Duality in AdS/CFT

Sebastian de Haro (University of Amsterdam and

University of Cambridge)

PSA, Chicago, 7 November 2014Partly based on PhilSci 10606 with: D. Dieks, J. van Dongen

AdS/CFT

•𝐷-dim. anti-de Sitter space • Can be extended to (AL)AdS

• In local coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 − d𝑡2 + d𝐱2

• Fields 𝜙 𝑟, 𝑥• Mass 𝑚

• CFT on ℝ𝐷−1

• QFT with a fixed point, other backgrounds

•Operators 𝒪 𝑥• Dimension Δ

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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 −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

•Physical equivalence, mathematical structure different• Large distance ↔ high energy divergences

(1)

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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.

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Renormalization Group

• Radial integration: • Wilsonian renormalization:

Λ𝑏Λ0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

AdS𝑟

𝜕AdS𝑟 𝜕AdS𝜖

new boundary condition

integrate out

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT

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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

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?

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𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1)

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.

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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

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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)

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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

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Thank you!