An Introduction to AdS/CFT Correspondence

(for Non-Experts)

Tadashi Takayanagi (IPMU)

IPMU Colloquium, Nov.17th, 2011

Contents

① Introduction ⇒ What are AdS and CFT ?

② What is AdS/CFT ?

③ How does AdS/CFT work? ⇒ Holographic Principle

(for gravity people ??)

④ AdS/CFT from String Theory ? ⇒ Large N limit

(for mathematicians ??)

⑤ Developments of AdS/CFT

Note: I will try to give a conceptual guide to AdS/CFT , rather than a practical one.

① Introduction

The AdS/CFT correspondence argues the equivalence:

AdS = CFT . [1997 Maldacena]

More precisely,

Gravity in Anti de-Sitter Space = Conformal Field Theory .

d+1 dimension d dimension

(1-1) What is the AdS ?

Our spacetime is well described by the 3+1 dimensional

Minkowski spacetime .

If we express its coordinate by ,

then its metric is given by

It is clear that this spacetime has the SO(1,3) symmetry.

i.e. Lorentz symmetry

3,1R

),,,( 321 xxxt

.)()()( 23

22

21

22 dxdxdxdtds +++−=

(Space dim. + Time dim.)

This is easily extended to higher dimensions i.e. ,

which has the symmetry SO(1,d).

These are examples of flat spacetime. In other words,

they have the vanishing curvature

dR ,1

.0=µνR

To define AdS (anti-de Sitter) space, we start from :

This has the symmetry SO(2,d). To eliminate one time,

we consider the hypersurface defined by

This defines the d+1 dimensional AdS space .

(=Lorentzian version of d+1 dim. hyperbolic space)

dR ,2

.)()()()()( 222

21

21

20

2dd dxdxdxdxdxds ++++−−= +

. )()()()()( 2222

21

21

20 Rxxxxx dd ++++=+ +

1+dAdS

Global AdS

( ).)(sinh)(cosh

),..,2,1( sinh,sincosh

,coscosh

)()()()()(

2222222

1

0

2222

21

21

20

Ω++−=⇒

=Ω⋅==

=⇒

++++=+

+

+

dddtRds

diRxtRx

tRx

Rxxxxx

ii

d

dd

ρρρ

ρρ

ρ

1−dS

A Sketch of Global AdS

Time t ρ

1−dS

11)( −+ ×=∂ d

d SRAdS

( )2222222 )(sinh)(cosh Ω++−= dddtRds ρρρ

Poincare AdS

.

,12

,12

),1,..,2,1,0(

)()()()()(

2

222

2

2

1

2

2

2222

21

21

20

zdydydz

Rds

zyyRzx

zyyRzx

dz

Ryx

Rxxxxx

d

d

dd

µµ

µµ

µµ

µµ µ

+⋅=⇒

++=

−−=

−==

⇒

++++=+

+

+

z

Poincare AdSd+1

Boundary

1,11)( −+ =∂ d

d RAdS

1,1 −dR

. 2

222

zdydydz

Rds µµ+

⋅=

Summary:

(1) has the SO(2,d) symmetry.

(2) is the maximally symmetric space with

the negative curvature:

(3) has a d dim. time-like boundary.

1+dAdS

1+dAdS

.2 µνµν gRdR

−=

1+dAdS

(1-2) What is the CFT ?

CFT (conformal field theory)

= quantum field theory which has the conformal sym.

``Scale invariant theory’’

E.g. Electromagnetism (Maxwell theory)

TFT CFT QFT

µµ dxxAxAxBxE )()())(),(( =⇔

• Gravity = Theory of the dynamical metric .

• QFT = Theory of particles with a fixed metric .

• CFT = QFT which is invariant under

• TFT = QFT which does not depend on the metric .

µνg

µνφ

µν geg →

µνg

µνg

Conformal Symmetry

Diffeomorphism such that

)( νµµ xXX =

.)()()( µνµν xgxXg ⋅Λ=

+⋅−⋅−

=

⋅=⋅=+=

⇒

22

2

)(21:cft. Special

:Dilatation : trf.Lorentz

:nTranslatio

xbxbxbxX

xXxLX

axX

µµµ

µµ

νν

µµ

µµµ

λ

1,1on −dd RCFT

SO(1,d-1)

Total conformal sym. = SO(2,d)

② What is AdS/CFT ? The AdS/CFT is summarized as

Gravity (String Theory) on

=

Note: Both has the same symmetry SO(2,d).

×=∂

−

−

+ )Global( )Poincare(

)(AdSon CFT 1

1,1

1d d

d

d SRR

1AdS +d

Remarks:

(1) Quantum gravity effects are suppressed if

⇔ CFT = large N gauge theory (SU(N) Yang-Mills theory).

(2) Stringy effects are suppressed if (if so, we can approximate the gravity by general relativity)

⇔ CFT gets strongly interacting. (= the gauge theory coupling constant gets large)

If the conditions (1) and (2) are satisfied, then

String Theory (Quantum Gravity) ⇒ General Relativity. (classical diff. geometry)

pllR >>

stringlR >>

What does the `equivalence’ mean ?

AdS/CFT argues the equivalence of partition function:

In other words, the (free) energy agrees with each other.

The partition functions are functionals of boundary

metric etc. :

. CFTGravity ZZ =

. ],...,,[,...],,[ )0()0()0()0( φφ µµνµµν AgZAgZ CFTGravity =

Ex. Wilson loop (Quark-anti quark potential)

Open String

Quark

+Q

Anti-quark

-Q

Time

1AdS +d

1AdS +d

t∆

q

CFTtE

Gravity

E

WeeZ q

energy quark anti-Quark

)Area(

⇒

=== ∆⋅−Σ−

surface area Minimal :Σ

③ How does the AdS/CFT work ?

To understand this, we explain the holographic principle.

Q. How much information can we keep in a given

region in the presence of gravity ?

A lot of massive objects in a small region

Horizon

Black Holes (BHs)

Collapse

In physics, the amount of (hidden) information is

measured by entropy. (= entropy in thermodynamics)

The entropy of a black hole is given by the so called

Bekenstein-Hawking formula

constantNewton :NG

NBH G

S4

on)Area(Horiz=

It is known that black holes follow the laws of

thermodynamics.

Temperature T ⇔ Strength of surface gravity on a BH

Energy E ⇔ Mass of a BH

Entropy S ⇔ Area of a BH

The first law : TdS=dE

The second law : ΔS≧0

Geometrical origin of BH entropy (ex. 4D BH)

cf. In flat space, we have a cylinder instead of disk.

.11

11

2221

22

2221

22

Ω+

−+

−=→

Ω+

−+

−−=

−

−

drdrrmdt

rmds

drdrrmdt

rmds

EEonWickRotati

⊗

1 :Disk =χ 2S

1−= Tβ

The black hole is the maximally entropic state in the

presence of gravity.

This leads to the idea of entropy bound:

4

A)Area()(NG

AS ∂≤ A

The degrees of freedom in gravity are proportional to the area, instead of the volume ! (Cf. In the absence of gravity, they are proportional to the volume.)

Motivated by this, holographic principle has been

proposed: [’t Hooft 93 ,Susskind 94]

This is a very intuitive argument,

but later, an explicit example is

provided by the AdS/CFT.

d+1 dim. Gravity on M = d dim. non-gravitational theory on ∂Md+1

1+∂ dM

2+dM

Boun

dary

Bulk

What is the meaning of the extra one dimension ?

Consider the AdS/CFT:

The radial direction z corresponds to the length scale

in CFT under the RG flow. (1/z ~ Energy Scale)

1dAdS :Bulk +

dCFT :Boundary

1z IR >> UV

. 2

222

zdydydz

Rds µµ+

⋅=

AdS/CFT as a ``Gravitational Sieve’’

zLength Scale UV

IR

Black holes and AdS/CFT

AdS BH solution

This is dual to a CFT at finite temperature T.

T is the same as the black hole temperature .

BH

.)/(1)(

. )(

)(22

22

222

dH

ii

zzzf

zdydy

zfzdzdt

zzfRds

−≡

++−=

Hzz0

.1

HBH z

T ∝

The AdS/CFT argues BH thermodynamics = CFT thermodynamics BH entropy = thermal entropy Remark: It is well-known that in flat spacetime, the black hole typically has the negative specific heat (→unstable). However, the AdS BH (=BH in a box) has the positive specific heat.

④ AdS/CFT from String Theory ?

(4-1) Large N Limit and String Theory

It is useful to look at a toy model of non-abelian gauge

theories. Consider a matrix model:

Note: This simplification still keeps essential combinatorics.

),..,2,1,( )]([ NbadxxAA ababab =Φ⇒= µµ

SU(N) Connection (1-form) (SU(N) gauge field)

N×N Hermitian Matrix (0 dim. matrix model)

In this matrix model, the partition function is given by

the integral:

Now we take so called the large N limit:

coupling) (gaugeconstant coupling:g

].Tr[1)(

,)](exp[ ][

322 Φ+Φ=Φ

Φ−Φ= ∫

gS

SdZmatrix

finite. with 2 =≡∞→ NgN λ’t Hooft coupling

Let us perform perturbative expansions.

The propagator is given by

( )

].Tr[)( ],Tr[)(

, )()](exp[ ][

31

20

0 10

Φ≡ΦΦ≡Φ

Φ−⋅Φ−Φ= ∑∫∞

=

λλNSNS

SSdZn

nmatrix

. ~)](exp[ ][ 0 bcadcdabcdab NSΦΦdΦΦ δδλ

∫ Φ−Φ=

aedx

xedxfc

x

x

21

..

2

2 2

=⋅

∫∫

∞

∞−

−

∞

∞−

−

α

α

Perturbative expansions are expressed by using the

Feynman diagrams:

n=0 n=2 n=4

Interaction Vertex 3Φ

Rules

Propagator

Vertex

Loop

Thus we find

I

N

λ

VN

λ

LN

for propagators I

V

L

for vertices

for loops

.~graphs allgraphs all

VIIVLV

LI

matrix NNNN

Z −−+ ⋅=

⋅⋅

∑∑ λ

λλ

Triangulation and Feynman Diagrams

Dual Graphs

cycle)-(2 Face Vertex cycle)-(1 line Propagatorcycle)-(0point Loop

⇔⇔⇔

(V) (I)

(L)

Therefore,

In this way,

Planar ex. Non-planar ex.

number).(Euler )faces(#)lines(#)points(# χ=+−=+− VIL

.~graphs all

VImatrix NZ −⋅∑ λχ

(Sphere) 22 ,3 ,3

=⇒===

χVIL

(Torus) 04 ,6 ,2

=⇒===

χVIL

2S 2T

String Theory

Particle String

Closed string ⇒ (Quantum) Gravity

Open string ⇒ Electromagnetism

World-sheet (Riemann surface)

+Q

-Q

Actually, the same topological expansion appears in

string theory:

world-sheets

Thus we expect the correspondence:

length string : constant coupling string :

)(SurfacesRiemann

s

s

sgsString

lg

R/lfgZ ⋅= ∑ −χ

⇔ AdS/CFT.in find alsocan we

slRλ

1−⇔ sgNexpansion expansion large theTherefore, sgN ⇔

+ +⋅⋅⋅+

(4-2) Matrix Model and 2D String Theory

Before we go back to AdS/CFT, let us briefly introduce

the simplest example of holography: 2D string theory.

Consider a matrix quantum mechanics:

sym.). (gauge

energy), (potential )(

, ])()(Tr[

1

22

2

gΦgΦlU

UL

s

tMQM

−

−

⋅⋅→

Φ⋅−=Φ

Φ−Φ∂=Φ

)(ΦU

Using the U(N) gauge sym. , we diagonalize the matrix

This shows that the system is described by N free fermions

=⋅⋅ −

N

gΦg

λ

λλ

2

1

1

[ ]

−∝Ψ

+∂=

∏

∑

<

=

jijiN

N

i it UH

)(),....,,(

, )()(

21

12

λλλλλ

λλ

∏∏<

=−=Φ

jiji

N

i idd 21

)(][ λλλ

Pauli’s exclusion principle requires that there is

only one fermion on each energy level.

Fermi energy (Fermi surface)

In our matrix model, we find the following structure:

λ

E

Fermi Surface Fermi Surface

Fermi Sea Fermi Sea

This matrix model is known to be equivalent to

the two dim. string theory.

Note: Usually, string theory is defined only in 10 dim.

To obtain 2D string, we turn on non-trivial dilaton

field (described by the Liouville CFT).

Strong coupling

Weak coupling

Potential Wall (Liouville Wall)

`Gravitational wave’ reflected by the wall !

11) ( :spacetime ,Rρt, ∈

ρρQ

s eg =

λ

E

The waves on the fermi surface = `Gravitational waves’

Fermi Sea

Reflect wave

In this way, we find the following simple holography:

Matrix Quantum Mechanics = 2D (super) string theory

(2D quantum gravity)

0+1 dim. 1+1 dim.

In this example, the space dimension in gravity can be

regarded as the direction along the Fermi surface.

⇒ A mechanism of emergent space dimension !

model)matrix 1ˆ1( == c or c

Bosonic string [Polyakov, ….Gross-Miljkovic, Berezin-Kazakov-Zamolodchikov, Ginsparg-ZinnJustin 90], D-brane interpretation [Mcgreevy-Verlinde 03] Especially, superstring (type 0 string) version becomes non-perturbatively stable. [Toumbas-TT 03, Douglas-Klebanov-Kutasov-Maldacena-Martinec-Seiberg 03]

(4-3) AdS/CFT from String Theory

Now let us go back to the AdS/CFT correspondence.

Originally, the AdS/CFT has been found in string theory

systems with so called D-branes.

Open strings can end

on D-branes.

Dp-brane ⇒ (p+1) dim. charged object in string theory

Open string

D-brane D-brane

Consider N Dp-branes:

・・・・・

1 2 3 N

N×N matrix

23Φ abΦ

N Dp-branes ⇒ a p+1 dim. U(N) gauge theory (theory of `matrix values functions’) )(xabΦ

≈Φ

)(

)2(

)1(

Nλ

λλ

λ

A basic example of AdS/CFT correspondence is

obtained by looking at the low energy limit of D3-branes.

Open string viewpoint = Closed string viewpoint

N D3-branes (very heavy) 5 dim. AdS space

AdS)on (Gravity on String IIB Type 5

5 SAdS ×(CFT) Mills-YangSuper

SU(N) 4 .dim13 =+ N

Closed String

Open String

Note: Open-Closed Duality

55on String IIB Type SAdS ×

Einstein Manifolds with negative curvature (AdS BH, AdS Soliton etc.) ⇒ Breaks SO(2,4) conformal sym.

Einstein Manifolds with positive curvature ⇒ Breaks SO(6) R-sym. of N=4 Supersym. ∃N=1 Susy if Sasaki-Einstein.

5N 5M

Basic Modifications

Yau).-Calabi (i.e.holonomy SU(3) a has :)C(

Einstein -Sasaki is 2

5222

6

5

MdsrdrdsMM

+=⇔

⑤ Developments of AdS/CFT

In summary, the AdS/CFT argues an equivalence

between theories with gravity and those without gravity.

Gravity on AdS = CFT

`Geometry’ `Quantum (algebra?)’

Interestingly, it often happens that when either side is

difficult to analyze, the other side gets more tractable.

This enables us to apply the AdS/CFT to many subjects !

AdS/CFT

Integrability

Hydrodynamics

Hadron Physics

Condensed Matter Physics

Quantum Gravity

Quantum Information

Geometry

Gauge Theories

Example 1. Viscosity in Strongly Coupled Systems

The viscosity is a fundamental property for fluids.

In the classical gravity (= large N strongly coupled CFT),

the AdS/CFT predicts the universal result

[Kovtun-Son-Starinets 05]

)(for water 300~ n⋅⋅ηv

.4 Bks π

η = small)(very 0.1~

density)entropy :~(n

kns B

⋅⋅⇒⋅

η

Interestingly, the same order of viscosity has been

observed in two different strongly coupled systems:

(1) Quark-gluon plasma (QGP)

(2) Cold Atoms (Fermi 6Li Gas)

Bks

⋅1.0~η

.10 ~ at 6 KT −

.10 at 12 K~T

Bks

⋅2.0~η

Example 2. Entanglement Entropy

～ Hidden information in A [Ryu-TT 06] is the minimal area surface (codim.=2) such that homologous

. ~ and AA AA γγ∂=∂

2dAdS + z

A

)direction. timeomit the (We

BAγ

1dCFT +

off)cut (UV

ε>z

]logTr[S AA Aρρ−=

Holographic Proof of Strong Subadditivity [Headrick-TT 07]

We can easily derive the strong subadditivity, which is known as the most important inequality satisfied by EE.

[Lieb-Ruskai 73]

B A C

BCBACBBA SSSS +≥+⇒ ∪∪∪∪

A B C

= A B C

≥A

B C

CACBBA SSSS +≥+⇒ ∪∪

A B C

= A B C

≥A B C

Tripartite Information [Hayden-Headrick-Maloney 11]

Recently, the holographic entanglement entropy is shown

to have a special property called monogamy.

AdS/CFT predicts that large N gauge theories are monogamous.

):():():( BCAICAIBAISSSSSSS ABCCBAACBCAB

≤+⇔+++≥++

A

B C

≥A

B C

Future Problems

• Proof of AdS/CFT

• Holography for Cosmological spacetimes ?

(e.g. de Sitter space)

• Search of New States of Matter ?

• More universal predictions ?

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