Transcript Dynamical solutions in intersecting brane systems

Dynamical solutions in intersecting
brane systems
Kunihito Uzawa
Osaka City University Advanced Mathematical
Institute
[1] Introduction
・Time dependent solution of Einstein equations in higher
dimensional theory
target
・Analysis of the early universe, higher dimensional BH
String theory, supergravity theory : higher dimension (D>4)
↓
4-dim (compactification)
It is necessary to obtain the dynamical solution of the field equations
(including the Einstein equations) , and to extract information of
the cosmological behavior.
・Dynamics of 4d or internal space, symmetry breaking
◆ Compactification
Expansion (4-dimension)
scale of extra dimension
・Dynamics of extra dimension and moduli stabilization
(Alan Chodos, Steven Detweiler, Phys.Rev.D21:2167,1980.)
(Philip Candelas, Steven Weinberg, Nucl.Phys.B237:397,1984.)
(S. Kachru, R. Kallosh, A. Linde, S. P. Trivedi; Phys.Rev.D68:046005,2003)
V(φ)
Run away potential
φ：moduli
V(φ)
Stabilization of moduli
Field strength, quantum corrections
φ：moduli
◆ KKLT model
(S. Kachru, R. Kallosh, A. Linde, S. P. Trivedi; Phys.Rev.D68:046005,2003)
・ Flux compactification of ten-dimensional type IIB supergravity
・ Flux and quantum corrections ⇒ All moduli are potentially stabilised.
・ Inclusion of anti-D3-brane ― AdS4 ⇒ dS4
⇒ de Sitter space in IIB SUGRA
Calabi-Yau space
☆ Theoretical issues
・ 4-dimensional feature is described by …
→ effective theory in large radius limit
◎ It is NOT exact solution of IIB supergravity.
・ 4-dimensional effective theory : (classical framework)
(S. B. Giddings, S. Kachru, J. Polchinski; Phys.Rev.D66:106006,2002 [arXiv: hep-th/0105097] )
Throat
Cosmological solution in the higher dimensional theory
Higher dimensional gravity
String theory
・ Inflation model
・ Stabilization of the scale
of internal space
・Low energy SUGRA
・Dynamical solution of
Einstein equation
complementary
cosmological model
Construction of the cosmological scenario
Overview
☆ 4 dimensional Gravity
・ Reissner & Nordstrøm solution
(Charged BH )
・ Dynamical solution in Einstein-Maxwell
theory (Kastor & Traschen)
★ Higher dimensional Gravity
・ p-brane solution of SUGRA
(Horowitz & Strominger)
・ Intersecting brane
(Guven, Papadopoulos & Townsend, Ohta)
・ Dynamical p-brane solution
(Gibbons, Binetruy, Kodama,
Sasaki, Uzawa)
・ Dynamical solution of
intersecting brane
・4-dim Maxwell theory:
A charged particles (0-dim) couples to 1-form gauge field.
⇒ 2-form field strength
・The preceding can be generalized to an (p+1)-form gauge field in D-dim
(p+2)-form field strength
⇒A charged objects (p-dim) couples to (p+1)-form gauge field.
・String theory, supergravity theory :
There are anti-symmetric tensor fields of higher rank.
p-(mem)brane
0-brane
1-brane
2-brane
These higher dimensional objects (p-brane) intersects each
other in D-dim.
・moduli stabilization without quantum correction
(O.DeWolfe, A. Giryavets, S. Kachru, W. Taylor, JHEP 0507:066,2005.)
(S. Acharya, F. Benini, R. Valandro, JHEP 0702:018,2007. )
Known :
・single p-brane ⇒ dynamical p-brane
(Gibbons, Lu, Pope or H. Kodama, P.Binetruy, M.Sasaki, K. Uzawa)
・Klebanov & Strassler solution (10d IIB) or heterotic theory
⇒ dynamical case (H. Kodama & K. Uzawa)
・intersecting brane ⇒ dynamical intersecting brane
D4-D8 or D3- D7 brane (P. Binetruy, M.Sasaki, K. Uzawa)
Unknown:
・intersecting p-brane : more general case (R. Argurio, Phys. Lett. B 398 61 )
⇒ dynamical case (K. Maeda, N. Ohta, K. Uzawa)
[2] Dynamical solutoin of p-brane system
(G.W. Gibbons, H. Lu, C.N. Pope Phys.Rev.Lett.94:131602,2005)
(P. Binetruy, M. Sasaki, K. Uzawa, arXiv:0712.3615)
Let us consider the case of an arbitrary p-brane background
The dynamical background of the p-brane can be written by
In the
case, the field equations are reduced to
・ Internal and external space are Ricci flat.
In the
case, the field equations are reduced to
◆ For example, in the case of
the solution is
(G.W. Gibbons, H. Lu, C.N. Pope; Phys.Rev.Lett.94:131602,2005）
(H. Kodama & K. Uzawa ; JHEP 07 (2005) 061)
(P. Binetruy, M.Sasaki, K.Uzawa, arXiv:0712.3615 [hep-th])
: constant parameters
For example,
★ Dynamics of 4-dimensional spacetime (D3-brane case)
◇ Let us consider the case
in more detail.
In this case, the solution for the warp factor
explicitly as
can be obtained
In the following, we consider the simple case
If we introduce a new time coordinate
by
■ metric of ten-dimensional spacetime :
; 4d scale factor is proportional to
☆ Constants
are nonzero ；
★metric of 3-brane in ten dimension
For
, the metric exists inside a domain
； bounded by the level set
Small positive
,
is large.
Large positive
,
is small.
increases, domain
shrinks
Small positive
,
Is large.
Large positive
,
is small.
Universe splits into disconnect regions.
increases
D3-brane
Individual D3-brane
[3] Dynamical solution of intersecting p-brane system
(K. Maeda, N. Ohta, K. Uzawa)
D-dim action
Ansatz for the dynamical background
[3] Dynamical solution of intersecting p-brane system
(K. Maeda, N. Ohta, K. Uzawa)
D-dim action
Ansatz for the dynamical background
Let us assume
Then field equations then reduce to
Above Equations can be satisfied it only if there is only one function
depending on both
and
.
As a special example, we consider the case
In this case, the solution for
where
and
can be obtained explicitly as
are constant parameters.
★For example, M5-M5 intersecting brane
[4] Summary :
★ The solutions we found have the property that they are genuinely higherdimensional in the sense that one can never neglect the dependence on
say of
.
・ The same results hold for other intersecting p-brane model.
☆ Warped structure :
linear combination of the
and
→ 10-dimensional IIA, IIB and 11-dimensional supergravity
・The lower-dimensional effective theory for warped compactification allows
solutions that cannot be obtained from solutions in the original
higher-dimensional theory
★ Further calculations :
・ Construction of the special solution of Einstein equation
→ Classification of dynamical solutions
→ stabilization and application to cosmology
[4] lower dimensional effective theory
★(p+1)-dimensional effective theory (No flux case)
★ Ansatz for background
★ Field equations are reduced to
□ lower-dimensional effective action
・No flux and internal space is Ricci flat space
・Scalar field satisfies the equation of motion.
★ Ansatz for background
□ (p+1)-dimensional field equations
★(p+1)-dimensional effective theory with Flux
◇ D-dimensional model
● Ansatz for background
・Internal space is Einstein space
・ Gauge fields satisfy field equations.
◎ D-dimensional action
□ (p+1)-dimensional effective action
・No flux and internal space is Ricci flat space
Conformal transformation :
★ Field equations
□ (p+1)-dimensional effective action
・No flux and internal space is Einstein space
Conformal transformation :
★ Field equations
☆ p=3, D=10 case, ⇒ 4-dimensional moduli potential (Einstein frame)