Download - 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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
BOSONIC STRINGSBOSONIC STRINGS
SUPERSTRINGSSUPERSTRINGS
COMPACTIFICATIONS OF COMPACTIFICATIONS OF SIX DIMSIX DIM
D = 26
D = 10
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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”
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
AdS5 METRIC
, = 0, 1, 2, 3(brane world-volume indices)
e 2A(r) ≡ warp factor
ds52= e2A(r) dx dx + dr2
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
THE Randall-Sundrum SCENARIO
r
A (r)
r
e 2A (r)
SOLUTION:|5| = 12 k2 = σ2 / 12
A = - k |r|
branebulk
xdrRgxdS 55
5 )(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
h
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
h
)()()]([ 22 rmrrVr
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
H (r) = m2 (r) H = Q+ Q
Q = r + 3 r A(r)_2
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
r
o e -3/2 k |r|
Localization of Gravity!
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRAVITY FLUCTUATIONS
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|
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
z
V(z)
Massive modes
z
V(z)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
Massive modes
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
z
V(z)KK modes
Massive modes
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
z
V(z)
Massive modes
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
Massive modes
Correction of Newtonian Potential!
3521
40
52144 )(
|)0(|kRG
Rmm
GedmRG
Rmm
GU mmR
D
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRS SCENARIO
Massive gravity: metastable gravity
Gregory, Rubakov & Sibiryakov (2000)
222 )( drdxdxrads
crk
crk
rrae
rrera
c
0)(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRS SCENARIO
Massive gravity: metastable gravity
Gregory, Rubakov & Sibiryakov (2000)
222 )( drdxdxrads
crk
crk
rrae
rrera
c
0)(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
GRS SCENARIO Flat brane embeded into 5d Minkowski
bulk: infinite volume!
No zero modes
rc rcσ < 0 σ < 0σ > 0
0
A
r
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
ASYMMETRIC BRANESBrito & Gomes (work in progress)
2||2222/)3|(|2 )( dredxdxedteds rkiirkrrk
Finite volume massive modes
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
U (R) ~ 1 / R L
L
log R
1
2
R >> RcR << Rc
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
LOCALLY LOCALIZED GRAVITY
r
A (r)
AdS4 (Local localization)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
A = -k |r|
M4
LOCALLY LOCALIZED GRAVITY
r
A (r)
AdS4 (Local localization)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
LOCALLY LOCALIZED GRAVITY
r
A (r)
A = -k |r|
M4
dS4
“No global issues !”
e. g. infinite volume
AdS4 (Local localization)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SCHROEDINGER POTENTIAL
z
V (z) AdS4
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SCHROEDINGER POTENTIAL
z
V (z)
M4
AdS4
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SCHROEDINGER POTENTIAL
z
V (z)
M4
AdS4
dS4
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SCHROEDINGER POTENTIAL
z
V (z) AdS4
Quase-zeromode emerges M4
dS4
(Massive) GRAVITY LOCALIZATION : A LOCAL EFFECT !!
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
dS4M4AdS4
Susy Breaking
Λ = 0-
Λ < 0- Λ > 0
-
0T*∞ critical temperature
T
GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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!!!!
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
THE SUSY FLOW EQUATIONS
CRITICAL POINTS
i (r →∞) = i * ) (i)’ = 0
) j W (i* ) = 0
) kWaa
'
krera )(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
THE SUSY FLOW EQUATIONS
CRITICAL POINTS
i (r →∞) = i * ) (i)’ = 0
) j W (i* ) = 0
W
*
*Flow
) kWaa
'
krera )(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
RG EQUATION
Wg jiji 3)( '
X ara
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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?
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SPECIAL GEOMETRIES
Thus we find
Assuming perturbation as
)(32)( ** WgW ijji
iji
j
2)( *
)2(...)( ij
ji
jji
i
aa
ac ii ; ci = constant
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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)
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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”
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
“BENT” BRANE GEOMETRIES
2)(225 drdxdxgeds rA
3,2,1,0,
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
“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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM AND “BENT” BRANES:
NEW DEVELOPMENTSNEW DEVELOPMENTS
EQUATIONS OF MOTION
NNN
VA
VA
''''
1
'1
'''1 4,...,4
)...(32 2'2'
12''
NAeA
),...,(31)...(
61
12'2'
122'
NNA VeA
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
FIRST ORDER EQUATIONS
ii) “BENT” BRANES: 0
2
11 )(
31)3()(
81),...,( ZWZWZWV
N
iiiiiiiN
NiZW iii ,...,2,1;)(21
ZWA 31'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
FIRST ORDER EQUATIONS
ii) “BENT” BRANES: 0
2
11 )(
31)3()(
81),...,( ZWZWZWV
N
iiiiiiiN
034
)(2...
ZZWZZW iiiiiii
CONSTRAINTSCONSTRAINTS
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
FIRST ORDER EQUATIONS
ii) “BENT” BRANES: 0
2
11 )(
31)3()(
81),...,( ZWZWZWV
N
iiiiiiiN
034
)(2...
ZZWZZW iiiiiii
NiZW iii ,...,2,1;)(21
ZWA 31'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
iii) BETA FUNCTION
ZW
ZWa
a iii
i
)(
23
)(
*
2*'
)()()()(
23)(
ZW
ZWZWZW
ZW iiiiiii
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
r
EXAMPLESii) Z;0
)sinh(baW
A
r
rbabababh
b2222
41tanarctan2
)(cos26ln21
2222412
2222
rbababab
baA
0)( * i
023*)(
2'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
ii) EXAMPLES
r Z;0
)sinh(baW
A
r
rbabababh
b2222
41tanarctan2
)(cos26ln21
2222412
2222
rbababab
baA
0)( * i 023*)(
2'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
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