Emergence in Gauge/Gravity Dualities

Sebastian de Haro University of Cambridge and University of Amsterdam

Geneva, 14 January 2016

Based on:

• de Haro (2016). Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.08.004, to appear.

• de Haro, Teh, Butterfield (2016). Studies in History and Philosophy of Modern Physics, submitted.

• Dieks, Dongen, Haro (2015). Studies in History and Philosophy of Modern Physics, 52(B), pp. 203–216.

• de Haro, Mayerson, Butterfield (2016). Foundations of Physics, submitted.

Introduction

• In recent years, gauge/gravity dualities have been an important focus in quantum gravity research.•Gauge/gravity dualities relate a theory of gravity in (𝑑 + 1) dimensions to a quantum field theory (no gravity!) in 𝑑 dimensions.• Also called ‘holographic’.• Not just nice theoretical models: one of its versions

(AdS/CFT) successfully applied: RHIC experiment in Brookhaven (NY).

• It is often claimed that, in these models, spacetime and/or gravity ‘disappear/dissolve’ at high energies; and ‘emerge’ in a suitable semi-classical limit.• Analysing such claims can: (i) clarify the meaning of

‘emergence of spacetime/gravity’ (ii) provide insights into the conditions under which emergence can occur.

• It also prompts the more general question: how are dualities and emergence related?

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Aim of this Talk

• To distinguish two ways of emergence that arise when emergence is dependent on duality (as in the gauge/gravity literature).

• The focus will be on emergence of one spacelike direction, in gauge/gravity duality, and its relation to Wilsonian RG flow.• Thus, this is not emergence of the entire spacetime out of non-spatio-

temporal degrees of freedom. But it is an important first step!

3

Plan of the Talk

• The gauge/gravity dictionary

• Emergence vs. Duality• Two ways of emergence

• Back to the examples:• Holographic RG

• Diffeomorphism invariance

• Conclusion

4

Gauge/Gravity Dictionary

• (𝑑 + 1)-dim AdS

• Poincaré metrics:

• 𝑔 =ℓ2

𝑟2d𝑟2 + 𝑔 𝑟, 𝑥

• Boundary at 𝑟 = 0

• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯+ 𝑟𝑑𝑔 𝑑 𝑥

• Scalar field 𝜙 𝑟, 𝑥 , mass 𝑚• 𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + ⋯+ 𝑟𝑑𝜙 𝑑 𝑥

• Long-distance (IR) divergences

• Radial motion in 𝑟 (towards IR)

• CFT on ℝ𝑑

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

• 𝑇 𝑥 =ℓ𝑑−1

16𝜋𝐺𝑁𝑔 𝑑 𝑥 + ⋯

• Operator 𝒪 𝑥 with scaling dimension Δ 𝑚• 𝜙 0 𝑥 = coupling in action

• 𝒪 𝑥 = 𝜙 𝑑 𝑥

• High-energy (UV) divergences

• RG flow (towards UV)

5

Gravity (AdS) Gauge (CFT)

de Haro et al. (2001)

Maldacena (1997)

Witten (1998)

Gubser Klebanov Polyakov (1998)

• The limits are incompatible (it is a weak/strong coupling duality: which is why the duality is useful!)

• Only gauge invariant quantities can be compared

• The claim is that these two theories are dual• By ‘duality’, physicists mean the (unitary) equivalence between

quantum theories whose classical descriptions may be verydifferent. One theory lives in 5d, the other one in 4d.

Example: AdS5 × 𝑆5 ≅ SU 𝑁 SYM

AdS5 × 𝑆5

• String theory (type IIB)• Regard the 𝑆5 as internal

• Limit of small curvature:Einstein’s theory with specific matter fields

• Example: massless scalar

SU 𝑁 SYM

• 4d Yang-Mills theory: supersymmetric, with gauge group SU(𝑁)

• Limit of strong coupling(’t Hooft limit)

• 𝒪 𝑥 = Tr 𝐹2 𝑥

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Maldacena (1997)

Comments

• At present, no one knows how to rigorously define the theories involved in gauge/gravity dualities (except for lower-dimensional cases): not just the string theories, but also the conformal field theories involved.

• The evidence for the duality is perturbative (some non-perturbative results are available).

• Full establishment of duality within the current techniques/framework of string theory is unlikely. New formulations of these theories (M-theory?) are likely to be required.

• But we do not need to settle on these issues in order to discuss emergence. For we will be concerned with the low-energy theories.

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Emergence

• I endorse Butterfield’s (2011) notion of emergence as “properties or behaviour of a system which are novel and robust relative to some appropriate comparison class”• I will distinguish emergence of one theory from another, and then discuss

emergence of properties or behaviour.

8

See also: Crowther (2015)

Duality

• Duality is an isomorphism between two physical theories (for quantum theories: unitary equivalence). Therefore, it must satisfy the following, roughly:• Each side of the duality gives a complete, self-consistent theory of a given physical

domain (no requirement that a theory should be a final theory of any sort!).• But the two theories also agree with each other, i.e. they give identical results for their

physical quantities (in their proper domains of applicability).

• Thus, three conditions need to be satisfied:i. (Num) Numerically complete: the states and quantities are all the relevant states and

quantities in the given domain. E.g.: the theory is not missing any operators that are needed to describe the relevant physics.

ii. (Consistent) The dynamical laws and quantities satisfy all the mathematical and physical requirements expected from such theories in the given domain. Examples: (1) A candidate theory of gravity should be background-independent. (2) A CFT should be classically scale invariant.

iii. (Identical) The structures of the invariant physical quantities on either side are identical, i.e. the duality is exact. E.g.: if the given domain includes non-perturbative physics, the two theories should also agree in their non-perturbative terms.

• These requirements are strong, but this is what is needed if one is to speak of ‘duality’ as an isomorphism.• Duality in this sense is sometimes called the ‘strong version’ of the gauge/gravity

correspondence: and it is the one advocated by Maldacena (1997). Also in standard accounts: e.g. Polchinski (1998), Aharony et al. (1999), Ammon et al. (2015).

• Desirable as it would be to have a proof of duality for AdS/CFT: it is not needed in order to discuss emergence. 9

Duality vs. Emergence

• The very notion of duality suggests how emergent behaviour can arise:

(a) There is the emergent behaviour that can take place independently on each side of the duality, in particular via coarse-graining.

• The duality should then map one emergent behaviour to another.

(b) But emergence can also arise when the duality itself is only approximate, e.g. if the map fails to be a bijection at some order in perturbation theory. Then one theory can emerge from the other, when appropriate approximations and redefinitions are made.

• We can regard both cases as a weakening, or even a violation, of one of the conditions stated for duality, as follows.

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Two ways of emergence

• Recall the duality conditions (Num), (Consistent), (Identical). Any of the three can be weakened but I will only consider two of them:

• (BrokenMap): the duality map (Identical) breaks down at some level of fine-graining: it fails to be a bijection. (So there is no exact duality to start with).• E.g.: the map only holds up to some order in perturbation theory, and breaks down after

that, and so there is no duality of fine-grained theories.

• If F(fundamental) is the fine-grained theory and G(gravity) its approximate dual, then there may well be behaviour and physical quantities described by G that emerge, by perturbative duality, from F.

• (Approx): an approximation scheme is applied on each side of the duality. The approximated theories only describe the relevant physics approximately. Thus (Consistent) only holds approximately, perhaps in a restricted domain. (Approx) produces families of theories related pairwise by duality, at each level of coarse-graining.

• Failure of (Num) does not seem to give an independent third way of emergence; in this case, a subset of the quantities agree, but the numbers of quantities differ. • Taking a subset out of all the quantities, there is only a notion of belonging to that set or

not; but no notion of a successive approximation such that there can be robustness: there is no coarse graining. 11

Emergence

• So I argue that in the weakening, or even the violation, of two of the duality conditions, there can be novelty and robustness (autonomy).

• The comparison class, needed for emergent behaviour (i.e. answering the question: novel and robust with respect to what?), is provided by the duality itself:• Introducing coarse-graining gives us a measure for how robust the novel

behaviour is: since coarse-graining can be done in different steps, which can be compared to the ‘exact’ case.

• To allow for this quantitative comparison, coarse-graining is measured by a parameter that can be either continuous or discrete.

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See: Butterfield (2011)

Crowther (2015)

Two ways of emergence

𝑑′: 𝐺′ 𝐹′

𝑑: 𝐺 𝐹

𝐺′′ 𝐹′′

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𝑑′′

(BrokenMap)

(Approx)

Comparing the two ways of emergence

• (BrokenMap) is a clear case of emergence of one theory from another.• For instance, Newtonian gravity in 3d may emerge from a theory in which there are

only quantum mechanical degrees of freedom fitting a 2d surface (Verlinde (2011): Newtonian gravity is regarded as an approximation: it breaks down at some level of coarse-graining, at which the world should be described by quantum mechanical degrees of freedom.)

• The duality provides the relevant class to which novelty and robustness (autonomy) are compared: the class is the set of theories to which this approximate duality applies.

• (Approx) would seem to be trivial: structures emerge on both sides but their emergence is independent of the presence of duality.• (Approx) gives an interesting way of producing emergent properties or behaviour,

once a duality is given that depends on external parameters:• For dualities with external parameters (e.g. energy scales, boundary conditions),

consider a series of approximations, adjusted to specific values of those parameters. In many cases, novel structures arise as we carry out the approximations. Examples:

(1) Yang-Mills theories do not contain any strings, but, in the ’t Hooft limit, string-like structures emerge.

(2) The underlying string theory might not be a theory of gravity, but graviton-like excitations appear at low energies.

• The original duality may be replaced by a series of duals, each of them valid at the corresponding level of coarse-graining.

• If the duality maps the parameters: whatever emergence there is in G, is mirrored in F by the duality, even if it takes a completely different form.

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Dieks Dongen Haro (2015)

Emergence in gauge/gravity duality

• If gauge/gravity duality is an exact duality (as it is conjectured to be for Maldacena’s AdS/CFT correspondence), then there is no (BrokenMap). • But (BrokenMap) can be obtained by considering deformations of Maldacena’s

original case.

• Even in cases of exact duality, emergence can take place according to the second way: by a weakening of (Approx) that generates a series of duals.

• The full string theory is approximated (asymptotically) by a semi-classical supergravity theory:• The approximation is parametrised by the radial distance, which corresponds

to the energy scale in the boundary theory.

• The radial flow in the bulk geometry can be interpreted as the renormalization group flow of the boundary theory.

• Wilsonian renormalization group methods can be used. The gravity version of this is called the ‘holographic renormalization group’.

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Holographic Renormalization Group

• Radial integration: integrate out the (semi-classical) asymptotic geometry between two cut-offs 𝜖, 𝜖′

• Wilsonian renormalization: integrate out degrees of freedom between two cut-offs Λ, 𝑏Λ (𝑏 < 1)

Λ𝑏Λ0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT

AdS𝜖′

𝜕AdS𝜖′ 𝜕AdS𝜖

new boundary condition

integrate out

cut-off surface

Holographic Renormalization Group

• Integrating out the bulk degrees of freedom between 𝜖, 𝜖′ results in a boundary action 𝑆bdy 𝜖′ which provides boundary conditions for the bulk modes

• This effective action can be identified with the Wilsonian effective action of the boundary theory at scale 𝑏Λ , with the boundary conditions in 𝑆bdy 𝜖′ identified with the couplings for operators in the boundary theory

• Requiring that physical quantities be independent of the choice of cut-off scale 𝜖′ determines a flow equation for the Wilsonian action and the couplings

• Example: for a scalar field theory with mass 𝑚 in the bulk, the boundary coupling is found to obey the 𝛽-function equation found in QFT:

𝜖 𝜕𝜖𝑓 = −𝑓2 + 2𝜈𝑓

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Faulkner Liu Rangamani (2010)

Balasubramanian Kraus (1999)

de Boer Verlinde Verlinde (1999)

Holographic RG: the Conformal Anomaly

• CFTs in even dimensions are anomalous. This anomaly takes a universal form and can be reproduced from the bulk (in the field theory’s UV; take 𝑑 = 4):

𝑇𝑖𝑖𝑟=∞

=𝑁2

32𝜋2𝑅𝑖𝑗𝑅𝑖𝑗 −

1

3𝑅2

• 𝑁=number of gauge degrees of freedom (rank of gauge group).

• The classical gravity calculation of the anomaly matches the exact QFT result.

• For more general ‘domain wall’ solutions:

𝑇𝑖𝑖𝑟= 𝐶 𝑟 𝑅𝑖𝑗𝑅𝑖𝑗 −

1

3𝑅2

• 𝐶 𝑟 is monotonically decreasing when moving to the IR at 𝑟 → −∞. At both infinities, it approaches a (different) constant: the AdS radius.

• This mirrors the QFT renormalization group flow, where gauge degrees of freedom are expected to disappear/emerge on an energy scale.

• The coarse-graining is introduced by the holographic renormalization group. Two AdS regions disappear/emerge along the radial direction.

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domain wall

Freedman et al. (1999)

Henningson Skenderis (1998)

𝑟 → ∞ 𝑟 → −∞

Emergence of Diffeomorphism Invariance

• The previous analysis of emergence focused on the quantities that are related by duality. These are the physical quantities.

• But could emergence also concern the parts of the theory that are not related by duality?

• The diffeomorphism invariance of the gravity theory is one such phenomenon. There are diffeomorphisms that relate points in spacetime, but which are not seen by the physical quantities.• In so far as the existence of diffeomorphisms is a novel phenomenon that

appears in the low-energy limit (Einstein’s theory), with relative independence from the microscopic details of the theory, we can say that diffeomorphism invariance emerges.

• Horowitz and Polchinski (2006) introduce the notion of ‘invisibility’: “the gauge variables of AdS/CFT are trivially invariant under the bulk diffeomorphisms, which are entirely invisible in the gauge theory.”

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Which Diffeomorphisms?

• Not all diffeomorphisms are invisible, and not all of them emerge. For there is an important class of diffeomorphisms that acts on the physical quantities and is expected to act in the microscopic theory. These diffeomorphisms are visible, and they are ‘seen’ by the boundary theory.

• Roughly, they correspond to conformal transformations.

• So, let us see how to distinguish the diffeomorphisms that are invisible, from the ones that are visible.

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Defining the class of metrics

• A Poincaré metric 𝑔 on 𝑀 is, roughly speaking, a metric of signature (𝑝 + 1, 𝑞), such that:

(1) 𝑔 has conformal infinity 𝑀, 𝑔 : a conformal manifold 𝑀 of signature 𝑝, 𝑞 , and 𝑔 a representative of the conformal class.

(2) It satisfies Einstein’s equations in vacuum.

• Theorem (Fefferman and Graham (1985, 2012)): let 𝑔 be a Poincaré metric on 𝑀. Then there exists an open neighbourhood 𝑈 near the boundary of 𝑀 on which there is a unique diffeomorphism 𝜙:𝑈 → 𝑀such that 𝜙 boundary is the identity map, and 𝜙∗ 𝑔 is in normal form:

𝜙∗ 𝑔 =1

𝑟2d𝑟2 + 𝑔 𝑥, 𝑟

• In other words, when working with Poincaré metrics, we may, without loss of generality, consider those that are in normal form.

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Visible and Invisible Diffeomorphisms

• A diffeomorphism 𝜙:𝑈 → 𝑈 (𝑈 open in 𝑀) is said to be invisible relative to 𝑔,𝑀, 𝑔 , if it satisfies:

(i: invisible relative to 𝑔): 𝜙∗ 𝑔 is in normal form.

(ii: invisible relative to 𝑀): 𝜙 boundary = idboundary .

(iii: invisible relative to 𝑔): 𝜙 is an isometry of 𝑀 (the boundary manifold).

• Proposition. If a diffeomorphism 𝜙 is invisible relative to 𝑔, 𝑔 (i.e., it satisfies (i,iii)), then 𝜙boundary is a conformal transformation on 𝑀.

• Theorem. If a diffeomorphism 𝜙 is invisible relative to ( 𝑔,𝑀, 𝑔) (i.e., it satisfies (i)-(iii), then 𝜙 is the identity.

• Thus: there are no invisible diffeomorphisms relative to ( 𝑔,𝑀, 𝑔). The only diffeomorphisms that preserve the normal form, as well as 𝑀, are the conformal transformations: which are visible.

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Invisible Diffeomorphisms

• The condition that diffeomorphisms are not ‘seen’ by the duality amounts to the requirement that all the physical quantities (in the QFT: correlation functions) must be unaffected by them.

• Even though, given a Poincaré metric, there are no diffeomorphisms that are invisible relative to ( 𝑔,𝑀, 𝑔), there are diffeomorphisms that are invisible relative to the boundary quantities 𝑀,𝑔 .

• These are then good candidates for diffeomorphisms that are invisible relative to the duality: to prove their invisibility, one has to prove that all physical quantities are unaffected.

• The existence of such diffeomorphisms, and the invariance of the theory under them, is an emergent phenomenon of the gravity theory: they emerge together with the spacetime.

• This emergence is similar in kind to (Approx), in that it is applied on one of the two sides of the duality. But it does not need to be mirrored by a dual phenomenon of emergence in the QFT, because the physical quantities are not affected.

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Summary and conclusions

• Emergence can take place when duality is broken by coarse-graining:• Two ways of emergence, depending on which duality condition is violated or

weakened: (BrokenMap) vs. (Approx).

• In (BrokenMap), there is no exact duality to start with. But the presence of an approximate duality provides a natural comparison class, needed for emergence.

• In (Approx), there is a duality, but coarse graining replaces it by a series of dualities between approximated theories with reduced domains of applicability.

• It would be interesting to apply this schema to other dualities.

• Gauge/gravity duality was discussed as a case of (Approx) emergence.

• The mechanism for emergence discussed is the holographic renormalization group (and its dual RG flow in QFT). • Classical features of the spacetime appear after integrating out short-distance

degrees of freedom.

• An interesting class of diffeomorphisms can emerge.

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Thank you!

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• The basic physical quantities on both sides:

• Other physical quantities are calculated by differentiation:Π𝜙 𝑥 Π𝜙 𝑦 ≡ 𝒪Δ 𝑥 𝒪Δ 𝑦

• For instance: in the supergravity limit, the solution of the Klein-Gordon equation in the bulk with given boundary condition 𝜙 0 is:

𝜙 𝑟, 𝑥 = d𝑑𝑥𝑟Δ

𝑟2 + 𝑥 − 𝑦 2 Δ𝜙 0 (𝑦)

⇒ Π𝜙 𝑥 Π𝜙 𝑦 =1

𝑥 − 𝑦 2Δ

• This is precisely the two-point function of 𝒪Δ in a CFT

𝑍string 𝜙 0 : = 𝜙 0,𝑥 =𝜙 0 𝑥

𝒟𝜙 𝑒−𝑆bulk 𝜙 ≡ exp d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

=:𝑍CFT 𝜙 0

Gauge/Gravity Dictionary (Continued)

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Witten (1998)

Gubser Klebanov Polyakov (1998)

Gauge/Gravity Duality: Gravity Side

• AdS is the maximally symmetric space-time with constant negative curvature

• Useful choice of local ‘Poincaré’ coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝜂𝑖𝑗 d𝑥

𝑖d𝑥𝑗 , 𝑖 = 1, … , 𝑑

• 𝜂𝑖𝑗 = flat metric (Lorentzian or Euclidean signature)

• We will need less symmetric cases: generalized AdS (‘GAdS’)

• Fefferman and Graham (1985): for a space that satisfies Einstein's equations with a negative cosmological constant, and given a conformal metric at infinity, the line element can be written as:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 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

27

• This metric includes pure AdS, but also: AdS black holes (any solution with zero stress-energy tensor and negative cosmological constant). AdS/CFT is not restricted to the most symmetric case! Hence the name ‘gauge/gravity’

• So far we considered Einstein’s equations in vacuum. The above generalizes to the case of gravity coupled to matter. E.g.:

• Scalar field 𝜙 𝑟, 𝑥 : solve its equation of motion (Klein-Gordon 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 (as well as the metric coefficients)

Adding Matter

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The Gravity Side (cont’d)

Duality (more refined version)

• For the theories of interest, we will need some more structure

• Add external parameters 𝒞 (e.g. couplings, sources)

• The theory is given as a quadruple ℋ,𝒬, 𝒞, 𝐷

• Duality is an isomorphism ℋ𝐴, 𝒬𝐴, 𝒞𝐴 ≃ ℋ𝐵, 𝒬𝐵, 𝒞𝐵 . There are three bijections: • 𝑑ℋ:ℋ𝐴 → ℋ𝐵

• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵• 𝑑𝒞: 𝒞𝐴 → 𝒞𝐵

such that:

𝑂, 𝑠 𝑐 ,𝐷𝐴 = 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵 ∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴, 𝑐 ∈ 𝒞𝐴

• Need to preserve also triples 𝒪; 𝑠1, 𝑠2 𝑐 ,𝐷𝐴

𝒪, 𝑠 𝑐 ,𝐷𝐴 = 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠{𝑑𝒞(𝑐)} ,𝐷𝐵

(1)

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AdS/CFT Duality

• AdS/CFT can be described in terms of the quadruple ℋ,𝒬, 𝒞, 𝐷 : • Normalizable modes correspond to exp. vals. of operators (choice of state)

• Fields correspond to operators

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

• Formulation otherwise different (off-shell Lagrangian, different dimensions!)

• Two salient points of :• Physical quantities, such as boundary conditions, that are not determined by

the dynamics, now also agree: they correspond to couplings in the CFT

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

• It is also more general: while ℋ,𝒬, 𝐷 are a priori fixed, 𝒞 can be varied at will (Katherine Brading: ‘modal equivalence’). We have a multidimensional space of theories

• Dualities of this type are not isomorphisms between two given theories (in the traditional sense) but between two sets of theories

ℋ

𝒬

𝒞

𝐷

(1)

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Duality: a simple definition

• Regard a theory as a triple ℋ,𝒬, 𝐷 : states, physical quantities, dynamics • ℋ = states: in the cases I consider: a Hilbert space• 𝒬 = physical quantities: a specific set of operators: self-adjoint,

renormalizable, invariant under symmetries• 𝐷 = dynamics: a choice of Hamiltonian, alternately a Lagrangian

• A duality is an isomorphism between two theories ℋ𝐴, 𝒬𝐴, 𝐷𝐴 and ℋ𝐵, 𝒬𝐵, 𝐷𝐵 , as follows:

• There exist structure-preserving bijections: • 𝑑ℋ:ℋ𝐴 → ℋ𝐵 ,

• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵

and pairings (expectation values) 𝒪, 𝑠 𝐴 such that:𝒪, 𝑠 𝐴 = 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠

𝐵∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴

as well as triples 𝒪; 𝑠1, 𝑠2 𝐴 and 𝑑ℋ commutes with (is equivariantfor) the two theories’ dynamics

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Generalisations to de Sitter Spacetime

• Gauge/gravity duality has been conjectured to hold also for de Sitter spacetime. The conjectured duality goes under the name of ‘dS/CFT’.

• The status of dS/CFT is much less clear than that of AdS/CFT. Nevertheless there has been much progress in the past 5 years, and there is now a concrete proposal for the CFT dual of the ‘Vasiliev higher-spin theory’ in the bulk.

• The previous calculation generalises to dS: the radial variable 𝑟 is replaced by the time variable 𝑡. For a metric of Friedmann-Lemaitre-Robertson-Walker form (for simplicity: 𝑘 = 0): d𝑠2 = −d𝑡2 + 𝑎 𝑡 2 d 𝑥2, 𝑎 𝑡 has two different limits at early and late times (two Hubble parameters):

𝑎 −∞

𝑎 −∞= 𝐻init,

𝑎 ∞

𝑎 ∞= 𝐻fin

• At intermediate times, 𝑎 𝑡 satisfies the Friedmann equation

• Again, there is a c-theorem where 𝐻 𝑡 decreases with time

• If dS/CFT exists, bulk time evolution is dual to RG flow. The flow begins at a UV fixed point and ends at an IR fixed point.

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Strominger 2001