1. Dynamical solutions in intersecting brane systems Kunihito Uzawa Osaka City University Advanced Mathematical Institute

2. ・ 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

3. ◆ Compactification Expansion (4-dimension) scale of extra dimension

4. 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 moduli Field strength, quantum corrections

5. ◆ 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

6. 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

7. ☆ 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

8. 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.

9. 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. )

10. 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)

11. 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)

12. ◆ 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])

13. For example,

14. ★ 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

15. ☆ 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,

16. Large positive, Is large. is small. Universe splits into disconnect regions. Individual D3-brane D3-brane increases

17. [3] Dynamical solution of intersecting p-brane system (K. Maeda, N. Ohta, K. Uzawa) D-dim action Ansatz for the dynamical background

18. [3] Dynamical solution of intersecting p-brane system (K. Maeda, N. Ohta, K. Uzawa) D-dim action Ansatz for the dynamical background

19. 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.

20. As a special example, we consider the case In this case, the solution for can be obtained explicitly as whereand are constant parameters. ★ For example, M5-M5 intersecting brane

21. [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

22. ★ Ansatz for background ★ (p+1)-dimensional effective theory (No flux case) ★ Field equations are reduced to [4] lower dimensional effective theory

23. □ (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

24. ★ (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.

25. □ (p+1)-dimensional effective action ★ Field equations Conformal transformation : ・ No flux and internal space is Ricci flat space

26. □ (p+1)-dimensional effective action ★ Field equations Conformal transformation : ・ No flux and internal space is Einstein space

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