Talk on the philosophy of dualities, in particular AdS/CFT. Joint talk delivered with Jeremy Butterfield at the Biennial Meeting of the Philosophy of Science Association, Chicago, 7 Nov. 2014.

1. Duality in AdS/CFT
Sebastian de Haro
(University of Amsterdam and
University of Cambridge)
PSA, Chicago, 7 November 2014
Partly based on PhilSci 10606
with: D. Dieks, J. van Dongen

2. AdS/CFT
• 𝐷-dim. anti-de Sitter space
• Can be extended to (AL)AdS
• In local coordinates:
d𝑠2
=
ℓ2
𝑟2
d𝑟2
− d𝑡2
+ d𝐱2
• Fields 𝜙 𝑟, 𝑥
• Mass 𝑚
• CFT on ℝ 𝐷−1
• QFT with a fixed point,
other backgrounds
• Operators 𝒪 𝑥
• Dimension Δ
2

3. Duality Statement
•One-to-one map of states and quantities
(observables) between distinct theories preserving
certain structures (see Jeremy’s talk).
•String theory in (AL)AdS space = QFT on boundary
•Fields 𝜙 𝑟, 𝑥 ↔ Operators 𝒪 𝑥
•Partition function 𝑑 = 𝐷 − 1 :
𝑍string 𝑟Δ −𝑑 𝜙 𝑟, 𝑥
𝑟=0
= 𝜙 0 𝑥 = 𝑒 d 𝑑 𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
•Physical equivalence, mathematical structure
different
•Large distance ↔ high energy divergences
(1)
3

4. 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.
4

5. 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
5

6. 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, but 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 a choice of particular solutions
of Einstein’s equations?
• The equations of motion do not determine the boundary conditions, which 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 equations of motion simply do not contain the informtion about the
boundary/initial conditions.
• 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 by a choice of a
particular solution. 6

7. 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?
7
𝑍string 𝑟Δ −𝑑
𝜙 𝑟, 𝑥
𝑟=0
= 𝜙 0 𝑥 = 𝑒 d 𝑑 𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
(1)

8. 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.
8

9. 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
9

10. Remarks
•External view: meaning of observables externally
fixed. Duality relates different physical quantities
• No empirical equivalence, numbers correspond to
different physical quantities
• The symmetry is broken by the different physical
interpretation given to the symbols
•Internal point of view:
• The two descriptions are equivalent. Both theories
describe the same set of observables. No devisable
experiment could tell one from the other (each
observation can be reinterpreted in the ‘dual’ language)
• Cannot decide which description is superior. One
formulation may be superior on practical grounds (e.g.
computational simplicity in a particular regime)
10

11. Does AdS/CFT face the puzzling scenario?
•In the external view, the two theories are not equally
successful because they describe different physical
quantities: only one of them may describe this world.
•In 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. Currently there is no indication that the
two theories are notational variants of each other.
• If the two theories are not notational variants of each
other, then we do face the puzzling scenario! 11

12. 12
Thank you!