Dynamical solutions in intersecting brane systems

Kunihito Uzawa

Osaka City University Advanced Mathematical Institute

・ Time dependent solution of Einstein equations in higher dimensional theory

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

・ Analysis of the early universe, higher dimensional BH

String theory, supergravity theory : higher dimension (D>4) ↓ 4-dim (compactification)

[1] Introduction

target

◆ Compactification

Expansion (4-dimension)

scale of extra dimension

φ ：moduli

V(φ)

φ ：moduli

V(φ)

Run away potential

・ 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)

Stabilization of moduliField strength, quantum corrections

◆ 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

☆ 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] )

Calabi-Yau space

Throat

Cosmological solution in the higher dimensional theory

Higher dimensional gravity

・ Inflation model・ Stabilization of the scale of internal space

String theory

・ Low energy SUGRA ・ Dynamical solution of Einstein equation

cosmological model

Construction of the cosmological scenario

complementary

☆ 4 dimensional Gravity

・ p-brane solution of SUGRA (Horowitz & Strominger)

・ Dynamical p-brane solution (Gibbons, Binetruy, Kodama, Sasaki, Uzawa)

・ Dynamical solution in Einstein-Maxwell 　 theory (Kastor & Traschen)

・ Reissner & Nordstrøm solution 　 (Charged BH )

★ Higher dimensional Gravity

・ Intersecting brane(Guven, Papadopoulos & Townsend, Ohta)

Overview

・ Dynamical solution of intersecting brane

p-(mem)brane

・ The preceding can be generalized to an (p+1)-form gauge field in D-dim

1-brane 2-brane0-brane

・ 4-dim Maxwell theory:

A charged particles (0-dim) couples to 1-form gauge field. ⇒ 2-form field strength

(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.

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)

The dynamical background of the p-brane can be written by

Let us consider the case of an arbitrary p-brane background

In the case, the field equations are reduced to

・ Internal and external space are Ricci flat.

[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)

◆ For example, in the case of the solution is 　

In the case, the field equations are reduced to

　 (G.W. Gibbons, H. Lu, C.N. Pope; Phys.Rev.Lett.94:131602,2005 ）

: constant parameters

(H. Kodama & K. Uzawa ; JHEP 07 (2005) 061) (P. Binetruy, M.Sasaki, K.Uzawa, arXiv:0712.3615 [hep-th])

For example,

★ Dynamics of 4-dimensional spacetime (D3-brane case)

◇ Let us consider the case in more detail.

If we introduce a new time coordinate by

■ metric of ten-dimensional spacetime :

In this case, the solution for the warp factor can be obtained explicitly as

; 4d scale factor is proportional to

In the following, we consider the simple case

☆ Constants 　　　 are nonzero ；★metric of 3-brane in ten dimension

For 　 , the metric exists inside a domain

； bounded by the level set

Large positive ,

is large.

is small.

increases, domain shrinks

Small positive ,

Small positive ,

Large positive ,

Is large.

is small.

Universe splits into disconnect regions.

Individual D3-braneD3-brane

increases

[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 can be obtained explicitly as

where and are constant parameters.

★For example, M5-M5 intersecting brane

[4] Summary :

★ The solutions we found have the property that they are genuinely higher- dimensional in the sense that one can never neglect the dependence on say of .・ The same results hold for other intersecting p-brane model.

★ Further calculations : ・ Construction of the special solution of Einstein equation　 → Classification of dynamical solutions → stabilization and application to cosmology

☆ 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

★ Ansatz for background

★(p+1)-dimensional effective theory (No flux case)

★ Field equations are reduced to

[4] lower dimensional effective theory

□ (p+1)-dimensional field equations

□ 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 effective theory with Flux

◇ D-dimensional model

● Ansatz for background

◎ D-dimensional action

・ Internal space is Einstein space

・ Gauge fields satisfy field equations.

□ (p+1)-dimensional effective action

★ Field equations

Conformal transformation :

・ No flux and internal space is Ricci flat space

□ (p+1)-dimensional effective action

★ Field equations

Conformal transformation :

・ No flux and internal space is Einstein space

☆ p=3, D=10 case, ⇒ 　 4-dimensional moduli potential (Einstein frame)