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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 = 26
D = 10 D = 10
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
BOSONIC STRINGSBOSONIC STRINGS
SUPERSTRINGSSUPERSTRINGS
COMPACTIFICATIONS OF SIX DIMCOMPACTIFICATIONS OF SIX DIM
D = 26D = 26
D = 10D = 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 (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
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
Massive modes
Correction of Newtonian Potential!
3521
4
0
52144 )(
|)0(|kR
G
R
mmGedm
R
G
R
mmGU m
mRD
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
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
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 BreakingSusy 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
2
2
* )( WW
V
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 Wa
a
'
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
) kWa
a
'
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
) kWa
a
'
krera )(
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
RG EQUATION
where
Wg jiji 3)( '
Wa
rag
a
ra j
iji
3'
)(3 ijiji
W
Wg
aa
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)( '
Wa
rag
a
ra j
iji
3'
)(3 ijiji
W
Wg
aa
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)( '
Wa
rag
a
ra j
iji
3'
)(3 ijiji
W
Wg
aa
0)( * i
iii *...)()( **
i
jjj
i
)()( ** i
jjj
i
aa
** 3)(
W
Wg
jiiji
j
Restrictions on W?
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
SPECIAL GEOMETRIES
Thus we find
Assuming perturbation as
)(3
2)( ** WgW ijji
iji
j
2)( *
)2(...)( ij
ji
jji
i
aa
a
c 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 UVAdS5 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 mode
o 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
),...,(
2
1...
2
1
4
1|| 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
),...,(
2
1...
2
1
4
1|| 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
),...,(
2
1...
2
1
4
1|| 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
dxdxg
x
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
)...(3
2 2'2'1
2''N
AeA
),...,(3
1)...(
6
11
2'2'1
22'NN
A VeA
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
i) MINKOWSKI BRANES: 0
2
2
11 3
1
8
1),...,( W
WV
N
i iN
FIRST ORDER EQUATIONS
Wii 2
1' NiW
Wi
i ,...,2,1,
WA3
1'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
FIRST ORDER EQUATIONS
ii) “BENT” BRANES: 0
2
11 )(
3
1)3()(
8
1),...,( ZWZWZWV
N
iiiiiiiN
NiZW iii ,...,2,1;)(2
1
ZWA 3
1'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
FIRST ORDER EQUATIONS
ii) “BENT” BRANES: 0
2
11 )(
3
1)3()(
8
1),...,( ZWZWZWV
N
iiiiiiiN
03
4)(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 )(
3
1)3()(
8
1),...,( ZWZWZWV
N
iiiiiiiN
03
4)(2...
ZZWZZW iiiiiii
NiZW iii ,...,2,1;)(2
1
ZWA 3
1'
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
FIRST ORDER FORMALISM
iii) BETA FUNCTION
ZW
ZW
aa ii
ii
)(
2
3)(
*
2*'
)(
)()()(
2
3)(
ZW
ZWZW
ZW
ZW iiiiiii
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
EXAMPLES
r
A
r
32
32 babW
)(tanh1 2 rbab
0i) 0* W
)(tanh9
1)(secln
9
4 222
22 rab
brabh
bA )(tanh
9
1)(secln
9
4 222
22 rab
brabh
bA 0)( * i
02
9)(
2*'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
EXAMPLES
r
A
r
32
32 babW
)(tanh1 2 rbab
)(tanh9
1)(secln
9
4 222
22 rab
brabh
bA
0* W0i)
0)( * i 02
9)(
2*'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
r
EXAMPLES
ii) Z;0
)sinh(baW
A
r
rbabab
abh
b2222
4
1tanarctan
2
)(cos26ln2
12222
412
2222
rbababab
baA
0)( * i
02
3*)(
2'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
ii)
EXAMPLES
r Z;0
)sinh(baW
A
r
rbabab
abh
b2222
4
1tanarctan
2
)(cos26ln2
12222
412
2222
rbababab
baA
0)( * i 02
3*)(
2'
bi
BRANE SOLUTIONS AND RG FLOWBRANE SOLUTIONS AND RG FLOW
CONCLUSIONS
i) 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