Talk given at the conference Equivalent Theories in Physics and Metaphysics, 20-21 March 2015, Princeton University, NJ, USA.

1. 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

2. 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’

3. 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

4. • 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!

5. 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”

6. 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

7. 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

8. 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?

9. Plan
•Duality:
• Introduction to AdS/CFT
• Duality
• Renormalization group
• Diffeomorphism invariance and background
independence
• Interpretation
•Emergence

10. Geometry of AdS 𝐷
• Hyperboloid in 𝐷 + 1 dimensions:
−𝑋0
2
− 𝑋 𝐷
2
+
𝑖=1
𝐷−1
𝑋𝑖
2
= −ℓ2
• Constraint can be solved introducing 𝐷 coordinates:
𝑋0 = ℓ cosh 𝜌 cos 𝜏
𝑋 𝐷 = ℓ cosh 𝜌 sin 𝜏
𝑋𝑖 = ℓ sinh 𝜌 Ω𝑖 𝑖 = 1, … , 𝑑 = 𝐷 − 1 , Ω𝑖 = unit vector
• Leading to: d𝑠2
= ℓ2
− cosh2
𝜌 d𝜏2
+ d𝜌2
+ sinh2
𝜌 dΩ 𝐷−2
2
• Symmetry group SO 2, 𝑑 apparent from the construction
• Riemann tensor in terms of the metric (negative curvature):
𝑅 𝜇𝜈𝜆𝜎 = −
1
ℓ2
𝑔 𝜇𝜆 𝑔 𝜈𝜎 − 𝑔 𝜇𝜎 𝑔 𝜈𝜆
𝑋𝑖
𝑋0
𝑋 𝐷
ℓ

11. Geometry of AdS 𝐷
• Useful choice of local coordinates:
d𝑠2
=
ℓ2
𝑟2 d𝑟2
+ 𝜂𝑖𝑗 d𝑥 𝑖
d𝑥 𝑗
, 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 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

12. 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 + ⋯ + 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 (and the metric)

13. AdS/CFT
• 𝐷-dim AdS
• 𝐷-dim (AL)AdS
• d𝑠2 =
ℓ2
𝑟2 d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥 𝑖d𝑥 𝑗
• 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
• CFT on ℝ 𝐷−1
• QFT with a fixed point
• Metric 𝑔 0 (𝑥)
• 𝑇𝑖𝑗 𝑥 =
ℓ 𝑑−1
16𝜋𝐺 𝑁
𝑔 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

14. 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

15. 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, 𝒪 𝐴, 𝐷𝐴 and
𝒮 𝐵, 𝒪 𝐵, 𝐷 𝐵 .
• There exist bijections:
• 𝑑 𝒮: 𝒮𝐴 → 𝒮 𝐵,
• 𝑑 𝒪: 𝒪 𝐴 → 𝒪 𝐵
and pairings (vevs) 𝑂, 𝑠 𝐴 such that:
𝑂, 𝑠 𝐴 = 𝑑 𝒪 𝑂 , 𝑑 𝒮 𝑠 𝐵 ∀𝑂 ∈ 𝒪 𝐴, 𝑠 ∈ 𝒮𝐴

16. 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, 𝒪 𝐴, 𝒞 𝐴, 𝐷𝐴 and
𝒮 𝐵, 𝒪 𝐵, 𝒞 𝐵, 𝐷 𝐵 . There exist bijections:
• 𝑑 𝒮: 𝒮𝐴 → 𝒮 𝐵
• 𝑑 𝒪: 𝒪 𝐴 → 𝒪 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 ∈ 𝒪 𝐴, 𝑠 ∈ 𝒮𝐴, 𝑐 ∈ 𝒞 𝐴
𝑐 𝐵 = 𝑑 𝒞 𝑐 𝐴 𝑐 𝐴 ∈ 𝒞 𝐴, 𝑐 𝐵 ∈ 𝒞 𝐵
𝑂, 𝑠 𝑐 ,𝐷 𝐴
= 𝑑 𝒪 𝑂 , 𝑑 𝒮 𝑠 {𝑑 𝒞(𝑐)} ,𝐷 𝐵
𝑐 𝐵 = 𝑑 𝒞 𝑐 𝐴
(1)
(2)

17. 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 ,𝐷 𝐴
= 𝑑 𝒪 𝑂 , 𝑑 𝒮 𝑠 {𝑑 𝒞(𝑐)} ,𝐷 𝐵

18. 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 d 𝑑
𝑥 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′)
𝑐 𝐵 = 𝑑 𝒞 𝑐 𝐴

19. Renormalization Group
• Radial integration: • Wilsonian renormalization:
Λ𝑏Λ0
𝑘
integrate out
New cutoff 𝑏Λ
rescale 𝑏Λ → Λ until 𝑏 → 0
AdS 𝑟
𝜕AdS 𝑟 𝜕AdS 𝜖
new boundary condition
integrate out
IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT(2) 𝑐 𝐵 = 𝑑 𝒞 𝑐 𝐴

20. 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

21. 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

22. 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 −𝑑
𝜙 𝑟, 𝑥
𝑟=0
= 𝜙 0 𝑥 = 𝑒 d 𝑑 𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
(1′)

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. •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!

30. • 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

31. Analysing the process

32. • 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?

33. 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

34. 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

35. 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

36. Thank you!