Multidimensional Gravity with p-branes

3 Pages·0.08 MB·English

by  V. D. Ivashchuk

MULTIDIMENSIONAL GRAVITY WITH P-BRANES
V.D.IVASHCHUK and V.N. MELNIKOV
Center for Gravitation and Fundamental Metrology, VNIIMS,
3-1 M. Ulyanovoy Str., Moscow, 117313, Russia
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1 The model
n
a
J We consider the model governed by the action

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1 S = dDz |g|{R[g]−2Λ−h gMN∂ ϕα∂ ϕβ (1)
ZM p αβ M N
1 θ
v − na!exp[2λa(ϕ)](Fa)2g},
1 aX∈∆ a
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0 where g =g dzM ⊗dzN is the metric, ϕ=(ϕα)∈IRl is a vector from dilatonic
MN
1 scalar fields, (h ) is a non-degenerate l×l matrix (l ∈ IN), θ = ±1, Fa = dAa
0 αβ a
is a n -form (n ≥1) on a D-dimensional manifold M, Λ is cosmological constant
8 a a
9 and λa is a 1-form on IRl: λa(ϕ) = λaαϕα, a ∈ ∆, α = 1,...,l. In (1) ∆ is some
/ finite set. In the models with one time all θ =1 when the signature of the metric
c a
q is (−1,+1,...,+1).
- We consider the manifold
r
g
: M =M0×...×Mn (2)
v
i
X with the metric
n
r g =e2γ(x)g0+ e2φi(x)gi, (3)
a
X
i=1
where g0 = g0 (x)dxµ ⊗ dxν is an arbitrary metric with any signature on the
µν
manifoldM andgi =gi (y )dymi⊗dyni isametriconM satisfyingtheequation
0 mini i i i i
R [gi]=ξ gi , (4)
mini i mini
m ,n = 1,...,d ; ξ = const, (thus, (M ,gi) is Einstein space) i = 1,...,n. The
i i i i i
functions γ,φi :M →IR are smooth.
0
EachmanifoldM is assumedto be orientedandconnected, i=1,...,n. Then
i
the volume d -form τ = |gi(y )| dy1 ∧...∧dydi and the signature parameter
i i i i i
ε(i)=signdet(gi )=±1pare correctly defined for all i=1,...,n.
mini
Let Ω be a set of all subsets of I ≡{1,...,n}. For any I ={i ,...,i }∈Ω ,
0 0 1 k 0
i <...<i , we define a form
1 k
τ(I)≡τ ∧...∧τ , (5)
i1 ik
1
of rank d(I) = d +...+d , and a corresponding p-brane submanifold M ≡
i1 ik I
M ×...×M , where p=d(I)−1. We also define ε-symbolε(I)≡ε(i )...ε(i ).
i1 ik 1 k
For I =∅ we put τ(∅)=ε(∅)=1, d(∅)=0.
For fields of forms we adopt the following ”composite electro-magnetic”ansatz
Fa = F(a,e,I)+ F(a,m,J), (6)
I∈XΩa,e J∈XΩa,m
where
F(a,e,I) =dΦ(a,e,I)∧τ(I), (7)
F(a,m,J) =e−2λa(ϕ)∗ dΦ(a,m,J)∧τ(J) , (8)
(cid:16) (cid:17)
a ∈ ∆, I ∈ Ω , J ∈ Ω and Ω ,Ω ⊂ Ω . In (8) ∗ = ∗[g] is the Hodge
a,e a,m a,e a,m 0
operator on (M,g).
For the potentials in (7), (8) we put
Φs =Φs(x), (9)
s∈S, where
S =S ⊔S , S ≡ {a}×{v}×Ω , (10)
e m v a,v
aa∈∆
v =e,m.
For dilatonic scalar fields we put
ϕα =ϕα(x), (11)
α=1,...,l.
¿From(7)and(8)weobtaintherelationsbetweendimensionsofp-braneworld-
sheetsandranksofforms: d(I)=n −1, I ∈Ω ;d(J)=D−n −1, J ∈Ω ,
a a,e a a,m
in electric and magnetic cases respectively.
2 Sigma model representation
We consider the case d =dimM 6=2 and use generalized harmonic gauge
0 0
n
1
γ =γ (φ)= d φi (1)
0 2−d i
0 X
i=1
(d = dimM ). For the model under consideration the equations of motion and
i i
Bianchi identities dFs =0,s∈S, are equivalent to the equations of motion for the
σ-model with certain constraints imposed 3.
The σ-model action reads 3
S = dd0x |g0| R[g0]−G g0 µν∂ φi∂ φj −2V(φ)−L , (2)
σ ZM0 p n ij µ ν o
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where
d d
G =d δ + i j (3)
ij i ij d −2
0
arethe componentsofthe (”purelygravitational”)midisuperspacemetriconIRn 1,
i,j =1,...,n, and
n
1
V =V(φ)=Λe2γ0(φ)− ξ d e−2φi+2γ0(φ) (4)
i i
2
X
i=1
is the potential and
L=h g0 µν∂ ϕα∂ ϕβ + ε exp(−2Us)g0 µν∂ Φs∂ Φs. (5)
αβ µ ν s µ ν
aX∈∆Xs∈S
Here
Us =Us(φ,ϕ)=−χ λ (ϕ)+ d φi, (6)
s as i
iX∈Is
ε =(−ε[g])(1−χs)/2ε(I )θ (7)
s s as
fors=(a ,v ,I )∈S,ε[g]=signdet(g ),andχ =+1,forv =e,andχ =−1,
s s s MN s s s
for v =m.
s
3 Exact solutions
In 3,4 the Majumdar-Papapetrou type solutions were obtained in orthogonal case
(Us,Us′)=0,s6=s′. (For non-composite case see 1,2). These solutions correspond
to Ricci-flat (M ,gi), i=0,...,n, and were generalizedalso to the case of Einstein
i
internal spaces 3. In 5 the Toda-lattice generalizationof the ”orthogonal”intersec-
tion rules was obtained and cosmological and spherically symmetric (classical and
quntum) solutions were considered.
References
1. V.D.IvashchukandV.N.Melnikov,”Intersectingp-BraneSolutionsinMultidimen-
sional Gravity and M-Theory”, hep-th/9612089; Gravitation and Cosmology 2, No
4, 297 (1996).
2. V.D.Ivashchukand V.N.Melnikov, Phys. Lett. B 403, 23 (1997).
3. V.D. Ivashchuk and V.N. Melnikov, ”Sigma-Model for Generalized Composite p-
branes”, hep-th/9705036, Class. and Quant. Grav. 14 (11), 3001-3029 (1997).
4. V.D. Ivashchuk, V.N. Melnikov and M. Rainer, ”Multidimensional Sigma-Models
with Composite Electric Composite p-branes”, gr-qc/9705005.
5. V.D.IvashchukandV.N.Melnikov,”MultidimensionalClassicalandQuantumCos-
mology with Intersecing p-Branes”, hep-th/9708157, to appear J. Math. Phys..
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