This document discusses emergence and duality in gauge/gravity dualities. It begins by introducing gauge/gravity dualities, which relate a theory of gravity in (d+1) dimensions to a quantum field theory without gravity in d dimensions. The document then discusses two ways in which emergence can occur in theories related by duality: 1) the duality map breaks down at some level of fine-graining, and 2) an approximation scheme is applied to each side of the duality, only holding the duality approximately. Even if gauge/gravity duality is exact, emergence can occur through the second way by approximating the full string theory with a semiclassical gravity theory, where the radial direction corresponds to energy scale

1. 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

2. 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

3. 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

4. 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

5. Gauge/Gravity Dictionary
• (𝑑 + 1)-dim AdS
• GAdS
• d𝑠2
=
ℓ2
𝑟2 d𝑟2
+ 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥 𝑖
d𝑥 𝑗
• Boundary at 𝑟 = 0
• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯ + 𝑟 𝑑
𝑔 𝑑 𝑥
• Field 𝜙 𝑟, 𝑥 , mass 𝑚
• 𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + ⋯ + 𝑟 𝑑
𝜙 𝑑 𝑥
• Long-distance (IR) divergences
• Radial motion in 𝑟 (towards IR)
• CFT on ℝ 𝑑
• QFT with a fixed point
• Metric 𝑔 0 (𝑥)
• 𝑇𝑖𝑗 𝑥 =
ℓ 𝑑−1
16𝜋𝐺 𝑁
𝑔 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)

6. 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)

7. • The basic physical quantities on both sides:
• Other physical quantities are calculated by differentiation:
Π 𝜙 𝑥 Π 𝜙 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 = d 𝑑 𝑥
𝑟Δ
𝑟2 + 𝑥 − 𝑦 2 Δ
𝜙 0 (𝑦)
⇒ Π 𝜙 𝑥 Π 𝜙 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 d 𝑑 𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
=: 𝑍CFT 𝜙 0
Gauge/Gravity Dictionary (Continued)
7
Witten (1998)
Gubser Klebanov Polyakov (1998)

8. 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, 𝒬 𝐴, 𝐷𝐴 and
ℋ 𝐵, 𝒬 𝐵, 𝐷 𝐵 , as follows:
• There exist structure-preserving bijections:
• 𝑑ℋ: ℋ𝐴 → ℋ 𝐵,
• 𝑑 𝒬: 𝒬 𝐴 → 𝒬 𝐵
and pairings (expectation values) 𝒪, 𝑠 𝐴 such that:
𝒪, 𝑠 𝐴 = 𝑑 𝒬 𝒪 , 𝑑ℋ 𝑠 𝐵
∀𝒪 ∈ 𝒬 𝐴, 𝑠 ∈ ℋ𝐴
as well as triples 𝒪; 𝑠1, 𝑠2 𝐴 and 𝑑ℋ commutes with (is equivariant
for) the two theories’ dynamics
8

9. 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!
9

10. 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). 10

11. 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
11
See also: Crowther (2015)

12. 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
12

13. 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)

14. 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. 14

15. Two ways of emergence
𝑑′
: 𝐺′ 𝐹′
𝑑: 𝐺 𝐹
𝐺′′ 𝐹′′
15
𝑑′′
(BrokenMap)
(Approx)

16. 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

17. 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’.
17

18. 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
AdS 𝜖′
𝜕AdS 𝜖′ 𝜕AdS 𝜖
new boundary condition
integrate out
cut-off surface

19. 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 𝜕𝜖 𝑓 = −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)

20. 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):
𝑇𝑖
𝑖
𝑟=∞
=
𝑁2
32𝜋2
𝑅 𝑖𝑗 𝑅𝑖𝑗 −
1
3
𝑅2
• 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:
𝑇𝑖
𝑖
𝑟
= 𝐶 𝑟 𝑅 𝑖𝑗 𝑅𝑖𝑗 −
1
3
𝑅2
• 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 20
domain wall
Freedman et al. (1999)
Henningson Skenderis (1998)
𝑟 → ∞ 𝑟 → −∞

21. 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

22. 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

23. Thank you!
23

24. Gauge/Gravity Duality: Gravity Side
• AdS is the maximally symmetric space-time with constant negative curvature
• Useful choice of local ‘Poincaré’ coordinates:
d𝑠2
=
ℓ2
𝑟2
d𝑟2
+ 𝜂𝑖𝑗 d𝑥 𝑖
d𝑥 𝑗
, 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
𝑟2
d𝑟2
+ 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥 𝑖
d𝑥 𝑗
𝑔𝑖𝑗 𝑟, 𝑥 = 𝑔 0 𝑖𝑗 𝑥 + 𝑟 𝑔 1 𝑖𝑗 𝑥 + 𝑟2
𝑔 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

25. • 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 + ⋯ + 𝑟 𝑑
𝜙 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)

26. 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, 𝒬 𝐴, 𝒞 𝐴 ≃ ℋ 𝐵, 𝒬 𝐵, 𝒞 𝐵 . There are
three bijections:
• 𝑑ℋ: ℋ𝐴 → ℋ 𝐵
• 𝑑 𝒬: 𝒬 𝐴 → 𝒬 𝐵
• 𝑑 𝒞: 𝒞 𝐴 → 𝒞 𝐵
such that:
𝑂, 𝑠 𝑐 ,𝐷 𝐴
= 𝑑 𝒪 𝑂 , 𝑑 𝒮 𝑠 {𝑑 𝒞(𝑐)} ,𝐷 𝐵
∀𝒪 ∈ 𝒬 𝐴, 𝑠 ∈ ℋ𝐴, 𝑐 ∈ 𝒞 𝐴
• Need to preserve also triples 𝒪; 𝑠1, 𝑠2 𝑐 ,𝐷 𝐴
𝒪, 𝑠 𝑐 ,𝐷 𝐴
= 𝑑 𝒬 𝒪 , 𝑑ℋ 𝑠
{𝑑 𝒞(𝑐)} ,𝐷 𝐵
(1)
26

27. 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