Duality and Emergent Gravity in AdS/CFT

Sebastian de Haro University of Amsterdam and University of Cambridge

Equivalent Theories in Science and Metaphysics

Princeton, 21 March 2015

Motivating thoughts

•Duality and emergence of space-time have been a strong focus in quantum gravity and string theory research in recent years

•The notion of ‘emergence’ of space-time and/or gravity is often attached to the existence of a ‘duality’

Motivating thoughts

• An argument along the following lines is often made:

a) Theory F (‘fundamental’) and theory G (‘gravity’) are dual to one another

b) Theory F does not contain gravity (and/or space-time) whereas theory G does

c) Therefore space-time (and/or gravity) emerges in theory G. Theory F is to be regarded as more fundamental

•But this argument is problematic: it replaces ‘duality’ by ‘emergence’.• Duality is a symmetric relation, whereas emergence is

not symmetric• We need to explain what breaks the symmetry • Emergence of space-time requires more than simply

‘the space-time being dual to something that is not spatio-temporal’

• It might lead to bad heuristics for constructing new theories, e.g. when the argument is taken as a reason not to pursue theory G but to just work on theory F

• I will discuss the notions of duality and emergence in holographic scenarios, in particular AdS/CFT

• I will only discuss the possible emergence of gravity together with one, spatial dimension.• This is a non-trivial task: for obtaining the right classical

dynamics for the metric is hard!

Introduction: ’t Hooft’s Holographic Hypothesis

• The total number of degrees of freedom, 𝑛, in a region of space-time containing a black hole, is:

𝑛 =𝑆

log 2=

𝐴

4𝐺log 2

• Hence, “we can represent all that happens inside [a volume] by degrees of freedom on the surface”

• “This suggests that quantum gravity should be described entirely by a topological quantum field theory, in which all degrees of freedom can be projected on to the boundary”

• “We suspect that there simply are no more degrees of freedom to talk about than the ones one can draw on a surface [in bit/Planck length2]. The situation can be compared with a hologram of a three dimensional image on a two dimensional surface”

Introduction: ’t Hooft’s Holographic Hypothesis

• The observables “can best be described as if“ they were Boolean variables on a lattice, which suggests that the description on the surface only serves as one possible representation

• Nevertheless, 't Hooft's account more often assumes that the fundamental ontology is the one of the degrees of freedom that scale with the space-time's boundary. He argued that quantum gravity theories that are formulated in a four dimensional space-time, and that one would normally expect to have a number of degrees of freedom that scales with the volume, must be “infinitely correlated" at the Planck scale

• The explanatory arrow here clearly goes from surface to bulk, with the plausible implication that the surface theory should be taken as more basic than the theory of the enclosed volume

• There is no indication that a notion of emergence is relevant here

Introduction: ’t Hooft’s Holographic Hypothesis

• ’t Hooft’s paper wavers between boundary and bulk as fundamental ontologies

• There is an interpretative tension here, that resurfaces in other contexts where there are dualities

Philosophical concerns regarding holographic dualities:

•Can one decide which side of the duality is more fundamental?

•Is one facing emergence of space, time, and/or gravity?

Plan

•Duality: • Introduction to AdS/CFT•Duality•Renormalization group•Diffeomorphism invariance and background

independence• Interpretation

•Emergence

Geometry of AdS𝐷

• Hyperboloid in 𝐷 + 1 dimensions:

−𝑋02 − 𝑋𝐷

2 +

𝑖=1

𝐷−1

𝑋𝑖2 = −ℓ2

• Constraint can be solved introducing 𝐷 coordinates:𝑋0 = ℓ cosh𝜌 cos 𝜏𝑋𝐷 = ℓ cosh 𝜌 sin 𝜏𝑋𝑖 = ℓ sinh 𝜌 Ω𝑖 𝑖 = 1, … , 𝑑 = 𝐷 − 1 , Ω𝑖 = unit vector

• Leading to: d𝑠2 = ℓ2 − cosh2 𝜌 d𝜏2 + d𝜌2 + sinh2 𝜌 dΩ𝐷−22

• Symmetry group SO 2, 𝑑 apparent from the construction

• Riemann tensor in terms of the metric (negative curvature):

𝑅𝜇𝜈𝜆𝜎 = −1

ℓ2 𝑔𝜇𝜆𝑔𝜈𝜎 − 𝑔𝜇𝜎𝑔𝜈𝜆

𝑋𝑖

𝑋0

𝑋𝐷

ℓ

Geometry of AdS𝐷

• Useful choice of local coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝜂𝑖𝑗 d𝑥𝑖d𝑥𝑗 , 𝑖 = 1, … , 𝑑 = 𝐷 − 1

• 𝜂𝑖𝑗 = flat metric (Lorentzian or Euclidean signature)

• Can be generalised to (AL)AdS:

d𝑠2 =ℓ2

𝑟2 d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖d𝑥𝑗

𝑔𝑖𝑗 𝑟, 𝑥 = 𝑔 0 𝑖𝑗 𝑥 + 𝑟 𝑔 1 𝑖𝑗 𝑥 + 𝑟2𝑔 2 𝑖𝑗 𝑥 + ⋯

• Einstein’s equations now reduce to algebraic relations between 𝑔 𝑛 𝑥 𝑛 ≠ 0, 𝑑 and 𝑔 0 𝑥 , 𝑔 𝑑 𝑥

Adding Matter

• Matter field 𝜙 𝑟, 𝑥 (for simplicity, take 𝑚 = 0), solve KG equation coupled to gravity:

𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + 𝑟 𝜙 1 𝑥 + ⋯ + 𝑟𝑑𝜙 𝑑 𝑥 + ⋯

• Again, 𝜙 0 𝑥 and 𝜙 𝑑 𝑥 are the boundary conditions and all other coefficients 𝜙 𝑛 𝑥 are given in terms of them (and the metric)

AdS/CFT

•𝐷-dim AdS

•𝐷-dim (AL)AdS

• d𝑠2 =ℓ2

𝑟2 d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖d𝑥𝑗

• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯ + 𝑟𝑑𝑔 𝑑 𝑥

• Field 𝜙 𝑟, 𝑥 , mass 𝑚

• 𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + ⋯ + 𝑟𝑑𝜙 𝑑 𝑥

• CFT on ℝ𝐷−1

•QFT with a fixed point• Metric 𝑔 0 (𝑥)

• 𝑇𝑖𝑗 𝑥 =ℓ𝑑−1

16𝜋𝐺𝑁𝑔 𝑑 𝑥 + ⋯

•Operator 𝒪 𝑥 , scaling dimension Δ 𝑚• Coupling 𝜙 0 𝑥

• 𝒪 𝑥 = 𝜙 𝑑 𝑥

Fields Operators

Normalizable mode (sub-leading) Vev (state)

Non-normalizable mode (leading) Coupling

Example: AdS5 × 𝑆5 = SU 𝑁 SYM

AdS5 × 𝑆5

• Type IIB string theory

• Limit of small curvature:supergravity (Einstein’s theory + specific matter fields)

• Symmetry of AdS: diffeo’s that preserve form of the metric generate conformal transformations on the bdy

• Symmetry of 𝑆5

SU 𝑁 SYM

• Supersymmetric Yang-Mills theory with gauge group SU(𝑁)

• Limit of weak coupling: ’t Hooft limit (planar diagrams)

• Classical conformal invariance of the theory

• Symmetry of the 6 scalar fields

• Limits are incompatible (weak/strong coupling duality: useful!)• Only gauge invariant quantities (operators) can be compared• Symmetry:

SO 2,4 × SO 6

What is a Duality? (Butterfield 2014)

• Regard a theory as a triple 𝒮, 𝒪, 𝐷• 𝒮 = states (in Hilbert space)

• 𝒪 = operators (self-adjoint, renormalizable, invariant under symmetries)

• 𝐷 = dynamics (given by e.g. Lagrangian and integration measure)

• A duality is an isomorphism between two theories 𝒮𝐴, 𝒪𝐴, 𝐷𝐴 and 𝒮𝐵, 𝒪𝐵, 𝐷𝐵 .

• There exist bijections: • 𝑑𝒮: 𝒮𝐴 → 𝒮𝐵 ,

• 𝑑𝒪: 𝒪𝐴 → 𝒪𝐵

and pairings (vevs) 𝑂, 𝑠 𝐴 such that:𝑂, 𝑠 𝐴 = 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 𝐵 ∀𝑂 ∈ 𝒪𝐴, 𝑠 ∈ 𝒮𝐴

AdS/CFT Duality

• Slightly more general statement

• The theory is given by 𝒮, 𝒪, 𝒞, 𝐷 . Distinguish:• 𝒞 = external parameters (e.g. couplings, boundary conditions): variable

• 𝐷 = dynamics (given by e.g. Lagrangian and integration measure): fixed

• AdS/CFT is an isomorphism between 𝒮𝐴, 𝒪𝐴, 𝒞𝐴, 𝐷𝐴 and 𝒮𝐵, 𝒪𝐵, 𝒞𝐵 , 𝐷𝐵 . There exist bijections: • 𝑑𝒮: 𝒮𝐴 → 𝒮𝐵

• 𝑑𝒪: 𝒪𝐴 → 𝒪𝐵

• 𝑑𝒞: 𝒞𝐴 → 𝒞𝐵

such that:

𝑂, 𝑠 𝑐 ,𝐷𝐴= 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵

∀𝑂 ∈ 𝒪𝐴, 𝑠 ∈ 𝒮𝐴, 𝑐 ∈ 𝒞𝐴

𝑐𝐵 = 𝑑𝒞 𝑐𝐴 𝑐𝐴 ∈ 𝒞𝐴, 𝑐𝐵 ∈ 𝒞𝐵

𝑂, 𝑠 𝑐 ,𝐷𝐴= 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵

𝑐𝐵 = 𝑑𝒞 𝑐𝐴

(1)

(2)

AdS/CFT Duality

• AdS/CFT can be described this way: • Normalizable modes correspond to vevs of operators (choice of state)

• Fields correspond to operators

• Boundary conditions (non-normalizable modes) correspond to couplings

• Dynamics otherwise different (different Lagrangian, different dimensions!)

• Two salient points of :• Part of the dynamics now also agrees (couplings in the Lagrangian vs.

boundary conditions). This is the case in any duality that involves parameters that are not operators, e.g. T-duality (𝑅 ↔ 1/𝑅), electric-magnetic duality (𝑒 ↔ 1/𝑒)

• It is also more general: while 𝒮, 𝒪, 𝐷 are a priori fixed, 𝒞 can be varied at will. Thus we have a multidimensional space of theories

• Dualities of this type are not isomorphisms between two given theories, but between two sets of theories

𝒮

𝒪

𝒞

𝐷

𝑂, 𝑠 𝑐 ,𝐷𝐴= 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵

AdS/CFT Duality (Continued)

•String theory in (AL)AdS space = QFT on boundary•Formula 1 is generated by:

𝑍string 𝜙 0 = 𝜙 0,𝑥 =𝜙 0 𝑥

𝒟𝜙 𝑒−𝑆 𝜙 = exp d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

•The correlation functions of all operators match•Physical equivalence, mathematical structure different•Large distance ↔ high energy divergences 2•Strictly speaking, the AdS/CFT correspondence has the

status of a ‘conjecture’, though there is massive evidence for it (and it is usually called a ‘correspondence’: compare e.g. Fermat’s last ‘theorem’ before it was proven!)

(1′)

𝑐𝐵 = 𝑑𝒞 𝑐𝐴

Renormalization Group

• Radial integration: • Wilsonian renormalization:

Λ𝑏Λ0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

AdS𝑟

𝜕AdS𝑟 𝜕AdS𝜖

new boundary condition

integrate out

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT(2) 𝑐𝐵 = 𝑑𝒞 𝑐𝐴

Conditions for AdS/CFT Duality

• What could lead to the failure of AdS/CFT as a duality?

• Two conditions must be met for this bijection to exist. The observable structures of these theories should be:

i. Complete (sub-) structures of observables, i.e. no other observables can be written down than (1): this structure of observables contains what the theories regard to be ‘physical’ independently on each side of the duality.

ii. Identical, i.e. the (sub-) structures of observables are identical to each other.

If ii. is not met, we can have a weaker form of the conjecture: a relation that is non-exact. For instance, if the duality holds only in some particular regime of the coupling constants

• There are no good reasons to believe that i. fails.

• Whether ii. is met is still open, but all available evidence indicates that it is satisfied, including some non-perturbative tests. However: see later

Remarks on Background Independence

• Theories of gravity are usually required to be ‘background independent’. In Einstein’s theory of relativity, the metric is a dynamical quantity, determined from the equations of motion rather than being fixed from the outset

• The concept of ‘background independence’ does not have a fixed meaning, see Belot (2011)

• Here I will adopt a ‘minimalist approach’: a theory is background independent if it is generally covariant and its formulation does not make reference to a background/fixed metric. In particular, the metric is determined dynamically from the equations of motion

• In this minimalist sense, classical gravity in AdS is fully background independent: Einstein’s equations with negative cosmological constant• Quantum corrections do not change this conclusion: they appear perturbatively as

covariant higher-order corrections to Einstein’s theory

• Could background independence be broken by the asymptotic form of the metric?• This is just a choice of boundary condition. The equations of motion do not determine

them: they need to be specified additionally (de Haro et al. 2001)• But this is not a restriction on the class of solutions considered; as in classical mechanics,

the laws (specifically: the equations of motion) simply do not contain the informtion about the boundary/initial conditions

• Boundary conditions do not need to preserve the symmetries of the laws. Thus this does not seem a case of lack of background independence of the theory. At most, it may lead to spontaneous breaking of the symmetry in the sense of a choice of a particular solution

• Hence, the background independence of the theory is well established

Diffeomorphism Invariance of (1’)

• I have discussed background independence of the equations of motion. What about the observables?

• Partition function (1’):• It depends on the boundary conditions on the metric (as do the classical

solutions)• It is diffeomorphism invariant, for those diffeomorphisms that preserve the

asymptotic form of the metric

• Other observables obtained by taking derivatives of (1): they transform as tensors under these diffeomorphisms. These observables are covariant, for odd d (=boundary dimension):

• For odd 𝑑: • Invariance/covariance holds

• For even 𝑑:• Bulk diffeomorphisms that yield conformal transformations of the boundary

metric are broken due to IR divergences (holographic Weyl anomaly). Is this bad?

𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1′)

Diffeomorphism Invariance (even 𝑑)

• The breaking of diffeomorphism invariance exactly mirrors the breaking of conformal invariance by quantum effects in the CFT

• The partition function now depends on the representative of the conformal structure picked for regularization

• The observables (1’) such as the stress-tensor no longer transform covariantly, but pick up an anomalous term

• Anomalies are usually quantum effects, proportional to ℏ. Here, the anomaly is (inversely) proportional to Newton’s constant 𝐺

• The anomaly is robust: it is fully non-linear and it does not rely on classical approximations

• This anomaly does not lead to any inconsistencies because the metric is not dynamical in the CFT

Philosophical Questions

• Is one side of the duality more fundamental?• If QFT more fundamental, space-time could be ‘emergent’• If the duality is only approximate: room for emergence

(e.g. thermodynamics vs. atomic theory)

• If duality holds good: one-to-one relation between the values of physical quantities. In this case we have to give the duality a physical interpretation

Interpretation

•External view: meaning of observables is externally fixed. Duality relates different physical quantities• No empirical equivalence, numbers correspond to

different physical quantities• The symmetry of the terms related by duality is broken by

the different physical interpretation given to the symbols• Example: 𝑟 fixed by the interpretation to mean ‘radial

distance’ in the bulk theory. In the boundary theory, the corresponding symbol is fixed to mean ‘renormalization group scale’. The two symbols clearly describe different physical quantities. More generally, the two theories describe different physics hence are not empirically equivalent

• Only one of the two sides provides a correct interpretation of empirical reality

Interpretation

• Internal point of view: • The meaning of the symbols is not fixed beforehand• There is only one set of observables that is described by

the two theories. The two descriptions are equivalent. No devisable experiment could tell one from the other (each observation can be reinterpreted in the ‘dual’ variables)

• Cannot decide which description is superior. One formulation may be superior on practical grounds (e.g. computational simplicity in a particular regime)

• On this formulation we would normally say that we have two formulations of one theory, not two different theories

Interpretation

•The internal point of view seems more natural for theories of the whole world

•Even if one views a theory as a partial description of empirical reality, in so far as one takes it seriously in a particular domain of applicability, the internal view seems the more natural description.• Compare: position/momentum duality in QM. Equivalence of

frames in special relativity

Interpretation

•The internal point of view seems more natural for theories of the whole world

•Even if one views a theory as a partial description of empirical reality, in so far as one takes it seriously in a particular domain of applicability, the internal view seems the more natural description.• Compare: position/momentum duality in QM. Equivalence of

frames in special relativity.

• We should worry about the measurement problem, but it is not necessarily part of what is here meant by ‘theories of the whole world’, because the statement is still true in the classical limit, where we get Einstein gravity

•Butterfields’s puzzling scenario about truth (2014): Does reality admit two or more complete descriptions which• (Different): are not notational variants of each other; and yet• (Success): are equally and wholly successful by all epistemic

criteria one should impose?

•On the external view, the two theories are not equally successful because they describe different physical quantities: only one of them may describe this world•On the internal view, the two descriptions are equivalent

hence equally successful• If they turn out to be notational variants of each other (e.g.

different choices of gauge in a bigger theory) then the philosophical conclusion is less exciting, but new physics is to be expected. This is how dualities are often interpreted by physicists: as providing heuristic guidance for theory construction

• If the two theories are not notational variants of each other, then we do face the puzzling scenario!

•On the external view, the two theories describe different physics• The dual theory is only a tool that might be useful, but does

not describe the physics of our world• Here, the idea of ‘emergence’ does not suggest itself

because whichever side describes our world, it does not emerge from something else

•On the internal view there is a one-to-one relation between the values of physical quantities• Again emergence does not suggest itself: the two

descriptions are equivalent• If the duality is only approximate then there may be room

for emergence of space-time (analogy: thermodynamics vs. statistical mechanics)

Emergence

Analysing the process

• The holographic relation may well be a bijective map• There is no reason in this case to think that one side is

more fundamental than the other (left-right)• But the thermodynamic limit introduces the emergence

of gravity in an uncontroversial sense (top-bottom)

Does Gravity Emerge?

At which level does this require holography?

• The emergence of gravity only requires approximate holography• According to E. Verlinde, the microscopic bulk theory can be

dispensed with

Emergence of Space and Gravity

• Gravity could thus emerge in the same way (via coarse graining) in other situations where gauge/gravity duality does not hold exactly (e.g. cosmological scenarios: dS/CFT)

• But this idea can be applied more generally to AdS/CFT, where the renormalization group flow introduces coarse graining over high-energy degrees of freedom

• In this case, Einstein gravity may emerge from the fundamental bulk theory, whether the latter contains gravity or not

Conclusions

• In holographic scenarios with an exact duality, the microscopic surface theory is not necessarily more fundamental than the microscopic bulk theory• The bulk does not emerge from the boundary in such

cases

•However, the appearance of gravity in the thermodynamic limit makes it a clear case of emergence, connected with robustness and novelty of behavior

Thank you!