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:
โข de Haro, S. (2015), Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.08.004
โข de Haro, S., Teh, N., Butterfield, J. (2015), Studies in History and Philosophy of Modern Physics, submitted
โข 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
โข In recent years, gauge/gravity dualities have been an important focus in quantum gravity researchโขGauge/gravity dualities relate a theory of gravity in (๐ + 1) dimensions to a quantum field theory (no gravity!) in ๐ dimensionsโข Also called โholographicโโข Not just nice theoretical models: one of its versions
(AdS/CFT) successfully applied: RHIC experiment in Brookhaven (NY)
โข It is often claimed that, in these models, space-time and/or gravity โdisappear/dissolveโ at high energies; and โemergeโ in a suitable semi-classical limitโข Analysing these claims can: (i) clarify the meaning of
โemergence of space-time/gravityโ (ii) provide insights into the conditions under which emergence can occur
โข It also prompts the more general question: how are dualities and emergence related?
2
Aim of this Talk
โข To expound on the relation between emergence and duality
โข Distinguish two ways of emergence that arise when emergence is dependent on duality (as in the gauge/gravity literature)
โข The conceptual framework allows an assessment of the claims of emergence in gauge/gravity duality in the literature
โข The focus will be on emergence of one spacelike direction in gauge/gravity duality and its relation to Wilsonian RG flowโข 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!
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Plan of the Talk
โข An example: gauge/gravity dictionary
โข Definition of duality
โข Emergence vs. Dualityโข Two ways of emergence
โข Back to the examples:โข Holographic RG
โข de Sitter generalisation
โข Conclusion
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Gauge/Gravity Dictionary
โข (๐ + 1)-dim AdS
โข GAdS
โข d๐ 2 =โ2
๐2d๐2 + ๐๐๐ ๐, ๐ฅ d๐ฅ๐d๐ฅ๐
โข Boundary at ๐ = 0
โข ๐ ๐, ๐ฅ = ๐ 0 ๐ฅ + โฏ+ ๐๐๐ ๐ ๐ฅ
โข Field ๐ ๐, ๐ฅ , mass ๐โข ๐ ๐, ๐ฅ = ๐ 0 ๐ฅ + โฏ+ ๐๐๐ ๐ ๐ฅ
โข Long-distance (IR) divergences
โข Radial motion in ๐ (towards IR)
โข CFT on โ๐
โข QFT with a fixed pointโข Metric ๐ 0 (๐ฅ)
โข ๐๐๐ ๐ฅ =โ๐โ1
16๐๐บ๐๐ ๐ ๐ฅ + โฏ
โข Operator ๐ช ๐ฅ with scaling dimension ฮ ๐โข ๐ 0 ๐ฅ = coupling in action
โข ๐ช ๐ฅ = ๐ ๐ ๐ฅ
โข High-energy (UV) divergences
โข RG flow (towards UV)
5
Gravity (AdS) Gauge (CFT)
de Haro et al. (2001)
Maldacena (1997)
Witten (1998)
Gubser Klebanov Polyakov (1998)
Example: AdS5 ร ๐5 โ SU ๐ SYM
AdS5 ร ๐5
โข Type IIB string theory
โข Limit of small curvature:supergravity (Einsteinโs theory + specific matter fields)
โข Example: massless scalar
SU ๐ SYM
โข Supersymmetric, 4d Yang-Mills theory with gauge group SU(๐)
โข Limit of strong coupling: โt Hooft limit (planar diagrams)
โข ๐ช ๐ฅ = Tr ๐น2 ๐ฅ
โข Limits are incompatible (weak/strong coupling duality: useful!)โข Only gauge invariant quantities (operators) can be compared
โข The claim is that these two theories are dual. Let us make this more precise
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Maldacena (1997)
โข The basic physical quantities on both sides:
โข Other physical quantities are calculated by differentiation:ฮ ๐ ๐ฅ ฮ ๐ ๐ฆ โก ๐ชฮ ๐ฅ ๐ชฮ ๐ฆ
โข For instance: in the supergravity limit, the solution of the Klein-Gordon equation in the bulk with given boundary condition ๐ 0 is:
๐ ๐, ๐ฅ = d๐๐ฅ๐ฮ
๐2 + ๐ฅ โ ๐ฆ 2 ฮ๐ 0 (๐ฆ)
โ ฮ ๐ ๐ฅ ฮ ๐ ๐ฆ =1
๐ฅ โ ๐ฆ 2ฮ
โข This is precisely the two-point function of ๐ชฮ in a CFT
๐string ๐ 0 : = ๐ 0,๐ฅ =๐ 0 ๐ฅ
๐๐ ๐โ๐bulk ๐ โก exp d๐๐ฅ ๐ 0 ๐ฅ ๐ช ๐ฅ
CFT
=:๐CFT ๐ 0
Gauge/Gravity Dictionary (Continued)
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Witten (1998)
Gubser Klebanov Polyakov (1998)
Duality: a simple definition
โข Regard a theory as a triple โ,๐ฌ, ๐ท : states, physical quantities, dynamics โข โ = states: in the cases I consider: a Hilbert spaceโข ๐ฌ = physical quantities: a specific set of operators: self-adjoint,
renormalizable, invariant under symmetriesโข ๐ท = dynamics: a choice of Hamiltonian, alternately a Lagrangian
โข A duality is an isomorphism between two theories โ๐ด, ๐ฌ๐ด, ๐ท๐ด and โ๐ต, ๐ฌ๐ต, ๐ท๐ต , as follows:
โข There exist structure-preserving bijections: โข ๐โ:โ๐ด โ โ๐ต ,
โข ๐๐ฌ: ๐ฌ๐ด โ ๐ฌ๐ต
and pairings (expectation values) ๐ช, ๐ ๐ด such that:๐ช, ๐ ๐ด = ๐๐ฌ ๐ช , ๐โ ๐
๐ตโ๐ช โ ๐ฌ๐ด, ๐ โ โ๐ด
as well as triples ๐ช; ๐ 1, ๐ 2 ๐ด and ๐โ commutes with (is equivariantfor) the two theoriesโ dynamics
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Comments
โข I call the definition of duality โsimpleโ (even: โsimplisticโ) 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)โข In the case at hand, duality amounts to unitary equivalence. But this need
not be the case in more general cases
โข 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)
โข 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
โข Duality is an isomorphism between two physical theories. Therefore it must satisfy the following, roughly:โข Each side of the duality gives a complete and self-consistent theory that describes
the pertinent physical domain.โข 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).
โข 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.
โข These requirements are very stringent, but this is what one has to meet if one is to speak of โdualityโโข Duality as โisomorphismโ is sometimes called the โstrong versionโ 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
โข I endorse Butterfieldโs (2011) notion of emergence as โproperties or behaviour of a system which are novel and robust relative to some appropriate comparison classโโข 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
โข Incompatibility of duality and emergence:โข 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
โข Emergence is asymmetric: if F emerges from G, then G cannot emerge from F; it is also non-reflexive: G cannot emerge from itself
โข Therefore, emergence cannot be defined in terms of duality; in the absence of additional relations, duality precludes emergence
โข If we violate or weaken one of the three conditions for duality, then there can be emergence
โข 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
12
Emergence
โข It is in the violation or weakening of the duality conditions that there can be novelty and robustness (autonomy)
โข The comparison class is provided by the duality itself:โข 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 โexactโ case
โข To allow for this quantitative comparison, coarse-graining is measured by a parameter (or family of parameters) that can be either continuous or discrete
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See also: Crowther (2015)
Two ways of emergence
โข Recall the duality conditions (Num), (Consistent), (Identical). Any of the three can be weakened but only two of them lead to emergence:
โข (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).โข 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.
โข 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.
โข (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.
โข 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. โข 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
๐โฒ: ๐บโฒ ๐นโฒ
๐: ๐บ ๐น
๐บโฒโฒ ๐นโฒโฒ
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๐โฒโฒ
(BrokenMap)
(Approx)
Comparing the two ways of emergence
โข (BrokenMap) is a clear case of emergence of one theory from another.โข For instance, Newtonian gravity may emerge from a theory in which there are only
quantum mechanical degrees of freedom (cf. Verlindeโs (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.)
โข 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.
โข In this talk I will concentrate on cases of (Approx) in which RG plays an important role:
โข (Approx) would seem to be trivial: structures emerge on both sides but their emergence is independent of the presence of duality.โข However, (Approx) gives an interesting way of producing emergent properties or
behaviour, once a duality is given that depends on external parameters:
โข For dualities with external parameters (e.g. coupling constants, boundary conditions), consider a series of approximations adjusted to various values of those parameters.
โข The original duality may be replaced by a series of duals, each of them valid at the corresponding level of coarse-graining.
โข 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
โข If gauge/gravity duality is an exact duality (as it is conjectured to be for Maldacenaโs AdS/CFT correspondence), then there is no (BrokenMap). โข In other interesting examples (e.g. deformations of Maldacenaโs original case)
there may only be an approximate duality: I will not consider those here.
โข But even as the full theories are each otherโs duals, emergence can take place according to the second way: by a weakening of (Approx) producing a series of duals.
โข The full string theory is approximated (asymptotically) by a semi-classical supergravity theory:โข The approximation is parametrised by the radial distance, which corresponds
to the energy scale in the boundary theory.
โข The radial flow in the bulk geometry can be interpreted as the renormalization group flow of the boundary theory.
โข Wilsonian renormalization group methods can be used. The gravity version of this is called the โholographic renormalization groupโ.
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Holographic Renormalization Group
โข Radial integration: integrate out the (semi-classical) asymptotic geometry between two cut-offs ๐, ๐โฒ
โข Wilsonian renormalization: integrate out degrees of freedom between two cut-offs ฮ, ๐ฮ (๐ < 1)
ฮ๐ฮ0
๐
integrate out
New cutoff ๐ฮ
rescale ๐ฮ โ ฮ until ๐ โ 0
IR cutoff ๐ in AdS โ UV cutoff ฮ in QFT
AdS๐โฒ
๐AdS๐โฒ ๐AdS๐
new boundary condition
integrate out
cut-off surface
Holographic Renormalization Group
โข Integrating out the bulk degrees of freedom between ๐, ๐โฒ results in a boundary action ๐bdy ๐โฒ which provides boundary conditions for the bulk modes
โข This effective action can be identified with the Wilsonian effective action of the boundary theory at scale ๐ฮ , with the boundary conditions in ๐bdy ๐โฒ identified with the couplings for (single-trace and multiple-trace) operators in the boundary theory
โข Requiring that physical quantities be independent of the choice of cut-off scale ๐โฒ determines a flow equation for the Wilsonian action and the couplings
โข Example: for a scalar field theory with mass ๐ in the bulk, the boundary coupling is found to obey the double-trace ๐ฝ-function equation found in QFTs:
๐ ๐๐๐ = โ๐2 + 2๐๐โข Two fixed points: UV fixed point ๐ = 0 (๐ โ โ) and IR fixed point ๐ = 2๐
(๐ small)
19
Faulkner Liu Rangamani (2010)
๐ =๐2
4+ ๐2
Balasubramanian Kraus (1999)
de Boer Verlinde Verlinde (1999)
Holographic RG: the Conformal Anomaly
โข CFTs in even dimensions are anomalous. This anomaly takes a universal form and can be reproduced from the bulk (in the field theoryโs UV; take ๐ = 4):
๐๐๐๐=โ
=๐2
32๐2๐ ๐๐๐ ๐๐ โ
1
3๐ 2
โข ๐=number of gauge degrees of freedom (rank of gauge group)
โข The classical gravity calculation of the anomaly precisely matches the QFT result: which is non-perturbative!
โข For more general โdomain wallโ solutions:
๐๐๐๐= ๐ถ ๐ ๐ ๐๐๐ ๐๐ โ
1
3๐ 2
โข ๐ถ ๐ is monotonically decreasing when moving to the IR at ๐ โ โโ. At both infinities, it approaches a (different) constant: the AdS radius
โข This mirrors the QFT RG flow, where gauge degrees of freedom are expected to disappear/emerge on an energy scale
โข 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)
๐ โ โ ๐ โ โโ
Generalisations to de Sitter Spacetime
โข Gauge/gravity duality has been conjectured to hold also for de Sitter spacetime. The conjectured duality goes under the name of โdS/CFTโ.
โข 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 โVasiliev higher-spin theoryโ in the bulk.
โข The previous calculation generalises to dS: the radial variable ๐ is replaced by the time variable ๐ก. For a metric of Friedmann-Lemaitre-Robertson-Walker form (for simplicity: ๐ = 0): d๐ 2 = โd๐ก2 + ๐ ๐ก 2 d ๐ฅ2, ๐ ๐ก has two different limits at early and late times (two Hubble parameters):
๐ โโ
๐ โโ= ๐ปinit,
๐ โ
๐ โ= ๐ปfin
โข At intermediate times, ๐ ๐ก satisfies the Friedmann equation
โข Again, there is a c-theorem where ๐ป ๐ก decreases with time
โข 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
โข Emergence cannot follow from duality alone (incompatibility)
โข But emergence can take place when duality is broken by coarse-graining:โข Two ways of emergence, according to which duality condition is violated or
weakened: (BrokenMap) vs. (Approx)
โข 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
โข 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
โข 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):โข Radial integration corresponds to integrating out energy degrees of freedom
โข IR/UV connection: an IR gravity cut-off corresponds to UV cut-off in QFT
โข ๐ฝ-function equations can be derived from the bulk
โข Precise conformal anomaly matching (and c-function theorem from domain walls)
โข Generalisations to de Sitter require more work: itโs a field in progress!
โข Interesting to work out other cases 22
Thank you!
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Gauge/Gravity Duality: Gravity Side
โข AdS is the maximally symmetric space-time with constant negative curvature
โข Useful choice of local โPoincarรฉโ coordinates:
d๐ 2 =โ2
๐2d๐2 + ๐๐๐ d๐ฅ
๐d๐ฅ๐ , ๐ = 1, โฆ , ๐
โข ๐๐๐ = flat metric (Lorentzian or Euclidean signature)
โข We will need less symmetric cases: generalized AdS (โGAdSโ)
โข 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:
d๐ 2 =โ2
๐2d๐2 + ๐๐๐ ๐, ๐ฅ d๐ฅ๐d๐ฅ๐
๐๐๐ ๐, ๐ฅ = ๐ 0 ๐๐ ๐ฅ + ๐ ๐ 1 ๐๐ ๐ฅ + ๐2๐ 2 ๐๐ ๐ฅ + โฏ
โข Einsteinโs equations now reduce to algebraic relations between:
๐ ๐ ๐ฅ ๐ โ 0, ๐ and ๐ 0 ๐ฅ , ๐ ๐ ๐ฅ
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โข 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 โgauge/gravityโ
โข So far we considered Einsteinโs equations in vacuum. The above generalizes to the case of gravity coupled to matter. E.g.:
โข Scalar field ๐ ๐, ๐ฅ : solve its equation of motion (Klein-Gordon equation) coupled to gravity:
๐ ๐, ๐ฅ = ๐ 0 ๐ฅ + ๐ ๐ 1 ๐ฅ +โฏ+ ๐๐๐ ๐ ๐ฅ +โฏ
โข Again, ๐ 0 ๐ฅ and ๐ ๐ ๐ฅ are the boundary conditions and all other coefficients ๐ ๐ ๐ฅ are given in terms of them (as well as the metric coefficients)
Adding Matter
25
The Gravity Side (contโd)
Duality (more refined version)
โข For the theories of interest, we will need some more structure
โข Add external parameters ๐ (e.g. couplings, sources)
โข The theory is given as a quadruple โ,๐ฌ, ๐, ๐ท
โข Duality is an isomorphism โ๐ด, ๐ฌ๐ด, ๐๐ด โ โ๐ต, ๐ฌ๐ต, ๐๐ต . There are three bijections: โข ๐โ:โ๐ด โ โ๐ต
โข ๐๐ฌ: ๐ฌ๐ด โ ๐ฌ๐ตโข ๐๐: ๐๐ด โ ๐๐ต
such that:
๐, ๐ ๐ ,๐ท๐ด = ๐๐ช ๐ , ๐๐ฎ ๐ {๐๐(๐)} ,๐ท๐ต โ๐ช โ ๐ฌ๐ด, ๐ โ โ๐ด, ๐ โ ๐๐ด
โข Need to preserve also triples ๐ช; ๐ 1, ๐ 2 ๐ ,๐ท๐ด
๐ช, ๐ ๐ ,๐ท๐ด = ๐๐ฌ ๐ช , ๐โ ๐ {๐๐(๐)} ,๐ท๐ต
(1)
26
AdS/CFT Duality
โข AdS/CFT can be described in terms of the quadruple โ,๐ฌ, ๐, ๐ท : โข Normalizable modes correspond to exp. vals. of operators (choice of state)
โข Fields correspond to operators
โข Boundary conditions (non-normalizable modes) correspond to couplings
โข Formulation otherwise different (off-shell Lagrangian, different dimensions!)
โข Two salient points of :โข Physical quantities, such as boundary conditions, that are not determined by
the dynamics, now also agree: they correspond to couplings in the CFT
โข This is the case in any duality that involves parameters that are not expectation values of operators, e.g. T-duality (๐ โ 1/๐ ), electric-magnetic duality (๐ โ 1/๐)
โข It is also more general: while โ,๐ฌ, ๐ท are a priori fixed, ๐ can be varied at will (Katherine Brading: โmodal equivalenceโ). We have a multidimensional space of theories
โข Dualities of this type are not isomorphisms between two given theories (in the traditional sense) but between two sets of theories
โ
๐ฌ
๐
๐ท
(1)
27