Emergence and RG in Gauge/Gravity Dualities

Sebastian de Haro University of Cambridge and University of Amsterdam

Effective Theories, Mixed Scale Modeling, and Emergence

Center for Philosophy of Science

University of Pittsburgh, 3 October 2015

Based on:

• de Haro, S. (2015), Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.08.004

• de Haro, S., Teh, N., Butterfield, J. (2015), Studies in History and Philosophy of Modern Physics, submitted

• Dieks, D. van Dongen, J., de Haro, S. (2015), Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.07.007

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, space-time and/or gravity ‘disappear/dissolve’ at high energies; and ‘emerge’ in a suitable semi-classical limit• Analysing these claims can: (i) clarify the meaning of

‘emergence of space-time/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?

2

Aim of this Talk

• To expound on the relation between emergence and duality

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

• The conceptual framework allows an assessment of the claims of emergence in gauge/gravity duality in the 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 space-time out of non-spatio-

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

3

Plan of the Talk

• An example: gauge/gravity dictionary

• Definition of duality

• Emergence vs. Duality• Two ways of emergence

• Back to the examples:• Holographic RG

• de Sitter generalisation

• Conclusion

4

Gauge/Gravity Dictionary

• (𝑑 + 1)-dim AdS

• GAdS

• d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖d𝑥𝑗

• Boundary at 𝑟 = 0

• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯+ 𝑟𝑑𝑔 𝑑 𝑥

• 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)

• 𝑇𝑖𝑗 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)

Example: AdS5 × 𝑆5 ≃ SU 𝑁 SYM

AdS5 × 𝑆5

• Type IIB string theory

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

• Example: massless scalar

SU 𝑁 SYM

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

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

• 𝒪 𝑥 = Tr 𝐹2 𝑥

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

• The claim is that these two theories are dual. Let us make this more precise

6

Maldacena (1997)

• 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)

7

Witten (1998)

Gubser Klebanov Polyakov (1998)

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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Comments

• I call the definition of duality ‘simple’ (even: ‘simplistic’) because a notion of duality that is applicable in some of the physically interesting examples may need a more general framework (e.g. a Hilbert space may be too restrictive for higher-dimensional QFTs)• In the case at hand, duality amounts to unitary equivalence. But this need

not be the case in more general cases

• 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 (however: see Schwarz 27 Sept 2015)

• But if one is willing to enter a mathematically non-rigorous (physics) discussion, then a good case can be made that:

(i) AdS/CFT can be cast in the language of states, quantities, and dynamics(ii) When this is done, the AdS/CFT correspondence indeed amounts to conjecturing a duality between two theories thus construed!

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Duality

• Duality is an isomorphism between two physical theories. Therefore it must satisfy the following, roughly:• Each side of the duality gives a complete and self-consistent theory that describes

the pertinent physical domain.• 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).

• I will spell this out in terms of three conditions:i. (Num) Numerically complete: the states and quantities are all relevant states

and quantities. E.g.: the theory is not missing any local operators.ii. (Consistent) The dynamical laws and quantities satisfy all the mathematical

and physical requirements expected from such theories in a particular domain. E.g.: a candidate theory of gravity should be background-independent.

iii. (Identical) The structures of the invariant physical quantities on either side are identical, i.e. the duality is exact. E.g.: if the theories are non-perturbative, they agree not only in perturbation theory, but also in the non-perturbative terms.

• These requirements are very stringent, but this is what one has to meet if one is to speak of ‘duality’• Duality as ‘isomorphism’ 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).

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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

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See also: Crowther (2015)

Duality vs. Emergence

• Incompatibility of duality and emergence:• Duality is a symmetric relation (isomorphism): if F is dual to G then G is dual

to F; and it is reflexive: F is dual to itself

• Emergence is asymmetric: if F emerges from G, then G cannot emerge from F; it is also non-reflexive: G cannot emerge from itself

• Therefore, emergence cannot be defined in terms of duality; in the absence of additional relations, duality precludes emergence

• If we violate or weaken one of the three conditions for duality, then there can be emergence

• The current definition of duality has two advantages:i. It is incompatible with emergent behaviour, hence giving a clear criterion

for when a theory will not be emergent from another (claims of emergence in the literature will have to specify an additional relation)

ii. It almost immediately indicates how emergent behaviour can occur: when there is only an approximate duality. The notion of coarse-graining will do this job

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Emergence

• It is in the violation or weakening of the duality conditions that there can be novelty and robustness (autonomy)

• The comparison class is provided by the duality itself:• Introducing coarse-graining to break duality 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 (or family of parameters) that can be either continuous or discrete

13

See also: Crowther (2015)

Two ways of emergence

• Recall the duality conditions (Num), (Consistent), (Identical). Any of the three can be weakened but only two of them lead to emergence:

• (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 or in a restricted domain. (Approx) produces families of theories related pairwise by duality, at each level of coarse-graining.

• Failure of (Num) does not 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.

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Two ways of emergence

𝑑′: 𝐺′ 𝐹′

𝑑: 𝐺 𝐹

𝐺′′ 𝐹′′

15

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 may emerge from a theory in which there are only

quantum mechanical degrees of freedom (cf. Verlinde’s (2011) gauge/gravity scheme: Newtonian gravity is regarded as an approximation: it breaks down at some level of coarse-graining, at which the world should be described by the quantum mechanical degrees of freedom.)

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

• In this talk I will concentrate on cases of (Approx) in which RG plays an important role:

• (Approx) would seem to be trivial: structures emerge on both sides but their emergence is independent of the presence of duality.• However, (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. coupling constants, boundary conditions), consider a series of approximations adjusted to various values of those parameters.

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

• Whatever emergence there is in G, is mirrored in F by the duality, even if it takes a completely different form. 16

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). • In other interesting examples (e.g. deformations of Maldacena’s original case)

there may only be an approximate duality: I will not consider those here.

• But even as the full theories are each other’s duals, emergence can take place according to the second way: by a weakening of (Approx) producing 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 (single-trace and multiple-trace) 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 double-trace 𝛽-function equation found in QFTs:

𝜖 𝜕𝜖𝑓 = −𝑓2 + 2𝜈𝑓• Two fixed points: UV fixed point 𝑓 = 0 (𝑏 → ∞) and IR fixed point 𝑓 = 2𝜈

(𝑏 small)

19

Faulkner Liu Rangamani (2010)

𝜈 =𝑑2

4+ 𝑚2

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 precisely matches the QFT result: which is non-perturbative!

• 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 RG flow, where gauge degrees of freedom are expected to disappear/emerge on an energy scale

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

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domain wall

Freedman et al. (1999)

Henningson Skenderis (1998)

𝑟 → ∞ 𝑟 → −∞

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.

21

Strominger 2001

Summary and conclusions

• Emergence cannot follow from duality alone (incompatibility)

• But emergence can take place when duality is broken by coarse-graining:• Two ways of emergence, according to 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 it is broken by coarse graining. A series of dualities is left between theories with reduced domains of applicability

• Gauge/gravity duality was discussed as a case of (Approx) emergence. The mechanism for emergence is the holographic renormalization group (and its dual RG flow in QFT):• Radial integration corresponds to integrating out energy degrees of freedom

• IR/UV connection: an IR gravity cut-off corresponds to UV cut-off in QFT

• 𝛽-function equations can be derived from the bulk

• Precise conformal anomaly matching (and c-function theorem from domain walls)

• Generalisations to de Sitter require more work: it’s a field in progress!

• Interesting to work out other cases 22

Thank you!

23

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

24

• 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

25

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)

26

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)

27