2. Introduction to AdS/CFT - 200.145.112.249200.145.112.249/webcast/files/Lecture_2_AdSCFT.pdf ·...

2. Introduction to AdS/CFT

References on Introduction to AdS/CFT: [1] https://arxiv.org/abs/0709.1523v1 ( Chapter 1-3 ) [2] https://arxiv.org/abs/0712.0689 ( Chapters 8 & 9 ) [3] https://arxiv.org/abs/1101.0618 ( Chapter 4 & 5 ) [4] https://arxiv.org/abs/1106.6073 [5] https://arxiv.org/abs/1612.07324 ( Chapter 1 )

Óscar Dias

STAG RESEARCHCENTERSTAG RESEARCH

CENTERSTAG RESEARCHCENTER

Ernest Rutherford

School on AdS/CMT Correspondence, ICTP-SAIFR/IFT-UNESP, Brasil

➙ Large-N expansion of SU(N) ←→ genus (quantum) expansion of string theory

• QCD: gauge theory with gauge group SU(3) but NO expansion parameter (low E)

=> NO perturbative analysis at low E [ for E >>1 => gqcd is small: perturbative QCD ]

• ’t Hooft’s large-Nc limit ( generalisation of QCD ):

replace gauge group by SU(Nc), take limit Nc → ∞ (# colours) & perform an expansion in 1/Nc

• d.o.f.: ~ Nc 2 gluon fields (Αµ)i j, Nc Nf << Nc 2 quark fields qai (Nf # quark flavours).

• When Nc → ∞, dynamics dominated by gluons: Nc Nf ~0<< Nc 2

• Organize expansion in Feymann diagrams (FD):

Figure 2: Double-line notation.

Figure 3: Vertices in double-line notation.

Figure 4: Gluon self-energy diagram in double-line notation.

4




4

q-q-g vertex: gYM3-gluon vertex: gYM

N ! 1, gYM ! 0

so that � = g2YMN finite




4

NcgYM gYM

λ → t’Hoof coupling

Gluon self-energy FD:

Feynman diagrams naturally organise in a double series expansion in powers of 1/N and λ:

• Non-planar diagrams: supressed by 1 / N2 wrt planar (when N → ∞)

Figure 6: A non-planar diagram.

Figure 7: A Riemann surface associated to a planar diagram.

surface is obtained by gluing together these faces along their boundaries as indicated by

the Feynman diagram. In order to obtain a compact surface we add ‘the point at infinity’

to the face associated to the external line in the diagram. This procedure is illustrated

for a planar diagram in fig. 7, and for a non-planar diagram in fig. 8. In the first case

we obtain a sphere, and the same is true for any planar diagram. In the second case we

obtain a torus. It turns out that the power of Nc associated to a given Feynman diagram

is precisely Nχc , where χ is the Euler number of the corresponding Riemann surface. For

a compact, orientable surface of genus g with no boundaries we have χ = 2 − 2g. Thus

for the sphere χ = 2 and for the torus χ = 0. We therefore conclude that the expansion

of any gauge theory amplitude in Feynman diagrams takes the form

A =∞

!

g=0

Nχc

∞!

n=0

cg,nλn , (3)

where cg,n are constants. We recognise the first sum as the loop expansion in Riemann

6

Figure 5: Some planar diagrams.

power of λℓ−1, with ℓ the number of loops. The N2c scaling is not the same for all diagrams,

however. For example, the three-loop diagram in fig. 6 scales as λ2, and is thus suppressed

with respect to those in fig. 5 by a power of 1/N2c ; the reader is invited to draw other

diagrams suppressed by higher powers of 1/N2c . The difference between the diagrams in

fig. 5 and that of fig. 6 is that the former are planar, i.e., can be ‘drawn without crossing

lines’, whereas the latter is not. We thus see that diagrams are classified by their topology,

and that non-planar diagrams are suppressed in the large-Nc limit.

The topological classification of diagrams, which leads to the connection with string

theory, can be made more precise by associating a Riemann surface to each Feynman

diagram, as follows. In double-line notation, each line in a Feynman diagram is a closed

loop that we think of as the boundary of a two-dimensional surface or ‘face’. The Riemann

5













5













5

g# vertices

YM

N#loops . � = g2YM

N

= �1

2

(# vertices)N# loops -1

• Planar diagrams: ‘drawn without crossing lines’ → all scale as N2

String theory has 2 free parameters,

— string length: ls ( α΄= ls2 ) Mass or Tension of the string ~ 1 / ls — string coupling: gs measures interactions between strings

• In order to make contact with string theory first note that:

• We can make contact with string theory: fundamental objects are open and closed strings whose vibration modes are quantised and describe elementary particles including the graviton

• Topological classification of diagrams, leads to connection with string theory:

each Feynman diagram ←→ Riemann surface

Figure 6: A non-planar diagram.

Figure 7: A Riemann surface associated to a planar diagram.

surface is obtained by gluing together these faces along their boundaries as indicated by

the Feynman diagram. In order to obtain a compact surface we add ‘the point at infinity’

to the face associated to the external line in the diagram. This procedure is illustrated

for a planar diagram in fig. 7, and for a non-planar diagram in fig. 8. In the first case

we obtain a sphere, and the same is true for any planar diagram. In the second case we

obtain a torus. It turns out that the power of Nc associated to a given Feynman diagram

is precisely Nχc , where χ is the Euler number of the corresponding Riemann surface. For

a compact, orientable surface of genus g with no boundaries we have χ = 2 − 2g. Thus

for the sphere χ = 2 and for the torus χ = 0. We therefore conclude that the expansion

of any gauge theory amplitude in Feynman diagrams takes the form

A =∞

!

g=0

Nχc

∞!

n=0

cg,nλn , (3)

where cg,n are constants. We recognise the first sum as the loop expansion in Riemann

6

Figure 8: A Riemann surface associated to a non-planar diagram.

surfaces for a closed string theory with coupling constant gs ∼ 1/Nc. Note that the

expansion parameter is therefore 1/N2c . As we will see later, the second sum is associated

to the so-called α′-expansion in the string theory.

The above analysis holds for any gauge theory with Yang-Mills fields and possibly

matter in the adjoint representation, since the latter is described by fields with two colour

indices. In order to illustrate the effect of the inclusion of quarks, or more generally

of matter in the fundamental representation, which is described by fields with only one

colour index, consider the two diagrams in fig. 9. The bottom diagram differs from the

top diagram solely in the fact that a gluon internal loop has been replaced by a quark

loop. This leads to one fewer free colour line and hence to one fewer power of Nc. Since

the flavour of the quark running in the loop must be summed over, it also leads to an

additional power of Nf. Thus we conclude that internal quark loops are suppressed by

powers of Nf/Nc with respect to gluon loops. In terms of the Riemann surface associated

to a Feynman diagram, the replacement of a gluon loop by a quark loop corresponds to the

introduction of a boundary, as illustrated in fig. 10 for the diagrams of fig. 9. The power

of Nc associated to the Feynman diagram is still Nχc , but in the presence of b boundaries

the Euler number is χ = 2 − 2g − b. This means that in the large-Nc expansion (3) we

must also sum over the number of boundaries, and so we now recognise it as an expansion

for a theory with both closed and open strings. The open strings are associated to the

boundaries, and their coupling constant is gop ∼ Nf g1/2s = Nf/Nc.

The main conclusion of this section is therefore that the large-Nc expansion of a gauge

theory can be identified with the genus expansion of a string theory. Through this identi-

fication the planar limit of the gauge theory corresponds to the classical limit of the string

theory. However, the analysis above does not tell us how to construct explicitly the string

dual of a specific gauge theory. We will see that in some cases this can be ‘derived’ by

thinking about the physics of D-branes.

7

Planar diagram ←→ Riemann surface:

sphere with genus g = 0, Euler # χ = 2-2g = 2

Non-planar diagram ←→ Riemann surface:

torus with genus g = 1, Euler # χ = 2-2g = 0 ( g > 0, χ <2 )

�1

2

(# vertices)N# loops -1

= �1

2

(# vertices)N�

⇠ �2

⇠ �2N2

• Expansion of any gauge theory amplitude in FD: A =1X

g=0

1X

n=0

cg,nN��n

g2YM ⌘ (2⇡)p�2gs`p�3s ⌘ �

Np=3) gs ⇠

�

N

Σg: (quantum) loop expansion in Riemann surf. for a closed string theory with coupling const

Σn: (string loop) α΄-expansion (α΄= ls2: string length )

gs ⇠1

N⇠ ‘ ~ ’

( understand later )Later: ↵0 2 ⇠ 1/�

CONCLUSION:

Large-Nc expansion of a gauge theory ←→ genus expansion of a string theory.

In this identification: planar limit of gauge theory ←→ classical limit of string theory

Classical string theory corresponds to:

Low energy Classical string theory if:

Low energy Classical string theory = Supergravity

16⇡G10 ⌘ (2⇡)7g2s`8s

gs ⇠1

N⇠ ‘ ~ ’ ! 0 (to suppress genus (quantum) loop corrections)

`4s = ↵0 2 ⇠ 1

�! 0 (to suppress string loop corrections)

if we identify:

exercise

E`s ! 0 � better later

Pokémon

➙ Type II string theory & Dp-branes

➙ Type II supergravity & p-branes

Low energy Classical string theory = Supergravity

Next, look independently into 2 perspectives:

• Quantum mechanics: dynamics of point particles

• String theory (ST): quantum theory of interacting relativistic 1-dim objects (strings) with tension T

Spp = �m

Zd⌧ = �m

length of worldlinez }| {Z pgabdx

adx

b

➙ Type II string theory & Dp-branes & gauge theory

Sstr = �T

area of world sheetz }| {Zd

2�

pdet g ,

T =1

2⇡↵0 , ↵

0 ⌘ `

2s (fundamental string length)

2-dim g induced on world sheet: g↵� = @↵xa@�x

bgab

Proper time (dτ2 = - ds2) is an invariant

x1

t

x2,3…

xa(σ)

x1

t

x2,3…

σ0,1

xa(σ0,σ1)

• ST also has (non-perturbative) Dp-brane solutions where open strings can end (BCs):

• Quantisation of closed strings in a fixed background:

close string spectrum describes fluctuations of spacetime (graviton)

• Quantisation of open strings ending on Dp-branes (BC):

open string spectrum describes fluctuations of the D-brane

• Single Dp-brane open string spectrum:

ms ⇠ 1/`s

Describe excitations (δ shape, δy)

of Dp along transverse directions

(Goldstone modes )

massless gauge field Aa , 9-p scalars X i + super partners + infinite tower massive fields ( )fermonic

Describe excitations of Dp

along worlvolume

• For classical electrodynamics ( at low E ):

• Action for a Dp-brane:

S = �m

Zd⌧

| {z }Spp

� e

ZAadx

a

| {z }Sint (e-,�)

� 2

4e2

Zd

4xFµ⌫F

µ⌫

| {z }Sbulk

• Dirac-Born-Infeld action for single D3-brane (p=3)

• Expansion around ls → 0:

• When ls → 0 also have:

S = Sbrane| {z }

exc. open strings

+Sinteraction| {z }open-closed

+ Sbulk| {z }

exc. closed str.

SDBI = �TDp

Zd

p+1x e

��q

�det (g↵� + 2⇡`

2sF↵�) + fermions ,

(tension:Q/Vp)z }| {TDp = (2⇡)

�p`

�(p+1)s g

�1s

v = t+f

r) v = const : t %, r &) ingoing null rays

|K|H = Kµ

Kµ

��H = g

µ⌫

K⌫Kµ

��H = 0 . For Schw.: K = Kµ@

x

µ= @

t

v = t+f

r) v = const : t %, r & ) ingoing null rays

u = t� f

r) u = const : t %, r % ) outgoing null rays

ds2 = �✓1 +

r2

L2

◆dt2 +

dr2�1 +

r

2

L

2

�+ r2(d✓2 + sin

2 ✓d�2)

gµ⌫

! egµ⌫

= ⌦

2(r) g

µ⌫

=

L2

r2gµ⌫

des 2=

L2

r2ds2 =

⇥� dt2 + L2(d✓2 + sin

2 ✓d�2)

⇤| {z }

Einstein Static Universe

+

L4

r4+ subleading terms

2 AdS/CFT

16⇡G10 ⌘ (2⇡)7g2s

`8s

! 0 ) Grav. interaction ! 0 ) Sint

! 0, Sbulk

! S

ds2 =L2

z2⇥dz2 +G

ab

(z, x)dxadxb

⇤, . AdS

d+1, odd d

Gab

��z!0

= g(0)ab

(x) + · · ·+ zdg(d)ab

(x) + · · · with hTab

(x)i ⌘ d

16⇡GN

g(d)ab

(x)

• Then partition functions of 2 theories should agree:

ZCFT [�] = Zstring

h�

��@AdS

i. N ! 1, � � 1

ZCFT [�] ' e�Ssugra[�]

� = | |ee', | | ⇠ ⇢1/2C

(Cooper pair charge 2e, mass m), e' is macroscopic SC phase (1)

4

free closed str. in Mink10

g↵� = ⌘↵� +�2⇡`2s

�2@↵X

i@�Xi

e� ⇠ gstransv. def.

exercise

SDBI ! Sbrane = � 1

2⇡gs

Zd4�

✓1

4Fµ⌫F

µ⌫ +1

2@µX

i@µXi + fermions

◆+O(↵0)

• Low-energy Action for stack of N coincident D3-branes:

• When ls → 0 also have:

G10 ! 0 ) �

Newton

=

G10E

r7! 0 , E ! 0 ) low E expansion

• Low-energy Action for single Dp-brane:

^^

Aµ ! (Aµ)IJ , I, J = 1, · · ·N

Xi ! (Xi)IJ & N ⇥N matrices

ms ⇠ r/`s ! 0��r!0

S��Dp

low�E= SDBI| {z }

exc. open str.

+ Sbulk| {z }closed str. in Mink

10

+ O (↵0)| {z }O(G

10

E)

LDBI ! Lbrane = Tr

0

BBB@

@µ!Dµ=@µ+i[Aµ, . ]z }| {� 1

4⇡ gsFµ⌫F

µ⌫ �X

i

DµXiDµXi

+

Vintz }| {⇡ gs

X

i,j

⇥Xi, Xj

⇤2

1

CCCA+ fermions

v = t+f

r) v = const : t %, r &) ingoing null rays

|K|H = Kµ

Kµ

��H = g

µ⌫

K⌫Kµ

��H = 0 . For Schw.: K = Kµ@

x

µ= @

t

v = t+f

r) v = const : t %, r & ) ingoing null rays

u = t� f

r) u = const : t %, r % ) outgoing null rays

ds2 = �✓1 +

r2

L2

◆dt2 +

dr2�1 +

r

2

L

2

�+ r2(d✓2 + sin

2 ✓d�2)

gµ⌫

! egµ⌫

= ⌦

2(r) g

µ⌫

=

L2

r2gµ⌫

des 2=

L2

r2ds2 =

⇥� dt2 + L2(d✓2 + sin

2 ✓d�2)

⇤| {z }

Einstein Static Universe

+

L4

r4+ subleading terms

2 AdS/CFT

16⇡G10 ⌘ (2⇡)7g2s

`8s

! 0 ) Grav. interaction ! 0 ) Sint

! 0, Sbulk

! S

ds2 =L2

z2⇥dz2 +G

ab

(z, x)dxadxb

⇤, . AdS

d+1, odd d

Gab

��z!0

= g(0)ab

(x) + · · ·+ zdg(d)ab

(x) + · · · with hTab

(x)i ⌘ d

16⇡GN

g(d)ab

(x)

• Then partition functions of 2 theories should agree:

ZCFT [�] = Zstring

h�

��@AdS

i. N ! 1, � � 1

ZCFT [�] ' e�Ssugra[�]

� = | |ee', | | ⇠ ⇢1/2C


4

free closed str. in Mink10

• Low-energy Action for stack of N coincident D3-branes:

But Lbrane = LSYM if g2YM ⌘ (2⇡)gs [ g2YM ⌘ (2⇡)p�2gs`p�3s ]

L��Dp

low�E= Tr

0

@� 1

4⇡ gsFµ⌫F

µ⌫ �X

i

DµXiDµXi + ⇡ gs

X

i,j

⇥Xi, Xj

⇤2

1

A+ f.+ Lclstr.Mink10 +O(↵0

)

L��Dp

low�E= Tr

0

@� 1

2 g2YM

Fµ⌫Fµ⌫ �

X

i

DµXiDµXi +

1

2g2YM

X

i,j

⇥Xi, Xj

⇤2

1

A+ f.

| {z }-LSYM

+Lclstr.Mink10 +O(↵0

)

• CONCLUSION: Low-energy Action for stack of N coincident D3-branes

is

SU(N) N=4 Super-Yang-Mills (SYM) theory in 4 dimensions

+ closed strings propagating freely in Minkowski 10

SYM is a non-abelian SU(N) gauge theory (like QCD) but with conformal invariance (i.e. gYM is scale-independent and does not change with E)

Mink10

Pokémon

➙ Type II supergravity & p-branes

• Type II supergravity (with some fields switched-off)

I

(E)II =

1

16⇡G10

Zd10x

p�g

⇣R�1

2@µ�@

µ�� 1

2(p+ 2)!g

12 (p+1)s e

12 (3�p)�(dA(p+1))

2⌘

• Charge of 3-brane solution:

Q3 =V3

G10

Z

S5

?dA(4) =V3⌦5

4⇡G10L4

ds2 = H� 12

⇣�fdt 2 + dx2

k

⌘+H

12

✓dr2

f+ r2d⌦2

S

5

◆, dx2

k =3X

i=1

dx2i

e� = gs

, A(4) = g�1s

�1�H�1

�dt ^ dx1 ^ dx2 ^ dx3,

where H = 1 +L4

r4⇠ 1 + �

Newton

• 3-brane solution ( p = 3):

Figure 12: Spacetime around D3-branes.

Figure 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings.

10

Mink10

L

=> L sets lengthscale of grav. effects:

• r � L ) H ⇠ 1 )⇠ Mink10

• r ⌧ L ) H ⇠ L4

r4) Horizon throat

the other perspective

• Identify 3-brane with D3-brane:

Grav. radius in string units

Q3 ⌘ QD3 () V3⌦5

4⇡G10L4 = N T3 V3 () L4 =

N

4⌦5

T3=···z }| {1

(2⇡)3`(3+1)s gs

16⇡G10z }| {(2⇡)7g2s`

8s

, L4

`4s= 4⇡ gsN

If L � `s , � ⇠ gsN � 1 :

3-brane is massive ) grav. backreacts on spacetime

If L ⌧ `s , � ⇠ gsN ⌧ 1 :

3-brane is ⇠ zero thickness object in ⇠ Mink10

Mink10



10

Mink10

L

Pokémon

➙ Decoupling (low-energy, Maldacena) limit

`s ! 0 with

⇢r

`2s,L

`2s, � ⇠ gsN

�kept fixed

� `sEr : energy (in string units) of string excitation at a fixed r

� energy measured by an Obs1 : E1 =p�g00Er = H(r)�1/4Er

� deep in the throat, r ⌧ L ) `sE1 ⇠ r

L`sEr ! 0

=> interacting closed strings in bottom of throat

ds

2 =

"r

2

L

2

�dt

2 +3X

i=1

dx

2i

!+

L

2

r

2dr

2

#+ L

2d⌦2

5

= ds

2AdS5

+ L

2ds

2S5

• Maldacena lim is a near-horizon lim:

throat geometry decouples from asymptotic region and yields



10

Mink10

L AdS5 ⇥ S5

RAdS5 = � 20

L2, RS5 =

20

L2�! RAdS5⇥S5 = RAdS5 +RS5 = 0

• But arbitrary E string exc. deep in the throat can survive:

ds2@AdS = �dt2 + d⇢2 + ⇢2d'2 = ⇢2✓�dt2 + d⇢2

⇢2+ d'2

◆(26)

L2IR =

3

4L2 (27)

Simp = lim✓⇤!⇡

2

⇡L2IR

2GN

✓Z ✓⇤

0

d✓sin ✓

cos2 ✓

pH(✓)G(✓)�

Z ✓⇤

0

d✓sin ✓

cos2 ✓

◆(28)

S = ⇢s

Zd3x (r✓)2 . (29)

E ⇠ ⇢s

ZdR

1

Rn2 ⇠ ⇢s n

2 log

✓RIR cutoff

Rcore

◆(30)

A' = A(0)' + J' z + . . . , (31)

hJxi = �GxxR Ax with Gxx

R = �iZ +1

�1d4x ei!t�ik·x ✓(t)

⌦Jx(t,x), Jx(0,0)

↵

This is given by Ohm’s law: hJxi = �Ex � � is the AC conductivity

• In the gauge At = 0 one has: Ex = �@tAx

Fourier

GGGGGGGGGGA

transf.

i!Ax ) hJxi = � i!Ax

�(!) = �GxxR (!, 0)

i! � Kubo formula: relation between a transport coe�cient & a Green’s function

• Want low E string exc. as measured by Obs1 ) free closed str. in ⇠Mink10

9

r

`2s= finite|`s!0 ) r ! 0

Mink10



10

Mink10

L AdS5 ⇥ S5

=> free closed str. in Mink10 +

interacting open strings excitations on Dp worlvolume

( N=4 SYM in R1,3 )

➙ Two equivalent descriptions of the system

=> free closed str. in Mink10 +

interacting closed strings excitations in AdS5 x S5

( IIB supergravity in AdS5 x S5)

•Open string perspective, L ⌧ `s , � ⇠ gsN ⌧ 1 :

3-brane is ⇠ zero thickness object in ⇠ Mink10

Two perspectives should be equivalent descriptions of the same physics, and type IIB supergravity on Mink10 is present in both perspectives … so identify 2 other theories!

•Closed string perspective, L � `s , � ⇠ gsN � 1 :

3-brane is massive ) grav. backreacts on spacetime

Mink10

➙ The AdS/CFT conjecture



10

Mink10

LAdS5 ⇥ S5

IIB supergravity in AdS5 × S5 is dual to

N=4 SYM in R1,3

Trivial

match

• SYM is a non-abelian SU(N) gauge theory (like QCD)

but with conformal invariance (i.e. gYM is scale-independent and does not change with E):

hence SYM is a CFT (Conformal Field Theory)

• Relaxing the low energy limit:

N = 4 SYM theory in R1,3 with gauge group SU(N)and YM coupling gYM

is equivalent to type IIB string theory with string length ls and coupling constant gs on AdS5 × S5 with radius of

curvature L and N units of flux F(5) on S5

SYM

➙ Matching parameters ( the Pokémons )

g2YM ⌘ 2⇡gs =�

N( L

��Dp

low�E= LSYM L4

`4s= 4⇡ gsN = 2� ( Q3 ⌘ QD3

Free parameters of field theory {gYM , N} are mapped to free parameters { gs , L/ ls } on the string theory via:

N=4 SYM theory () IIB theory on AdS5 ⇥ S5

Strongest form: any N and � () Quantum string theory: gs

6= 0, `s

/L 6= 0

Strong form: N ! 1, any � () Classical string theory: gs

! 0, `s

/L 6= 0

Weak form: N ! 1, � � 1 () Classical supergravity: gs

! 0, `s

/L ! 0

Table 1: Di↵erent forms of the AdS/CFT correspondence.



6= 0,

`sL

6= 0

� = | |ee', | | ⇠ ⇢1/2C


Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 + 1

2µ0B2

(2)

Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 (3)

Minimize Fs

, �! �+ ��, �⇤ ! �

⇤+ ��⇤

: (4)

Minimize Fs

, �Fs

/�� = 0 : (5)

1

2m|(�ihr� 2eA)�|2 + ↵�+ �|�|2� = 0 (6)

r⇥B = µ0J ) µ0J = �r2A (London gauge:r⇥A = 0) & Minimize F

s

, A ! A+ �A : (7)


s

, �Fs

/�A = 0 : (8)



! 0,

`sL

6= 0

1



6= 0, `s

/L 6= 0


! 0, `s

/L 6= 0


! 0, `s

/L ! 0




6= 0,

`sL

6= 0

� = | |ee', | | ⇠ ⇢1/2C


Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 + 1

2µ0B2

(2)

Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 (3)

Minimize Fs

, �! �+ ��, �⇤ ! �

⇤+ ��⇤

: (4)

Minimize Fs

, �Fs

/�� = 0 : (5)

1

2m|(�ihr� 2eA)�|2 + ↵�+ �|�|2� = 0 (6)


s

, A ! A+ �A : (7)


s

, �Fs

/�A = 0 : (8)



! 0,

`sL

6= 0

1

• gs ⇠1

N⇠ ‘ ~ ’ ! 0

(to suppress genus (quantum) loops)

• L4

`4s⇠ gsN ⇠ � finite

• gYM ! 0 : ( planar lim of SYM )

• � = g2YMN finite

t’Hooft limit

N ! 1

➙ Matching parameters



6= 0, `s

/L 6= 0


! 0, `s

/L 6= 0


! 0, `s

/L ! 0


� = | |ee', | | ⇠ ⇢1/2C


Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 + 1

2µ0B2

(2)

Fs

(r, T ) = Fn

(r, T ) + ↵|�|2 + �

2

|�|4 + 1

2m|(�ihr� 2eA)�|2 (3)

Minimize Fs

, �! �+ ��, �⇤ ! �

⇤+ ��⇤

: (4)

Minimize Fs

, �Fs

/�� = 0 : (5)

1

2m|(�ihr� 2eA)�|2 + ↵�+ �|�|2� = 0 (6)

r⇥B = µ0J ) µ0J = �r2A (London gauge:r⇥A = 0) & Minimize Fs

, A ! A+ �A : (7)

r⇥B = µ0J ) µ0J = �r2A (London gauge:r⇥A = 0) & Minimize Fs

, �Fs

/�A = 0 : (8)

J =

e

m[�

⇤(�ihr� 2eA)�+ c.c.] (9)

1

• gs ⇠1

N⇠ ‘ ~ ’ ! 0


• `4sL4

⇠ 1

�! 0

(to suppress string loops)


• � = g2YMN � 1

) SYM is strongly coupled

t’Hooft limit

N ! 1



! 0,

`sL

! 0

J =

e

m[�

⇤(�ihr� 2eA)�+ c.c.] (9)

S =

Zd

4xp�g

�1

2

Fab

F ab � 2(Da

�)(Da

�)

† � 2V (|�|2)�

(10)

S =

Zd

4xp�g

⇥R +

6

L2� 1

2

Fab

F ab � 2(Da

�)(Da

�)

† � 2V (|�|2)⇤ (11)

S =

Zd

4xp�g

⇥R +

6

L2� 1

2

Fab

F ab � 2(Da

�)(Da

�)

† � 2V (|�|2)⇤ (12)

�|z=0 =

↵

r��+

�

r�++ · · · (13)

V (⌘) = ⌘ µ2

✓1� ⌘ µ2

4V0

◆, ⌘ = ��

†(14)

Sbdry

! Sbdry

�

Zd3xO†O (15)

µ2BF < µ2 < µ2

Unit (16)

� = | |ee', e' = in' (17)

ra

F ab

= gcJb

(18)

2

• gs ⇠1

N⇠ ‘ ~ ’ ! 0


• `4sL4

⇠ 1

�! 0

(to suppress string loops)


• � = g2YMN � 1

) SYM is strongly coupled

t’Hooft limit

N ! 1

➙ Strong / weak coupled duality !

pointlike regime of string theory:

IIB supergravity on weakly curved AdS5 × S5

Strong / weak coupled duality !

➙ Symmetries on both sides do match

Global symmetries ofN = 4 SYM theory in R1,3

Isometries of IIB string theory on AdS5 × S5

• Conf(1, 3): ie conformal group of Mink1,3

Dilatation + Poincare +special conf. transf.

• SO(6)~SU(4): R-symmetry (see review Maldacena)

• Fermionic symmetries: (…)

• SO(2, 4) ~ Conf(1, 3): isometry group of AdS5

• SO(6)~SU(4): isometry group of S5.

• Fermionic symmetries: (…)

• diffeomorphisms = gauge transf. • large gauge transf. that leave the asymptotic form

of the metric invariant: SO(2, 4) × SO(6)

SYM is CFT

=> Invariant to Dilatation:

x

µ ! �x

µ

Also symmetry in grav.:

ds

2AdS5

= r2

L2 ⌘µ⌫dxµdx

⌫ + L2

r2 dr2 invariant if

x

µ ! �x

µ & r ! r�

�(x,⌦) =X

`

�`(x)Y`(⌦)

change M

Pokémon

tattoo

AdS5

@AdS5⇠ Mink1,3

QFT

➙ UV / IR connection. RG flow

ds

2 = ds

2AdS5

+ L

2ds

2S5

=

✓r

2

L

2⌘µ⌫dx

µdx

⌫ +L

2

r

2dr

2

◆+ L

2d⌦2

5 . r =L

2

z

=L

2

z

2

⇣⌘µ⌫dx

µdx

⌫ + dz

2⌘+ L

2d⌦2

5

const-z slice of AdS5 is isometric to Mink1,3. {xµ } coord. QFT.

• z-direction is identified with renormalization group (RG) scale in the gauge theory: [ Wilson… 60’s ]

z ! 1 : Poincare Horizon (IR)

z = 0 : boundaryAdS (UV)

Bulkz ⇠ 1

EYM

RG flow

• SYM: {EYM , dYM} warp factor L2/z2

�! in the bulk: d =

L

zdYM , E =

z

LEYM

• Same E but di↵erent z ) di↵erent EYM .

Fix EL ⌘ 1. SYM process with EYM $ Bulk process localised at: z ⇠ 1

EYM

• UV: EYM ! 1 , z ! 0 , IR: EYM ! 0 , z ! 1

Want properties at length scale z >>ε ? Integrate out short-distance dof a effective theory at length scale z.

Repeat process if interested on z’>>z a effective theory at length scale z’

Defines a renormalization group (RG) flow a continuous family of EFTs in Mink 1,3 labeled by RG scale z.

May visualize continuous family of EFTs in Mink 1,3 as

a single theory in a (4 + 1)-dim spacetime with the RG scale z being a radial coord.

➙ gauge/gravity duality is a geometrization of the RG evolution of a QFT

z ! 1 : Poincare Horizon (IR)

z = 0 : boundaryAdS (UV)

Bulkz ⇠ 1

EYM

RG flow

Computational Methods in Holography

The Gauge-Gravity Duality

The Gauge-Gravity Duality

UVIR

!d!2,1

AdSd

UVIR

z " L ! uz

2

Motivation:

t’Hooft original arguments: the maximum entropy in a regionof space is the area of its boundary, in Planck units.

Weinberg-Witten: graviton can’t be a composite particle.

RG equations for coupling g are local in the energy scale u:

u@ug(u) = �(g(u)) .3 / 22

Bulk

Boundary

Wilson renormalization group (RG): integrating out short distance d.o.f. the coupling constant g of the theory changes under scale transformations according RG flow ODE which is local in the RG scale u.

• That is, given an eqn, β(g)=0, we can always form the Ricci flow eqn:

u ~ (fictitious) flow `time’. Ricci flow is the one-parameter family of couplings g(u) .

Solve it as as a geometric evolution eqn for g (after some initial guess) .

The system evolves towards a fixed point g that solves β(g)=0 ( CFT )

• A QFT is defined by an UV fixed point and a RG flow: SYM is a fixed point living at UV boundary.

• Dilatation is exact isometry of AdS => RG flow is trivial for AdS/SYM.

• For generic asymptotically AdS: Dilatation is only asymptotic isometry => non-trivial RG towards UV SYM

u@g(u)

@u= �(g(u))

(u=r )

2. Introduction to AdS/CFT - 200.145.112.249200.145.112.249/webcast/files/Lecture_2_AdSCFT.pdf ·...

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Transcript of 2. Introduction to AdS/CFT - 200.145.112.249200.145.112.249/webcast/files/Lecture_2_AdSCFT.pdf ·...