Holographic dualities and applications · gravitational physics: spacetime is emergent,...

IntroductionHolography: a primerHolographic RG flows

Domain-wall/Cosmology correspondenceHolography for cosmology

Conclusions

Holographic dualities and applications

Kostas Skenderis

University of Amsterdam

National seminarTheoretical High Energy Physics

27 March 2009

Kostas Skenderis Holographic dualities and applications



Conclusions

Outline

1 Introduction

2 Holography: a primer

3 Holographic RG flows

4 Domain-wall/Cosmology correspondence

5 Holography for cosmology

6 Conclusions




Conclusions

Introduction

Gauge/gravity duality has been one of most far reachingdevelopments in recent years. It introduced a new theoreticalframework to address fundamental conceptual questions:

On one hand, it opens a window into strong coupling dynamics ofQFTs.On the other hand it provides a qualitatively new paradigm forgravitational physics: spacetime is emergent, reconstructed fromgauge theory data.

In recent times the holographic dualities have found applications thatrange from mathematics to phenomenology to condensed matterphysics and as I will argue today to cosmology as well.




Conclusions

Introduction

The plan of this talk is

Explain what holography is.Discuss how holography applies to cosmology.




Conclusions

Reference

The second part is based onPaul McFadden, KS,Holography for Cosmology,arXiv: 0904.????




Conclusions

Holography

Holography originated from black hole physics [’t Hooft (1994)] as theanswer to the question:

Why does the entropy of a black hole behaves as an area?Usually, entropy is extensive, and scales with volume.

Definition

Holography states that a theory which includes gravity can bedescribed by a theory with no gravity (just forces likeelectromagnetism) is one fewer spatial dimension.

By proposing that the real degrees of freedom in gravity arethose of a field theory in one less dimension one automaticallygets entropies to scale with area, rather than volume.




Conclusions

A new paradigm

The holographic principle represents a fundamental change inparadigm whose consequences are only beginning to beappreciated.It suggests that one of the macroscopic spatial dimensions andone of the forces of Nature, gravity, that we perceive in everydaylife are emergent phenomena.




Conclusions

Holography and string theory

The initial proposal was received with some skepticism, tillconcrete realizations of holography were found in string theoryfew years later [Maldacena (1997)] [Gubser, Klebanov, Polyakov(1998)] [Witten (1998)] ... now 5000+ papersThe most famous example is the so-called AdS/CFTcorrespondence, relating gravity in a negatively curved (Anti-deSitter) backgrounds to conformal field theories.The AdS/CFT correspondence is an example of a holographicduality. Such dualities, which are also referred to asgauge/gravity dualities, are conjectured exact equivalencesbetween certain gravitational theories and certain QFT (ofteninvolving gauge fields).




Conclusions

Holographic dualities

Over the last 12 years there has been an enormous amount workon holography and there is a very impressive list of non-trivialtests.The support to the duality includes checks of the duality inexamples with many symmetries, such as the duality betweenAdS5 × S5 and N = 4 SYM:→ matching of the spectra of chiral primaries and correlation functions→ matching of entire spectrum of the planar theory using integrability

as well as structural support in the generic case→ Does the gravitational side exhibit the analytic structure of a

quantum field theory?




Conclusions

Structural support

Quantum field theories exhibit the following short distance structure:

UV divergences are local.There is a separation of scales. For example, cancelation of UVdivergences does not depend on IR physics.Symmetries give rise to Ward identities, whose form can beestablished without having to compute the correlators explicitly.Ward identities may be anomalous with the anomaly being localand computable from UV data.




Conclusions

Structural support

Any theory which is claimed to be dual to a QFT should exhibitsuch structure.In holographic dualities the UV structure of QFT is identified withthe IR structure of the gravitational theory (UV-IR connection)[Susskind, Witten (1998)].Holographic dualities indeed exhibit the required structure,[Henningson, SK (1998)], [de Haro, Solodukhin, KS (2000)], [Bianchi,Freedman, KS (2001)], showing that the duality is not an accidentalproperty of the very symmetric examples studied initially.




Conclusions

Outline

1 Introduction





6 Conclusions




Conclusions

How does it work?

Gauge-gravity duality relates string theory on certain backgrounds tonon-gravitational QFTs. For such a relation to be a well-posed oneshould specify:

How the variables and parameters of the two differentdescriptions are related to each other.How to compute.




Conclusions

Basic Dictionary

Roughly speaking, the dictionary is the following:1 There is 1-1 correspondence between local gauge invariant

operators O of the boundary QFT and bulk supergravity modesΦ. Non-local observables, such as Wilson loops etc, correspondto probe strings and branes in the bulk.→ The bulk metric corresponds to the energy momentum tensor of

the boundary theory.→ Bulk gauge fields correspond to boundary symmetry currents.→ Bulk scalar fields correspond to boundary scalar operators, i.e.

FµνFµν , ψ̄ψ, etc.

2 Correlation functions of gauge invariant operator can beextracted from the asymptotics of bulk solutions.




Conclusions

Precision holography

Extensive work over the years resulted in a very preciseholographic framework. Given a gravitational solution, there isprecise algorithm that leads to correlation functions.This information can be used to extract detailed information fromphenomenological holographic models, quantified how good/badthey are, and moreover play a key role in understanding howholography works in general. They also play a key role inunderstanding black hole physics.




Conclusions

Holographic computations: A primer

In QFT the theory is defined by giving the Lagrangian, fromwhich one can extract the Feynman rules, etc. Typically,correlation functions are then computed perturbatively at weakcoupling, with due care to remove infinities (renormalization).In holography the corresponding information is encoded insolutions to the bulk equations of motion. As in QFT, one needsto remove infinities, now due to infinite volume of spacetime(holographic renormalization) to properly extract the correlators.These computations yield the correlators at strong coupling.




Conclusions

Asymptotic solutionsTo understand the holographic computations we need to know a fewthings about the structure of solutions of Einstein’s theory with anegative cosmological constant.

For the metric, the most general asymptotic form looks like[Fefferman, Graham (1985)]

ds2 = dr2 + e2r gij(x , r)dx idx j

gij(x , r) = g(0)ij(x)+e−2r g(2)ij(x)+...+e−dr (rh(d)ij(x) + g(d)ij(x)

)+...

→ The metric with gij(x , r) = ηij is the AdSd+1 metric.→ The metric with g(0)ij(x) = ηij is an Asymptotically AdSd+1 metric.→ The metric with general g(0)(x) is an Asymptotically locally AdSd+1

metric.g(0)(x) is the metric of the spacetime where the boundary theorylives and (as such) it is also the source of the boundary energymomentum tensor.




Conclusions

Correlation functions

Using the formalism of holographic renormalization, we then finda precise relation between correlation functions and asymptotics[de Haro, Solodukhin, KS (2000)]

〈Tij〉 =d

16πG[g(d)ij + X (d)

ij (g(0))].

where X (d)ij (g(0)) are local functions of g(0).

→ Correlators satisfy all expected Ward identities,

∇i〈Tij〉 = 0, 〈T ii 〉 = A




Conclusions

Higher-point functions

Higher-point functions are obtained by differentiating the 1-pointfunctions w.r.t. sources and then set the sources to theirbackground value

〈Ti1 j1(x1)Ti2 j2(x2) · · ·Tin jn(xn)〉 ∼δ(n−1)g(d)i1 j1(x1)

δg(0)i2 j2(x2) · · · δg(0)in jn(xn)

∣∣∣g(0)=η

Thus to solve the theory we need to know g(d) as a function ofg(0).

→ This can be obtained perturbatively: 2-point functions areobtained by solving linearized fluctuations, 3-point functions bysolving quadratic fluctuations etc.




Conclusions

AlgorithmSuppose one is interested in computing the 2-point functions of Tij ina strongly coupled CFT that has a holographic dual.

1 For conformal field theories, CFTd , the corresponding bulksolution is AdSd+1.

2 To compute the 2-point function of the Tij we need to linearize thebulk gravitational equations around AdSd+1 and solve theresulting equations.

3 Expand asymptotically the exact solution of the linearizedequations. From our earlier discussion we know that this has theform

gLij (x , r) = g(0)ij

(1 + · · ·e−dr A(x) + · · ·

)4 The 2-point function is then given by

〈Tij(x)Tkl(0)〉 = ΠijklA(x)

where Πijkl is the transverse-traceless projection operator.




Conclusions

Outline

1 Introduction





6 Conclusions




Conclusions

Holographic RG flows

We just reviewed how to compute holographically correlationfunctions for a CFTd .We now describe now to extend the duality to non-conformaltheories.The main change is to replace AdSd+1 with a "domain-wall"spacetime,

ds2 = dr2 + e2A(r)dx idx i

Φ = Φ(r)

This configuration solves the field equations that follow from theaction

S =

∫dd+1x

√G(R + (∂Φ)2 + V (Φ))




Conclusions

Domain-wall spacetimes

The AdSd+1 metric is the unique metric whose isometry group isthe same as the conformal group in d dimensions. This is themain reason why the bulk dual of a CFT is AdS.The domain-wall spacetimes are the most general solutionswhose isometry group is the Poincaré group in d dimensions.Thus, if a QFT has a holographic dual the bulk solution must beof the domain-wall type.




Conclusions

Fake supersymmetry [Freedman, Nunez, Schnabl, KS (2003)]

Domain-wall spacetimes have remarkable properties. Provided thescalar field Φ(r) has only isolated zero’s, the following properties hold[KS, Townsend (2006)]:

1 The spacetime admits a covariantly constant spinor,

Dµε = 0, Dµ = Dµ + W (Φ)Γµ

where W (Φ), the fake superpotential, is determined by thesolution. The spinor ε is called fake Killing spinor.

2 The existence of fake Killing spinors guarantees perturbative andnon-perturbative stability of all non-singular domain-wallspacetimes.

3 All domain-wall spacetimes solve first order "BPS" equations.These follow from the fake Killing spinor equations.




Conclusions


There are two different types of domain-wall spacetimes whoseholographic interpretation is fully understood.

1 The domain-wall is asymptotically AdSd+1,

A(r) → r , Φ(r) → 0, as r →∞

This corresponds QFT that in the UV approaches a fixed point.The fixed point is the CFT which is dual to the AdS spacetimeapproached as r →∞.→ The rate at which Φ(r) approaches zero, signifies whether the QFT

is a relevant deformation of the CFT or the CFT in a non-conformalvacuum.




Conclusions


2 The domain-wall follows from AdSd+1 by dimensional reductionover the torus T σ and continuation in σ (i.e. σ may be nonintegral),

A(r) → n log r , Φ(r) →√

2n log r , as r →∞

where σ = (3n − 1)/2(n − 1). This case has only beenunderstood very recently [Kanitscheider, KS, Taylor (2008)][Kanitscheider, KS (2009)].

→ Specific cases of such spacetimes are ones obtained by takingthe near-horizon limit of the non-conformal branes (D0, D1, F1,D2, D4).

→ These solutions describe QFTs with a dimensionful couplingconstant in the regime where the dimensionality of the couplingconstant drives the dynamics.




Conclusions

Correlation functionsCorrelation functions for these strongly coupled QFT’s can now becomputed following the same steps as before. For 2-point functionsthis is done as follows:

1 We linearize the bulk gravitational equations around thedomain-wall solution,

ds2 = dr2 + e2A(r)(δij(1 + ψ(x , r)) + γij(x , r))dx idx i

Φ = Φ(r) + ϕ(x , r)

where γij is transverse traceless. In this case there are twoindependent modes: the transverse traceless γij and the scalargauge invariant combination ζ = −ψ/2 + ϕ(W/Φ̇), where W isthe fake superpotential, and the resulting equations can bediagonalized in complete generality [Bianchi, Freedman, KS (2001)][Papadimitriou, KS (2004)].




Conclusions


2 We now solve the linearized equations and expand themasymptotically

gLij (x , r) = g(0)ij

(1 + · · ·e−dr A(x) + · · ·

)ζ(x , r) = ζ(0)

(1 + · · ·e−∆r B(x) + · · ·

)3 The 2-point functions are given by

〈Tij(q)Tkl(−q)〉 ∼ πijklA(q2) + πijπklB(q2)

〈Tij(q)O(−q)〉 ∼ πijB(q2)

〈O(q)O(−q)〉 ∼ B(q2)

where πij = δij − qiqj/q2.




Conclusions


Such computations have been carried out explicitly for a numberof examples, i.e. the functions A(q2) and B(q2) (and othercorrelators) have been computed for specific domain-wallsolutionsThe correlators exhibit all expected properties of QFTcorrelators.→ satisfy the expected Ward identities.→ have correct short distance behavior.→ exhibit expected long distance behavior. For example, in the case

of spontaneous broken symmetries, one finds the correctGoldstone poles, etc.




Conclusions

Outline

1 Introduction





6 Conclusions




Conclusions

Domain-wall/Cosmology correspondence

The domain-wall spacetimes have a remarkable similarity to flatFRLW spacetime

ds2 = −dt2 + a(t)2dx idx i

Φ = Φ(t)

One can actually prove the following


For every domain-wall solution of a model with potential V there is aFRLW solution for a model with potential (-V). [Cvetic, Soleng (1994)],[KS, Townsend (2006)]




Conclusions


The correspondence also applies to open and closed FRLWuniverses which correspond to curved domain-walls.The correspondence can be understood as analytic continuationfor the metric. The flip in the sign of V guarantees that the scalarfield remains real.This seemingly trivial observation does have non-trivialimplications. For example,→ cosmologies admit covariantly constant spinors: they are

pseudo-supersymmetric.→ they solve first order equations.

The full set of implications of this unexpected fermionicsymmetry of cosmological solutions, namely ofpseudo-supersymmetry, is still be uncovered.




Conclusions

Domain-wall/Cosmology correspondence in SUGRA

One may wonder whether the correspondence and itsimplications are an accidental property of the very symmetricsolutions we consider.It turns out that one can embed the correspondence insupergravity [Bergshoeff etal, (2007)] [KS, Townsend, Van Proeyen(2007)]. This maps, in particular, AdS supergravity to dSsupergravity and cosmologies can be supersymmetric solutionsof the latter.dS supergravities are known to be contain fields with "wrong signkinetic terms". None of these "ghost fields" however participatein the cosmological solutions.




Conclusions

Examples

Let us see how the correspondence acts on the domain-wallsdescribing holographic RG flows.

1 Asymptotically AdS domain-walls are mapped to inflationarycosmologies that approach de Sitter spacetime at late times.

ds2 → ds2 = −dt2 + e2tdx idx i , as t →∞

2 The domain-walls obtained from AdS by generalized dimensionalreduction are mapped to solutions that approach power-lawscaling solutions at late times,

ds2 → ds2 = −dt2 + t2ndx idx i , as t →∞




Conclusions

Outline

1 Introduction





6 Conclusions




Conclusions

What is holography in the context of cosmology?

Recall that holography states that there should a QFT descriptionin one dimension less.So what we need to know is

1 what the QFT is2 what the holographic dictionary is

A test for such proposal would be to demonstrate that resultsobtained using a purely gravitational computation can berecovered doing a QFT computation.Then one can move on to use the new description in the regimewhere the gravitational description in not reliable to obtain newinformation.




Conclusions

The proposal for inflationary cosmology [McFadden, KS (2009)]

The dual QFT is obtained as follows:

1 A given inflationary model, based on a single scalar model, canbe mapped to a domain-wall via the domain-wall/cosmologycorrespondence.

2 As we discussed, these domain-walls are the ones withoperational gauge/gravity duality, i.e. there is a dual QFT via theusual gauge/gravity duality.

3 The analytic continuation that enters in the DW/cosmologycorrespondence can be expressed entirely in terms of QFTvariables.

4 We now apply this analytic continuation to the QFT dual of thedomain-wall to obtain the QFT dual of the inflationary model.




Conclusions

The proposal




Conclusions

Pseudo-QFT

We operationally define the pseudo-QFT as follows:

we do the computation in the QFT dual to the domain-wall andthen analytically continue parameters and momentaappropriately.

Perhaps a more fundamental perspective is to consider the QFTaction with complex parameters as the fundamental object.

Then the results on different real domains will be applicable toDW/cosmology as appropriate.

→ The supergravity realization of the DW/cosmologycorrespondence works this way.




Conclusions

Does it work?

In inflationary cosmology, interesting observables are thespectrum of primordial perturbations and non-gaussianities.These observables are computed by quantizing the bulkperturbations and computing two and higher point functions.We show that for all asymptotically dS and asymptotically powerlaw cosmologies based on a single scalar the power spectrumand non-gaussianities are reproduced exactly by the holographiccomputation.




Conclusions

Holography and the wavefunction of universe

This proposal can also be considered as providing a holographiccomputation of the wavefunction of the universe.As pointed out by [Maldacena (2002)] in the context of dScosmologies, a sum-of-the histories approach to cosmologicalperturbations naturally leads to a relation between bulk andputative boundary correlators.In our case, we led to this description via the DW/cosmologycorrespondence, generalized it to include power-lawcosmologies and provided a proposal about the dual QFT.




Conclusions

Outlook

Generalize to models with different matter content (e.g.multi-scalar models etc.)Understand implications of pseudo-supersymmetry.Perhaps most interesting however is to understand what the newdescription has to say about the regime where the gravitationaldescription is not reliable. E.g. what can we learn aboutsingularities, the physics of cosmological horizons etc etc?




Conclusions

Outline

1 Introduction





6 Conclusions




Conclusions

Holography is a very precise framework that can be used tocompute field theory properties from geometry and vice versa.It already has a wide range of application and one wouldanticipate the list to grow.I presented a proposal for a holographic reformulation ofinflationary cosmology.

→ It correctly reproduces the power spectrum andnon-gaussianities.

→ One can only anticipate new insights about "quantumcosmology" to emerge ...


Holographic dualities and applications · gravitational physics: spacetime is emergent,...

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