Free Yang-Mills 近似を用いた AdS/CFT 対応の解析

(An analysis of AdS/CFT using the free-field approximation)

高柳　匡 ( 京大理学部 )

　 Tadashi Takayanagi, Kyoto U.

based on hep-th/0611035 [JHEP 0701 (2007) 090]

hep-th/0702194

with 西岡 辰磨 (Tatsuma Nishioka)

KEK 理論研究会 2007

① 　 Introduction

AdS/CFT has been played a crucial role to understand the

non-perturbative properties of gauge and gravity theory.

This is owing to its S-duality nature:

IIB String on AdS5×S5 4D N=4 SU(N) YM

near horizon of N D3-branes SCFT

R: AdS radius λ=NgYM2

[J.Maldacena 98’]

4/1

'

R

In the large radius limit, we find IIB SUGRA Strongly coupled N=4 SYM

Easy ! An interesting theory

In the small radius limit, we obtain

IIB string Free N=4 SYM at the zero radius　　　　　　　　　　　　　　　　　　　　　　 What is this? (Unknown…) Very easy !

Setup: AdS/CFT correspondence in Poincare Coordinate

2dAdS

)Coordinate Poincare(AdS 2dNRM

on CFT

t

1d

-1energy)(~z

off)cut (UV az

1z IR UV

.2

211

20

222

z

dxdxdzRds i

di

To check the AdS/CFT duality, we need to compare the

supergravity (or semi-classical string) results with those of

strongly coupled Yang-Mills.

Recently this has been achieved for several quantities in

N=4 SYM by using the spin-chain descriptions.

However, if we want to consider AdS/CFT(QFT) for more

general examples without conformal invariance and SUSY,

we cannot resort such a `integrable’ method.

Now we would like to assume the zero-th order

approximation and to compare the following two theories.

IIB SUGRA Free Yang-Mills

in various backgrounds

Naively, this crude approximation does not seem to work.

However, there are several physical quantities which

we can compare semi-quantitatively between these two

theories.

A famous example will be the (thermodynamical) entropy.

We can compute the entropy in free N=4 SYM by counting

the number of bosons and fermions

In the gravity dual description, it is given by the

Bekenstein-Hawking entropy of AdS Schwarzschild BH

.3

2 322

VTNS free

.2

322

VTNSS AdSstrong

Therefore we find the semi-quantitative agreement [Gubser-Klebanov-Peet 96’]

up to 30 %.

In this talk we would like to check the AdS/CFT in various

backgrounds by confirming such a semi-quantitative

agreement.

Our Examples: AdS bubbles, Sasaki-Einstein Mfds.

....33.13

4

strong

free

S

S

② 　 AdS bubbles and Closed String Tachyon

(2-1) AdS bubbles

Compactify a space coordinate xi in AdS space and impose

the anti-periodic boundary condition for fermions.

Closed string tachyons in IR region

)Coordinate Poincare(AdS 2d

-1energy)(~z IR UV

Anti-periodic

Nishioka-Takayanagi hep-th/0611035

The end point of closed string tachyon condensation is conjectured to be the AdS bubbles (AdS solitons).

[Horowitz-Silverstein 06’]

Closed string

tachyon

Capped off !

2dAdS

z IR UV

Anti-Periodic

2dAdS

z IR UVAnti-Periodic

Explicit metric

AdS Schwarz-Schild (⇔ Finite temperature SYM)

AdS Bubble (AdS Soliton):

.1)(

),)(()(

4

0

22

21

222

2

2

222

r

rrf

dxdxddtrfR

r

rfr

drRds

Double Wick Rotationitit ,

).)(()(

22

21

222

2

2

222 dxdxdtdrf

R

r

rfr

drRds

χｒ

r=r0

)~ :ty(periodici L

The dual gauge theory is the 4D N=4 SYM compactified on a circle with the anti-periodic boundary condition for all fermions (i.e. thermal circle).

The supersymmetry is completely broken. Also there exists a mass gap in the IR. ~ a confinement from the viewpoint of 3D SYM. [Witten 98’]

Since this system is at zero temperature, the thermal entropy is zero. So we compare the following quantities: (i) Casimiar Energy = ADM energy [Horowitz-Myers 98]

(ii) Entanglement Entropy = Area of minimal surface [Ryu-Takayanagi 06]

(i) Casimir Eenergy

Free SYM side

Gravity side

Thus we again find

.6

720

78

908

4

22

4

22

4

22

00

L

N

LN

LNT

fermionsbosons

free

.816

4

22

4

33

00 L

N

LG

RπT

N

ADM

. 3

4

00

00

00

00 ADM

free

strong

free

T

T

T

T

Summing over KK modes

Free field computation of Casimir Energy

Ex. a massless scalar

.90)(

2

2

1

2

1

,)'()'()'()'(

1

4

1)'()(

4

2

04

220

22222

00 LnLT

ttnLzzyyxxxx

Zni

Zn

(ii) Entanglement Entropy

Free SYM side

We divide the space into two parts A and B. Then the

the total Hilbert space becomes factorized

We define the reduced density matrix for A by

taking trace over the Hilbert space of B .

Now the entanglement entropy is defined by the

von Neumann entropy

. BAtot HHH

A

, Tr totBA

AS

. log Tr AAAAS

A B

In the simplest case of the division by a straight line,

we can calculate the entropy exactly.

Consider the quantity

This is the same as the partition function on

We analytically continue the integer n to n=1/N<1.

Then we get the orbifold

A B

),( 2x

1x. )(ρ Tr A n

A

),(

2

),(

2

21

] of covers [xxt

RRn

E

),(

2

),(

N

21

ZCx

xt

R

E

The entanglement entropy is found to be

Note: this is essentially the same as the open string vacuum amplitude for a fractional D2-brane on C/ZN.

. 2)4(3

Zlog

Zlog)/1(

|)( trlog

42/1

1

/

1

22

2

Zk

s

kL

a d

N

CZC

nn

AAA

es

L

s

dsV

NN

nS

N

After the summation of KK modes, we obtain

We find from the area law formula

Thus we can conclude that

. 6

12

)()( L

VNSSS SUSYfree

AnonSUSYfree

Afree

A

Gravity side

. 44G

Area 12

N

L

VNS AdS

A

. 3

2

S

SAdSA

freeA

Holographic Computation

(1) Divide the space N is into A and B. (2) Extend their boundary to the entire AdS space. This defines a d dimensional surface. (3) Pick up a minimal area surface and call this .

(4) The E.E. is given by naively applying the Bekenstein-Hawking formula

as if were an event horizon.

A

A

.4

)Area()2(

A

dN

A GS

A

[Ryu-Takayanagi 06’]

Explicit proof via GKP-Witten relation: Fursaev 06’

)Coordinate Poincare(AdS 2d

N

z

B

A

A Surface Minimal

)direction. timeomit the (We

]98' Maldacena Yee,-[Reyn computatio loop Wilson cf.

In our case, the minimal surface looks like

r

),( 2x

r=r0

It is clear that the entropy decreases compared with the

supersymmetric AdS5 background.

Our conjecture

The entanglement entropy always decreases

under the closed string tachyon condensation.

Note: We neglect the radiations produced during

the tachyon condensation as we do so

for the Sen’s conjecture about the open string

tachyon condensation.

(2-2) Twisted AdS bubblesWe would like to generalize the above discussions to the twisted AdS bubbles, dual to the N=4 4D Yang-Mills with twisted boundary conditions:

Supersymmetries are broken except ζ ＝ 1.

The dual metric can be obtained from the double Wick rotation of the rotating 3-brane solution.

When ζ ＝ 0, the background becomes the AdS bubble. At ζ ＝ 1, it coincides with the supersymmetric AdS5.

).(e)( :fermions

),(eL)( :bosonsi

i2

L

The metric of the twisted AdS bubble

The smoothness of the metric at the tip of the bubble

requires the twisted periodicity

　　　　　　 We will again have a closed string tachyon

from the string wound around the twisted circle.

Closed string tachyon condensation

on a twisted circle (or Melvin background). [Review: Headrick-Minwalla-Takayanagi 04’]

).2,(~),( L

The result of Casimir Energy=ADM mass

Energy

Twist parameter

3

4

AdS

freeYM

E

E

8

9

AdS

freeYM

E

E

SUSY

Free Yang-Mills

AdS gravity

Cf. Y.Hikida hep-th/0610119: C2/ZN

The result of the entanglement entropy

This is a new quantitative evidence of AdS/CFT in a slightly susy breaking background.

Twist parameter

Entropy

Free Yang-Mills

AdS side (Strongly coupled YM)

Supersymmetric Point

3

2

AdSA

freeA

S

S

③ 　 Free Fields vs. Sasaki-Einstein

As we have seen, the free Yang-Mills approximation

of the entropy and Casimir energy to the SYM

semi-qualitatively agrees with the AdS gravity results.

This suggests that the degree of freedom of free Yang-Mills

is not so different from that of strongly coupled Yang-Mills.

[A comment]

This semi-quantitative agreements are very non-trivial and

may be special to QFTs which have their AdS duals.

We would like to test this speculation for infinitely many

examples of N=1 SCFTs which are dual to toric Sasaki-

Einstein manifolds X5.

[Examples of X5]

(i) [Klebanov-Witten 98’]

(ii) [Gauntlett-Martelli-Sparks-Waldram 04’]

(iii) [Cvetic-Lu-Page-Pope 05’]

Below we will assume X5 is a toric manifold.

)Einstein-Sasaki(XAdSon IIB SCFT 1N 4D 55

1,1TqpY ,

rqpL ,,Infinitely Many examples

We compare the thermal entropy in free Yang-Mills with the one in the strongly coupled YM.

The latter can be found as the black hole entropy

where a is the central charge of 4D N=1 SCFT.

It is related to the volume of the dual Sasaki-Einstein mfdvia

,24G

AreaHorizon 32

N

aVTSstrong

,)Vol(X4 5

32

Na

Now we define the ratio of the entropy

This index f can be found purely from the toric data of the (CY3 cone over) Sasaki-Einstein mfds, employing the Z-minimization [Martelli-Sparks-Yau] method.

Properties (1) f=1 for any orbifolds of C3

(2) f remains the same after orbifolding

X5 X5/Zn

. bosons)#(

, 3

4)Vol(X

43

43

52

B

B

strong

free

N

fN

N

S

S

Explicit Example (1):Ypq

X=q/p

f(x)

8/9 < f < 1.025

T1,1

S5/Z2

Explicit Example (2):Lpqr

x=p/q

y=r/qf(x,y)

8/9 < f < 1.025

Note the constraints: 0<x<y<1 y<(x+1)/2

Other examples

(3) Xpq: [Hanany-Kazakopoulos-Wecht 05’]

(4) Zpq: [Oota-Yasui 06’]

(5) Symmetric Pentagon

(6) Regular Polygon

8/9 < f < 1.037

8/9 < f < 1.048

8/9 < f < 1.032

,097.19

sin9

)(2

22

n

nnf

8/9 < f < 1.097 .

Del Pezzo Surfaces etc.

In this way, the ratio f takes values within a narrow range.

In other words, we can say that the N=1 SCFTs (quiver gauge theories) which have AdS duals are rather specialkinds of super Yang-Mills:

The degrees of freedom do not depend

on the coupling constant so much.

Notice also that 4f/3 is always greater than 1, which means the strongly coupled SYM has a smaller degree of freedom.

?2.1 )9/8( 889.0 f

④ 　 Conclusions and Discussions

In various example, we have seen that several physical

quantities agree semi-quantitatively between free Yang-

Mills and IIB supergravity.

It has not been known when a given CFT has its AdS dual. Therefore it would be useful to examine many examples even statistically and see if there exist any common physical properties.

It would be very nice if the narrow range of the index f offers us a criterion of the existence of AdS dual.