬁcationon S 7. It is a consistent truncation of the N = 8 4dsugra action where we only keep the metric and onescalar ﬁeld [Duﬀ,Liu 1999].• The coupling is then given by λ = 8πGN 6ℓ2 8
Potential ℓ2V (φ) in units where 8πGN = 1 for abackground with negative cosmological constant. 9
• There are two extrema: φ = 0 and φ = 6/8πGN .• Naively we would expect to roll down the hill bysmall perturbations. However, in the presence ofgravity such a picture is misleading: one needs totake into account kinetic terms and boundary terms[E. Weinberg ’86]• In fact, the φ = 0 point is known to be stable. It isthe well-known AdS4 vacuum with standard choiceof boundary conditions [Breitenlohner and Freedman,’82] 10
• But it is unstable for generic choice of boundaryconditions. One has to consider tunneling eﬀectsdue to the presence of kinetic terms [E. Weinberg’82]. Instanton eﬀects can mediate the decay.• I will construct such instantons explicitly for oneparticular choice of boundary conditions and computethe decay rate.• There is an interesting holographic dual descrip-tion of the decay that I will also analyze
InstantonsInstanton solutions: exact solutions of the Euclideanequations of motion with ﬁnite action.’t Hooft instantons have zero stress-energy tensor: T00 ∼ E 2 − B 2 = 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: ℓ2 ds2 = 2 dr 2 + dx2 , r>0 (4) rWe need to solve the Klein-Gordon equation in anAdS4 background: 1 φ − R φ − 2λ φ3 = 0 (5) 6
Unique solution with vanishing stress-energy tensor: 2 br φ= (6) ℓ |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• α = a/b labels diﬀerent boundary conditions: φ(r, x) = r φ(0)(x) + r 2 φ(1)(x) + . . . (7)It gives the relation between φ(1) and φ(0): φ(1)(x) = −ℓα φ2 (x) (0) (8) 12
Holographic analysis1) α is a deformation parameter of the dual CFT2) x0, a2 − b2 parametrize 3d instanton vacuum √• For α > λ (a > b) the eﬀective potential becomesunbounded from below. This is the holographicimage of the vacuum instability of AdS4 towardsdressing by a non-zero scalar ﬁeld with mixed bound-ary conditions discussed above.Similar conclusions were reached by Hertog & Horowitz (although only numerically). 13
Taking the boundary to be S 3 we plot the eﬀectivepotential 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. 14
• The instability region is φ → 6/8πGN , which cor-responds to the total squashing of an S 2 in the cor-responding 11d geometry. This signals a breakdownof the supergravity description in this limit.• In the Lorentzian Coleman-De Luccia picture, oursolution describes an expanding bubble centered atthe 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 truevacuum metric. 15
ζ z1 Ro(t) 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 z2 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 Expansion of the bubble towards the bulk in the Lorentzian.R0(t): radius of the bubblez1, z2: boundary coordinatesζ: radial AdS coordinate 16
The tunneling probability can be computed andequals 4π 2ℓ2 1P ∼ e−Γeﬀ , Γeﬀ = 2 2 − 1 (9) κ 1 − 6ℓκ α2 2The deformation parameter α drives the theory from √regime of marginal instability α = κ/ 6ℓ (P → 0) tototal instability at α → ∞ (P → 1). 17
Boundary description of the instability• We have seen that the bulk instability is mirroredby 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 largeN is obtained from the bulk on-shell sugra action: φ(r, x) = r ∆− φ(0)(x) + . . . bulk 3x φ e Son-shell[φ] = e W [φ(0)] ≡ e d (0) (x) O(x) (10) CFT 18
• The Wilsonian eﬀective action Γ[ O(x) ] can beobtained 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 betweenφ(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: 1 1Γeﬀ =√ d3 x g(0) φ−1 ∂iφ(0)∂ iφ(0) (0) + R[g(0)]φ(0) λ 2 √ √ +2 λ( λ − α) φ3 (12) (0)Redeﬁning φ(0) = ϕ2, we get 1 1 2 + 1 R[gΓeﬀ[ϕ, g(0)] = √ d3 x g(0) (∂ϕ) (0) ] ϕ2 λ 2 2 √ √ +2 λ( λ − α) ϕ6 (13) ⇒ 3d conformally coupled scalar ﬁeld with ϕ6interaction. [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 dimension1 is turned on. This CFT describes N coincidentM2-branes for large N away from the conformal ﬁxedpoint.This action is the matter sector of the U (1) N = 2Chern-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 andLambert arXiv:0712.3738 [hep-th]
Warming up: U (1) instantons• In the usual AdS/CFT interpretation, the boundaryvalues 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) := ∂r Ai(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 amore 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: 1 ai(x) + ǫijk ∂j aj (x) = 0 . (18) M Oi(x) J=0 = ai(x) . (19)
Oi satisﬁes the same dynamical equation as ai and itcan be integrated to an eﬀective action. In this case,this action turns out to be the massive topologicalgauge theory of Deser and Jackiw in 3d.
Gravitational instantons• The scalar ﬁeld described tunneling between twolocal minima of the scalar ﬁeld. One would like toﬁnd similar eﬀects for gravity. Should reveal extraholographic degrees of freedom.• Instanton solutions with Λ = 0 have self-dual Rie-mann tensor. However, self-duality of the Riemanntensor implies Rµν = 0.• In spaces with a cosmological constant we need 22
to choose a diﬀerent self-duality condition. It turnsout that self-duality of the Weyl tensor is compatiblewith a non-zero cosmological constant: 1 Cµναβ = ǫµν γδ Cγδαβ 2•This equation can be solved asymptotically. In theFeﬀerman-Graham coordinate system: ℓ2 ds2 = 2 dr 2 + gij (r, x) dxidxj rwhere gij (r, x) = g(0)ij (x) + r 2g(2)ij (x) + r 3g(3)ij (x) + . . .
We ﬁnd 1 g(2)ij = −Rij [g(0)] + g(0)ij R[g(0)] 4 2 kl ∇ 2 g(3)ij = − ǫ(0)i (0)k g(2)jl = C(0)ij 3 3Cij is the 3d Cotton tensor.• In general d, conformal ﬂatness is measured by theWeyl tensor [see e.g. Skenderis and Solodukhin, hep-th/9910023]• In d = 3, the Weyl tensor identically vanishes andconformal ﬂatness ⇔ Cij = 0
• The holographic stress tensor is Tij = 3ℓ2 g 16πGN (3)ij[SdH,Skenderis, Solodukhin 0002230]. We ﬁnd thatfor any g(0)ij the holographic stress tensor is givenby the Cotton tensor: ℓ2 Tij = C(0)ij 8πGN• Therefore, we can integrate the stress-tensor toobtain the boundary generating functional: 2 δW Tij g(0) =√ g δg ij (0)
• The Cotton tensor is the variation of the gravi-tational Chern-Simons action: k 2 b SCS = ωa ∧ dωb + ωa ∧ ωbc ∧c a b b 4π 3 k δSCS = δωαβ ∧ Rβ α 4π k 1 = δg ij ǫj kl ∇lRik − gil ∇lR . (20) 4π 4Therefore, the boundary generating functional is theChern-Simons gravity term.
GeneralizationConsider the bulk gravity action 1 √ 1 3x√γ 2KSEH = − 2 d4 x G (R[G] − 2Λ) − d 2κ 2κ2 1 3x√γ 4 + ℓ R[γ] + 2 d (21) 2κ ℓ 3where κ2 = 8πGN , Λ = − ℓ2 . γ is the induced metricon the boundary and K = γ ij Kij .We can add to it following bulk term: 2 R∧R= ω ∧ dω + ω ∧ ω ∧ ω M ∂M 3 23
• It doesn’t change the boundary conditions and givesno contribution to the equations of motion• It contributes to the holographic stress energy ten-sor, precisely by the Cotton tensor: 3ℓ2 Tij g(0) = g(3)ij + Cij [g(0)] 16πGNThis now aﬀects any state. Again the dual gen-erating functional contains the gravitational Chern-Simons term. One can show that such terms arisefrom R6 terms in M-theory [in progress].
Regularity of solutionsWe have solved a perturbative series for the metric: gij (r, x) = g(0)ij (x) + r 2g(2)ij (x) + r 3g(3)ij (x) + . . .We want regular the solutions in the interior r = ∞.We solve Einstein’s equations perturbatively: gij =δij + hij . The solution is d3p ip·x f (r, p) ¯ij (r, x) = h e Πijkl h(0)kl (p) (2π)3 f (0, p) 2 Πijkl = Πijkl (22) 24
Regular solutions have: π f (r, p) = (1 + |p|r ) e−|p|r . 2Expanding in powers of r, we get ¯ij (0, x) = ¯(0)ij (x) h h 1 3/2 ¯(3)ij (x) = h ¯(0)ij h (23) 3This is a Rubin-type boundary condition. Combinewith regularity condition: ¯(0)ij = α ǫikl ∂k 1/2¯(0)jl + (i ↔ j) h h (24)This can be rewritten as α δGij = − 1/2 δCij (25)
This is the linearized version of the 3d topologicallymassive gravity theory of Deser, Jackiw and Temple-ton!• Rubin-type boundary conditions for gravity are onlypossible 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 1 2 θ µνλσ S = Sgrav + d4 x − Fµν + ǫ Fµν Fλσ 4g 2 32π 2Solve eom: ℓ2 2 ds = 2 dr 2 + gij (r, x)dxidxj r 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 (0) (1) (2) Ai(r, x) = Ai (x) + rAi (x) + r 2Ai (x) + . . . 26
(n)Solve Einstein’s eqs: g(n) and Ai are determined interms of lower order coeﬃcients g(0), g(3), A(0), A(1)[SdH,K. Skenderis,S. Solodukhin hep-th/0002230] (0)Ai (x) ≡ Ai(x) = Ji (1)Ai (x) ≡ Ei(x) = O∆=2,i(x) electric ﬁeld FriEom can be exactly solved in this case: T T 1 Ai (r, p) = Ai (p) cosh(|p|r) + fi(p) sinh(|p|r) |p| 27
Impose regularity at r = ∞: 1 1 e|p|r + 1 1 TAi = AT(p) i + fi(p) AT(p) i − fi(p) e−|p|r 2 |p| 2 |p| 1 ⇒ AT(p) + |p| fi(p) = 0 regularity iS-duality acts as follows: E′ = B B ′ = −E ′ θ i τ = −1/τ , τ = 2 + 2 (26) 4π gThe action transforms as: S[A′, E ′] = S[A, E] + d3x (E − θ B) A 28
With usual boundary conditions, B is ﬁxed at theboundary 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 thesource.New source: J′ = ENew 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: 1 θ Oi(p) Oj (−p) A=0 = 2 |p| Πij + iǫ p 2)2 ijk k g (4π pipj Πij = δij − 2 (27) |p| g 4θ ′ ′ g2 4π 2Oi(p)Oj (−p) A′ = |p| Πij − i ǫijk pk g 4θ2 g 4θ2 1+ (4π 2)2 1 + (4π 2)2 (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 onRG ﬂows• Deforming the boundary conditions in a way thatbreaks conformal invariance (introducing mass pa-rameter)⇔ adding relevant operator that produces ﬂow to-wards new IR ﬁxed point 30
1Massive boundary condition: A + m (E − θB) = JIR: OiOj = |p|Πij + θ i ǫijk pk2-point function for conserved current of dim 2.UV: OiOj = m2 (|p|Πij − θ i ǫijk pk ) |p|2(1+θ 2)2S-dual current coming from dualizing gauge ﬁeld.See also [Leigh,Petkou hep-th/0309177;Kapustin]Such behavior has also been found in quantum Hallsystems [Burgess and Dolan, hep-th/0010246] 31
Conclusions• Instanton conﬁgurations in the bulk of AdS4 aredual to CFT’s whose eﬀective action is given by atopological term, typically a relative of the Chern-Simons action. They describe tunneling eﬀects inM-theory.• Bulk instantons modify the boundary conditions.Boundary degrees of fredom become dynamical. 32
Scalar ﬁelds• Generalized b.c. that correspond to multiple traceoperators destabilize AdS4 nonperturbatively bydressing of the scalar ﬁeld. The Lorentzian pictureis in terms of tunneling to a new vacuum. Thetunneling rate was computed.• Boundary eﬀective action was computed and itagrees with related proposals: 3d conformally cou-pled scalar with ϕ6 interaction. Boundary instantonsmatch 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 toa linearized graviton that satisﬁes a dynamical equa-tion. For the case of the Chern-Simons term we gettopologically massive gravity [Deser, Jackiw, Tem-pleton]. 34
Gauge ﬁelds• Electric-magnetic duality interpolates between D& N. On the boundary it interchanges the source andthe conserved current (dual CFT3’s).• Massive deformations generate RG ﬂow of the two-point function. One ﬁnds the conserved current inthe 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