UNIVERSIDADE FEDERAL DE CAMPINA GRANDE

BRANE SOLUTIONS AND RG FLOW

UNIVERSIDADE FEDERAL DE CAMPINA GRANDE

September 2006

FRANCISCO A. BRITO

BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW

INTRODUCTION

i) Compactification

- Factorizable

- Non-factorizable

(phenomenology d=4)

* Other interests (BTZ black holes, gravity in 2d string theory, and sugra 10 and 11 to lower dimensions > 4)

ii) Dualidade gauge/gravity (e.g. AdS/CFT)

- gravity duals (brane solutions): D - dimensions

- RG flow of a dual field theory: (D-1) - dimensions


BOSONIC STRINGSBOSONIC STRINGS

SUPERSTRINGSSUPERSTRINGS

COMPACTIFICATIONS OF COMPACTIFICATIONS OF SIX DIMSIX DIM

D = 26

D = 10


BOSONIC STRINGSBOSONIC STRINGS

SUPERSTRINGSSUPERSTRINGS

COMPACTIFICATIONS OF SIX DIMCOMPACTIFICATIONS OF SIX DIM

D = 26

D = 10

M10 = M4 X K6“factorizable geometry”

Compact 6-manifold

Our four dim universe


OUR UNIVERSE ON A 3-BRANE

Randall & Sundrum, (1999)

AN ALTERNATIVE TO COMPACTIFICATION

3-BRANEr

NON-COMPACT DIMENSION

M4 ½ AdS5

NON-FACTORIZABLE

“WARPED GEOMETRY”


AdS5 METRIC

, = 0, 1, 2, 3(brane world-volume indices)

e 2A(r) ≡ warp factor

ds52= e2A(r) dx dx + dr2


THE Randall-Sundrum SCENARIO

r

A (r)

r

e 2A (r)

SOLUTION:|5| = 12 k2 = σ2 / 12

A = - k |r|

branebulk

xdrRgxdS 55

5 )(


GRAVITY FLUCTUATIONS

h



h

)()()]([ 22 rmrrVr



H (r) = m2 (r) H = Q+ Q

Q = r + 3 r A(r)_2



SOLUTION:

Zero Mode: m = 0

H (r) = m2 (r) H = Q+ Q

Q = r + 3 z A(r)_2

H o = 0 ) Q o = 0 ) o e 3/2 A(r)



SOLUTION:

Zero Mode: m = 0

H (r) = m2 (r) H = Q+ Q

Q = r + 3 r A(r)_2

H o = 0 ) Q o = 0 ) o e 3/2 A(r)


r

o e -3/2 k |r|

Localization of Gravity!



SOLUTION:

r

Zero Mode: m = 0

Localization of gravity!

H (r) = m2 (r) H = Q+ Q

Q = r + 3 r A(r)_2

H o = 0 ) Q o = 0 ) o e 3/2 A(r)

o e -3/2 k |r|


z

V(z)

Massive modes

z

V(z)


Massive modes


z

V(z)KK modes

Massive modes


z

V(z)

Massive modes


Massive modes

Correction of Newtonian Potential!

3521

40

52144 )(

|)0(|kRG

Rmm

GedmRG

Rmm

GU mmR

D


GRS SCENARIO

Massive gravity: metastable gravity

Gregory, Rubakov & Sibiryakov (2000)

222 )( drdxdxrads

crk

crk

rrae

rrera

c

0)(


GRS SCENARIO Flat brane embeded into 5d Minkowski

bulk: infinite volume!

No zero modes

rc rcσ < 0 σ < 0σ > 0

0

A

r


ASYMMETRIC BRANESBrito & Gomes (work in progress)

2||2222/)3|(|2 )( dredxdxedteds rkiirkrrk

Finite volume massive modes


U (R) ~ 1 / R L

L

log R

1

2

R >> RcR << Rc


LOCALLY LOCALIZED GRAVITY Karch & Randall (2001)

ds2= eA(r) gdx dx + dr2-ds2= eA(r) gdx dx + dr2-

Λ > 0-

Λ = 0-

Λ < 0-

dS4

M4

AdS4

Λ → four dimensional-

cosmological constant

gR


LOCALLY LOCALIZED GRAVITY

r

A (r)

AdS4 (Local localization)


A = -k |r|

M4


r

A (r)




r

A (r)

A = -k |r|

M4

dS4

“No global issues !”

e. g. infinite volume



SCHROEDINGER POTENTIAL

z

V (z) AdS4



z

V (z)

M4

AdS4



z

V (z)

M4

AdS4

dS4



z

V (z) AdS4

Quase-zeromode emerges M4

dS4

(Massive) GRAVITY LOCALIZATION : A LOCAL EFFECT !!


GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY

Brito, Bazeia & Gomes (2004)Λ = L-2 [ σ (T)2 – σ* ]-Λ = L-2 [ σ (T)2 – σ* ]-

4 dim cosmological constant

Brane tension depending on temperature

T

σ


dS4M4AdS4

Susy Breaking

Λ = 0-

Λ < 0- Λ > 0

-

0T*∞ critical temperature

T

GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY


SUPERGRAVITY ACTION

5 dim cosmological constant

→ critical points

W - superpotential

5*2 0)()( WV

; *

FermionsVgRexdS NmMN )(5

22

* )( WWV

0*

W

Cvetic et al (2000)Brito & Cvetic (2001)Bazeia, Brito & Nascimento (2003)


SUPERGRAVITY ACTION

CONTENTS TURNED ON

Supergravity multiplet: (eam, i

m)

Scalar super multiplet:( , i

m)S = 0

im ea

m ;;;

UNDER SUSY TRANSFORMATIONS!!!!UNDER SUSY TRANSFORMATIONS!!!!


THE SUSY FLOW EQUATIONS

= 0

n = 0

ds2= a2 (r) dx dx + dr2

KILLING EQUATIONS

))

(i)’ = ± 3 g i j j W

g i j - metric definied on moduli space

energy scale (AdS/CFT))(22 )( rAera

WrA )(' or Waa

'

Skenderis & Townsend (1999)Freedman et al (1999)Kallosh & Linde (2000)Cvetic & Behrndt (2000)



CRITICAL POINTS

i (r →∞) = i * ) (i)’ = 0

) j W (i* ) = 0

) kWaa

'

krera )(



CRITICAL POINTS

i (r →∞) = i * ) (i)’ = 0

) j W (i* ) = 0

W

*

*Flow

) kWaa

'

krera )(


RG EQUATION

Wg jiji 3)( '

X ara


RG EQUATION

where

Wg jiji 3)( '

Warag

ara j

iji

3'

)(3 ijiji

WW

ga

a

0)( * i

a – energy scalei - couplings

RG EQUATION ON THE FIELD THEORY SIDE


RG EQUATION

where

Wg jiji 3)( '

Warag

ara j

iji

3'

)(3 ijiji

WW

ga

a

0)( * i

iii * ...)()( **

i

jjj

i

)()( **

i

jjj

i

aa

j


RG EQUATION

where

Wg jiji 3)( '

Warag

ara j

iji

3'

)(3 ijiji

WW

ga

a

0)( * i

iii *...)()( **

i

jjj

i

)()( **

i

jjj

i

aa

** 3)(

WW

g jiiji

j

Restrictions on W?


SPECIAL GEOMETRIES

Thus we find

Assuming perturbation as

)(32)( ** WgW ijji

iji

j

2)( *

)2(...)( ij

ji

jji

i

aa

ac ii ; ci = constant


SPECIAL GEOMETRIES

STABLE CRITICAL POINT

i) SUGRA D = 5

Not good for Not good for localizing gravity!localizing gravity!

) UV FIXED POINT (QFT)

QFT on AdS boundary

r

e 2 A ( r)

IR UV AdS5 solution: a (r) = e k r

UNSTABLE IR

> 0 r →∞

a →∞ i → 0 ;0

i

j


SPECIAL GEOMETRIES ii) GRAVITY LOCALIZATION < 0

AdS5 solution:

a (r) = e -k r

i = ci a ||:

“IR FIXED POINT”STABLE CRITICAL POINT r →∞

a → 0 i → 0 ;

0

i

j

r

e 2 A ( r)


SPECIAL GEOMETRIES

STABLE CRITICAL POINT r →∞

a → 0 i → 0 ;

INTRODUCING A BRANE: a (r) = e –k |r|

zero modeo e-k|r|

Two copies of AdS5 pasted

together

LOCALIZATION OF GRAVITY!!(Massless)

r

e 2 A ( r)


FIRST ORDER FORMALISM AND “BENT” BRANES: Freedman et al. (2004)Bazeia et al. (2006)Brito, Bazeia, Losano (work in progress)

NEW DEVELOPMENTSNEW DEVELOPMENTS

),...,(

21...

21

41|| 111

4NNN VRgdrxdS

““fake sugra”fake sugra”




),...,(

21...

21

41|| 111

4NNN VRgdrxdS

“BENT” BRANE GEOMETRIES

2)(225 drdxdxgeds rA

3,2,1,0,




),...,(

21...

21

41|| 111

4NNN VRgdrxdS

“BENT” BRANE GEOMETRIES

3,2,1,0, 2)(225 drdxdxgeds rA

gR

0;)(

0;

0;)(

23

22

21

22

23

22

21

2

23

22

21

22

3 dxdxdxdte

dxdxdxdt

dxdxdxedt

dxdxgx

t


FIRST ORDER FORMALISM AND “BENT” BRANES:


EQUATIONS OF MOTION

NNN

VA

VA

''''

1

'1

'''1 4,...,4

)...(32 2'2'

12''

NAeA

),...,(31)...(

61

12'2'

122'

NNA VeA


FIRST ORDER FORMALISM

i) MINKOWSKI BRANES: 0

22

11 3

181),...,( WWV

N

i iN

FIRST ORDER EQUATIONS

Wii 21' NiWW

ii ,...,2,1,

WA31'




ii) “BENT” BRANES: 0

2

11 )(

31)3()(

81),...,( ZWZWZWV

N

iiiiiiiN

NiZW iii ,...,2,1;)(21

ZWA 31'





2

11 )(

31)3()(

81),...,( ZWZWZWV

N

iiiiiiiN

034

)(2...

ZZWZZW iiiiiii

CONSTRAINTSCONSTRAINTS





2

11 )(

31)3()(

81),...,( ZWZWZWV

N

iiiiiiiN

034

)(2...

ZZWZZW iiiiiii

NiZW iii ,...,2,1;)(21

ZWA 31'



iii) BETA FUNCTION

ZW

ZWa

a iii

i

)(

23

)(

*

2*'

)()()()(

23)(

ZW

ZWZWZW

ZW iiiiiii


EXAMPLES

r

A

r

32

32 babW

)(tanh1 2 rbab

0i) 0* W

)(tanh91)(secln

94 22

22

2 rabb

rabhb

A )(tanh91)(secln

94 22

22

2 rabb

rabhb

A 0)( * i

029)(

2*'

bi


EXAMPLES

r

A

r

32

32 babW

)(tanh1 2 rbab

)(tanh91)(secln

94 22

22

2 rabb

rabhb

A

0* W0i)

0)( * i 029)(

2*'

bi


r

EXAMPLESii) Z;0

)sinh(baW

A

r

rbabababh

b2222

41tanarctan2

)(cos26ln21

2222412

2222

rbababab

baA

0)( * i

023*)(

2'

bi


ii) EXAMPLES

r Z;0

)sinh(baW

A

r

rbabababh

b2222

41tanarctan2

)(cos26ln21

2222412

2222

rbababab

baA

0)( * i 023*)(

2'

bi


CONCLUSIONSi) D=4 is phenomenologically motivated

ii) Infinite volume implies no zero modes

iii) Warp factor regarded as energy scale on dual theory

iv) Bent branes may give a dual gravitational description of RG flows in susy field theories in a curved spacetime

v) Theories in AdS spaces exhibit improved infrared behavior

Th e Th e E n d E n d

UNIVERSIDADE FEDERAL DE CAMPINA GRANDE

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