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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
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