Dilatonic quantum multi-brane-worlds

28 Pages·0.25 MB·

by  S. Nojiri

September 2000
Dilatonic quantum multi-brane-worlds.
1 Shin’ichi NOJIRI1, Sergei D. ODINTSOV , 2,
0 ♠♣
0 and K.E. OSETRIN 3
♠
2

n
Department of Applied Physics
a
J National Defence Academy, Hashirimizu Yokosuka 239, JAPAN

5
1
Tomsk State Pedagogical University, 634041 Tomsk,RUSSIA
♠
3
v
9 Instituto de Fisica de la Universidad de Guanajuato
♣
5 Apdo.Postal E-143, 37150 Leon, Gto., MEXICO
0
9
0 ABSTRACT
0
0
/ d5 dilatonic gravity action with surface counterterms motivated by
h
t AdS/CFT correspondence and with contributions of brane quantum CFTs is
-
p considered around AdS-like bulk. The effective equations of motion are con-
e
structed. They admit two (outer and inner) or multi-brane solutions where
h
: brane CFTs may be different. The role of quantum brane CFT is in inducing
v
i of complicated brane dilatonic gravity. For exponential bulk potentials the
X
number of AdS-like bulk spaces is found in analytical form. The correspon-
r
a dent flat or curved (de Sitter or hyperbolic) dilatonic two branes are created,
as a rule, thanks to quantum effects. The observable early Universe may
correspond to inflationary brane. The found dilatonic quantum two brane-
worlds usually contain the naked singularity but in couple explicit examples
the curvature is finite and horizon (corresponding to wormhole-like space)
appears.
PACS number: 04.50.+h; 04.70.-s; 98.80.Cq; 12.10.Kt
1email: [email protected]
2 email: [email protected],[email protected]
3 e-mail: [email protected]
1
1 Introduction
Recent booming activity in the study of brane-worlds is caused by several
reasons. First, gravity on 4d brane embedded in higher dimensional AdS-
like Universe may be localized [1, 2]. Second, the way to resolve the mass
hierarchy problem appears[1]. Third, the new ideas on cosmological constant
problem solution come to game [7, 8]. Very incomplete list of references[3, 4]
(and references therein) mainly on the cosmological aspects of brane-worlds
is growing every day.
The essential element of brane-world models is the presence in the the-
ory of two free parameters (bulk cosmological constant and brane tension, or
brane cosmological constant). The role of brane cosmological constant is to
fix the position of the brane in terms of tension (that is why brane cosmologi-
cal constant and brane tension are almost the same thing). Being completely
consistent andmathematically reasonable, such way ofdoingthings maylook
not completely satisfactory. Indeed, the physical origin (and prediction) of
brane tension in terms of some dynamical mechanism may be required.
The ideology may be different, in the spirit of refs.[6, 5]. One consid-
ers the addition of surface counterterms to the original action on AdS-like
space. These terms are responcible for making the variational procedure
to be well-defined (in Gibbons-Hawking spirit) and for elimination of the
leading divergences of the action. Brane tension is not considered as free
parameter anymore but it is fixed by the condition of finiteness of spacetime
when brane goes to infinity. Of course, leaving the theory in such form would
rule out the possibility of consistent brane-world solutions existance. Fortu-
nately, other parameters contribute to brane tension. If one considers that
there isquantum CFT living onthebrane (which is moreclose tothe spirit of
AdS/CFT correspondence[9] ) then such CFT produces conformal anomaly
(or anomaly induced effective action). This contributes to brane tension.
As a result dynamical mechanism to get brane-world with flat or curved (de
Sitter or Anti-de Sitter) brane appears. The curvature of such 4d Universe is
expressed in terms of some dimensional parameter l which usually appears in
AdS/CFT set-up and of content of quantum brane matter. In other words,
brane-world is the consequence of the fact (verified experimentally by ev-
erybody life) of the presence of matter on the brane! For example, sign of
conformal anomaly terms for usual matter is such that in one-brane case the
de Sitter (ever expanding, inflationary ) Universe is preferrable solution of
2
brane equation4.
The scenario ofrefs.[6,5]may be extended to the presence ofdilaton(s) as
it was donein ref.[10]or to formulation of quantum cosmology in Wheeler-De
Witt form [11]. Then whole scenario looks even more related with AdS/CFT
correspondence as dilatonic gravity naturally follows as bosonic sector of d5
gauged supergravity. Moreover, the extra prize-in form of dynamical deter-
mination of 4d boundary value of dilaton-appears. In ref.[10] the quantum
dilatonic one brane Universe has been presented with possibility to get in-
flationary or hyperbolic or flat brane with dynamical determination of brane
dilaton. The interesting question is related with generalization of such sce-
nario in dilatonic gravity for multi-brane case. This will be the purpose of
present work.
In the next section we present general action of d5 dilatonic gravity with
surface counterterms and quantum brane CFT contribution. This action is
convenient for description of brane-worlds where bulk is AdS-like spacetime.
There could be one or two (flat or curved) branes in the theory. As it was
already mentioned the brane tension is fixed in our approach, instead of it
the effective brane tension is induced by quantum effects. Section three is
devoted to formulation of effective bulk-brane field equations. The explicit
analyticalsolutionofbulk equationfornumber ofexponential bulkpotentials
is presented. The lengthy analysis of 4d brane equations shows the possi-
bility to have two (inner and outer) branes associated with each of above
bulk solutions. It is interesting that quantum created branes can be flat,
or de Sitter (inflationary) or hyperbolic. The role of quantum brane matter
corrections in getting of such branes is extremely important. Nevertheless,
there are few particular cases where such branes appear on classical level, i.e.
without quantum corrections. In section four we briefly describe how to get
generalization of above solutions for quantum dilatonic multi-brane-worlds
with more than two branes. Brief summary of results is given in final sec-
tion where also the study of character of singularities for proposed two-brane
solutions is presented. In most cases, as usually occurs in AdS dilatonic grav-
ity, the solutions contain the naked singularity. However, in couple cases the
scalar curvature is finite and there is horizon. The corresponding 4d branes
may be interpreted as wormhole.
4Similar mechanism for anomaly driven inflation in usual 4d world has been invented
by Starobinsky[15] and generalized for dilaton presence in refs.[17]
3
2 Dilatonic gravity action with brane quan-
tum corrections
LetuspresenttheinitialactionfordilatonicAdSgravityunderconsideration.
The metric of (Euclidean) AdS has the following form:
4
ds2 = dz2 + g dxidxj , g = e2A˜(z)gˆ . (1)
(4)ij (4)ij ij
i,j=1
X
Here gˆ is the metric of the Einstein manifold, which is defined by r = kgˆ ,
ij ij ij
where r is the Ricci tensor constructed with gˆ and k is a constant. One
ij ij
can consider two copies of the regions given by z < z and glue two regions
0
putting a brane at z = z . More generally, one can consider two copies of
0
regions z˜ < z < z and glue the regions putting two branes at z = z˜ and
0 0 0
z = z . Hereafter we call the brane at z = z˜ as “inner” brane and that at
0 0
z = z as “outer” brane.
0
Let us first consider the case with only one brane at z = z and start
0
with Euclidean signature action S which is the sum of the Einstein-Hilbert
action S with kinetic term and potential V(φ) = 12 +Φ(φ) for dilaton φ,
EH l2
the Gibbons-Hawking surface term S , the surface counter term S and the
GH 1
trace anomaly induced action W5:
S = S +S +2S +W, (2)
EH GH 1
1 1 12
S = d5x√g R φ µφ+ +Φ(φ) , (3)
EH 16πG (5) (5) − 2∇µ ∇ l2
Z (cid:18) (cid:19)
1
S = d4x√g nµ, (4)
GH 8πG (4)∇µ
Z
1 6 l
S = d4x√g + Φ(φ) , (5)
1 −16πGl (4) l 4 !
Z
W = b d4x gFA
Z q
+b d4xeeg A 2 22 +R 4R 22 + 2( µR) A
′ µν µ ν µ
( " ∇ ∇ − 3 3 ∇ ∇ #
Z q
e e
5For the introduction toeanomaly induceed efffectfive actionein curvedfspacee-tfime (with
torsion), see section 5.5 in [12]. This anomaly induced action is due to brane CFT living
on the boundary of dilatonic AdS-like space.
4
2
+ G 2R A (6)
− 3
(cid:18) (cid:19) (cid:27)
1 e b +e2(eb+b) d4x g R 6 2A 6( A)( µA) 2
′′ ′ µ
−12 3 − − ∇ ∇
(cid:26) (cid:27)Z q (cid:20) (cid:21)
+C d4x gAφ 22 +2R e e 2eR 22 +f1( µRf) φ .
µν µ ν µ
" ∇ ∇ − 3 3 ∇ ∇ #
Z q
e e
e f f e f e f
e
Here the quantities in the 5 dimensional bulk spacetime are specified by the
suffices and those in the boundary 4 dimensional spacetime are specified
(5)
by . The factor 2 in front of S in (2) is coming from that we have two
(4) 1
bulk regions which are connected with each other by the brane. In (4), nµ is
the unit vector normal to the boundary. In (4), (5) and (6), one chooses the
4 dimensional boundary metric as
g = e2Ag˜ , (7)
(4)µν µν
We should distinguish A and g˜ with A˜(z) and gˆ in (1). We will specify
µν ij
gˆ later in (27). We also specify the quantities given by g˜ by using ˜. G
ij µν
(G˜) and F (F˜) are the Gauss-Bonnet invariant and the square of the Weyl
tensor, which are given as 6
G = R2 4R Rij +R Rijkl,
ij ijkl
−
1
F = R2 2R Rij +R Rijkl , (8)
ij ijkl
3 −
In the effective action (6) induced by brane quantum matter, with N
scalar, N spinor, N vector fields, N (= 0 or 1) gravitons and N higher
1/2 1 2 HD
derivative conformal scalars, b, b and b are
′ ′′
N +6N +12N +611N 8N
b = 1/2 1 2 − HD
120(4π)2
6We use the following curvature conventions:
R = gµνR
µν
R = Rλ
µν µλν
Rλ = Γλ +Γλ Γη Γλ +Γη Γλ
µρν − µρ,ν µν,ρ− µρ νη µν ρη
1
Γη = gην(g +g g ) .
µλ 2 µν,λ λν,µ− µλ,ν
5
N +11N +62N +1411N 28N
b = 1/2 1 2 − HD ,
′
− 360(4π)2
b = 0 . (9)
′′
Usually, b may bechanged by the finiterenormalization of localcounterterm
′′
in gravitational effective action. As it was the case in ref.[10], the term
proportional to b + 2(b+b) in (6), and therefore b , does not contribute
′′ 3 ′ ′′
to the equationsnof motion. Noote that CFT matter induced effective action
may be considered as brane dilatonic gravity.
For typical examples motivated by AdS/CFT correspondence[9] one has:
a) = 4 SU(N) SYM theory
N
C N2 1
b = b = = − , (10)
′
− 4 4(4π)2
b) = 2 Sp(N) theory
N
12N2 +18N 2 12N2 +12N 1
b = − , b = − . (11)
′
24(4π)2 − 24(4π)2
Onecanwrite thecorresponding expression for dilatoncoupled spinor matter
[14] which also has non-trivial (slightly different in form) dilatonic contribu-
tion to CA than in case of holographic conformal anomaly[13] for = 4
N
super Yang-Mills theory.
Let us consider the case where there are two branes at z = z˜ and z = z ,
0 0
adding the action corresponding to the brane at z = z˜ to the action in (2):
0
S = S +S˜ +2S˜ +W˜ , (12)
two branes GH 1
1
S˜ = d4x√g nµ, (13)
GH 8πG (4)∇µ
Z
1 6 l
S˜ = d4x√g + Φ(φ) , (14)
1 16πGl (4) l 4 !
Z
W˜ = ˜b d4x gFA (15)
Z q
+˜b d4x g eAe 2 22 +R 4R 22 + 2( µR) A
′ µν µ ν µ
( " ∇ ∇ − 3 3 ∇ ∇ #
Z q
e e
2 e f f e f e f
+ G 2eR A
− 3
(cid:18) (cid:19) (cid:27)
e
e e
6
1 2 2
˜b + (˜b+˜b) d4x g R 6 2A 6( A)( µA)
′′ ′ µ
− 12 3 − − ∇ ∇
(cid:26) (cid:27)Z q (cid:20) (cid:21)
2 e 2e 2 f1 f
+C˜ d4x gAφ 2 +2R e R 2 + ( µR) φ .
µν µ ν µ
" ∇ ∇ − 3 3 ∇ ∇ #
Z q
e e
e e f f e f e f
We should note that the relative sign of S˜ is different from S . The param-
1 1
eters ˜b, ˜b, ˜b and C˜ correspond to the matter which may be different from
′ ′′
the outer brane one on the inner brane as in (9). Hence, the situation with
different CFTs on the branes may be considered. Having the action at hands
one can study its dynamics.
3 Dilatonic quantum brane-worlds
Let us start the consideration of field equations for two-branes model. First
of all, one defines a new coordinate z by
z = dy f(y), (16)
Z q
and solves y with respect to z. Then the warp factor is e2Aˆ(z,k) = y(z). Here
one assumes the metric of 5 dimensional spacetime as follows:
4
ds2 = g dxµdxν = f(y)dy2+y gˆ (xk)dxidxj. (17)
(5)µν ij
i,j=1
X
Here gˆ is the metric of the 4 dimensional Einstein manifold as in (1). From
ij
the variation over g in the Einstein-Hilbert action (3), we obtain the
(5)µν
following equation in the bulk
1 1 l2
0 = R g R +Φ(φ) g
(5)µν (5)µν (5)µν
− 2 − 2 12 !
1 1
∂ φ∂ φ g gρσ∂ φ∂ φ (18)
−2 µ ν − 2 (5)µν (5) ρ σ
(cid:18) (cid:19)
and from that over dilaton φ
0 = ∂µ √g(5)g(µ5ν)∂νφ +Φ′(φ) . (19)
(cid:16) (cid:17)
7
Assuming that g is given by (17) and φ depends only on y: φ = φ(y), we
(5)µν
find the equations of motion (18) and (19) take the following forms:
2kf 3 1 1 l2 1 dφ 2
0 = + +Φ(φ) f + (20)
y − 2y2 2 12 ! 4 dy!
kf 3 df 1 l2 1 dφ 2
0 = + + +Φ(φ) f (21)
y 4fydy 2 12 ! − 4 dy!
d y2 dφ
0 = +Φ(φ)y2 f . (22)
′
dy √f dy!
q
Eq.(20) corresponds to (µ,ν) = (y,y) in (18) and Eq.(21) to (µ,ν) = (i,j).
The case of (µ,ν) = (y,i) or (i,y) is identically satisfied.
On the other hand, on the (outer) brane, we obtain the following equa-
tions:
48l4 1 l
0 = ∂ A Φ(φ) e4A +b 4∂4A 16∂2A
16πG z − l − 24 ! ′ σ − σ
(cid:16) (cid:17)
4(b+b) ∂4A+2∂2A 6(∂ A)2∂2A
− ′ σ σ − σ σ
(cid:16) (cid:17)
+2C ∂4φ 4∂2φ , (23)
σ − σ
l4(cid:16) (cid:17)l3
0 = e4A∂ φ e4AΦ(φ)
−8πG z − 32πG ′
+C A ∂4φ 4∂2φ +∂4(Aφ) 4∂2(Aφ) . (24)
σ − σ σ − σ
For inner brane, onne g(cid:16)ets (cid:17) o
48l4 1 l
0 = ∂ A Φ(φ) e4A +˜b 4∂4A 16∂2A
−16πG z − l − 24 ! ′ σ − σ
(cid:16) (cid:17)
4(˜b+˜b) ∂4A+2∂2A 6(∂ A)2∂2A
− ′ σ σ − σ σ
+2C˜ ∂4φ(cid:16) 4∂2φ , (cid:17) (25)
σ − σ
l4 (cid:16) l3(cid:17)
0 = e4A∂ φ+ e4AΦ(φ)
z ′
8πG 32πG
+C˜ A ∂4φ 4∂2φ +∂4(Aφ) 4∂2(Aφ) . (26)
σ − σ σ − σ
n (cid:16) (cid:17) o
In(23)and(24),usingthechangeofthecoordinate: dz = √fdy andchoosing
l2e2Aˆ(z,k) = y(z) one uses the form of the metric as
ds2 = dz2 +e2A(z,σ)g˜ dxµdxν , g˜ dxµdxν l2 dσ2 +dΩ2 . (27)
µν µν ≡ 3
(cid:16) (cid:17)
8
Here dΩ2 corresponds to the metric of 3 dimensional unit sphere. Then for
3
the unit sphere (k = 3)
A(z,σ) = Aˆ(z,k = 3) lncoshσ , (28)
−
for the flat Euclidean space (k = 0)
A(z,σ) = Aˆ(z,k = 0)+σ , (29)
and for the unit hyperboloid (k = 3)
−
A(z,σ) = Aˆ(z,k = 3) lnsinhσ . (30)
− −
We now identify A and g˜ in (27) with those in (7). Then we find F˜ = G˜ = 0,
R˜ = 6 etc.
l2
Using (20) and (22), one can delete f from the equations and can ob-
tain an equation that contains only the dilaton field φ (and, of course, bulk
potential):
5k k dφ 2 3 y3 dφ 2 6 1 dφ
0 = y2 + y + Φ(φ)
 2 − 4 dy! 2 − 6 dy! (cid:18)l2 2 (cid:19) dy
y2 2k 6 1  d2φ 3 y2 dφ 2 
+ + + Φ(φ) + Φ(φ) . (31)
′
2 y l2 2 ! dy2 4 − 8 dy! 
 
Our choice for dilaton and bulk potential admitting the analytical solution
is
φ(y) = p ln(p y) (32)
1 2
12
Φ(φ) = +c exp(aφ)+c exp(2aφ) , (33)
−l2 1 2
where a, p , p , c , c are some constants. When p = 1 , Eq.(31) is always
1 2 1 2 1 ±√6
satisfied but from Eq.(22), we find that f(y) identically vanishes. Therefore
we should assume p = 1 . Then we find the following set of exact bulk
1 6 ±√6
solutions
6kp p2 1
case 1 c = 2 1 , c = 0 , a = , p = √6
1 3 2p2 2 −p 1 6 ±
− 1 1
9
3 2p2
f(y) = − 1 (34)
4ky
1
case 2 c = 6kp , a = , p = √3
1 2 1
− ±√3 ∓
3
f(y) = (35)
2c2 4ky
p22 −
1 √3
case 3 c = 3kp , a = , p =
2 2 1
±√3 ∓ 2
21√p
2
f(y) = . (36)
8√y c y +7k√p y
1 2
(cid:16) (cid:17)
We can check that the above solutions satisfy (21).
In the coordinate system in (17), Eq.(24) for an outer brane has the
following form:
y2 y2
0 = 0 ∂ φ 0 Φ(φ )+6Cφ , (37)
−8πG f(y ) y − 32πGl ′ 0 0
0
q
and (25) for inner brane
y˜2 y˜2
0 = 0 ∂ φ+ 0 Φ(φ˜ )+6C˜φ˜ . (38)
8πG f(y˜ ) y 32πGl ′ 0 0
0
q
Here φ (φ˜ ) is the value of the dilaton φ on the outer (inner) brane. We also
0 0
find Eq.(23) for an outer brane has the following form:
3y2 1 1 l
0 = 0 Φ(φ ) +8b (39)
16πG 2y f(y ) − l − 24 0  ′
0 0
 q 
for k = 0 case and
6
3y2 1 1 l
0 = 0 Φ(φ ) (40)
16πG 2y f(y ) − l − 24 0 
0 0
 q 
for k = 0 case. For the inner brane (25) for k = 0 has the form of
6
3y˜2 1 1 l
0 = 0 + + Φ(φ˜ ) +8˜b . (41)
0 ′
−16πG 2y˜ f(y˜ ) l 24 
0 0
 q 
10