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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. t arget. ・ Analysis of the early universe, higher dimensional BH

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Dynamical solutions in intersecting brane systems

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

  2. [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

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

  4. V(φ) φ：moduli φ：moduli ・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) Run away potential V(φ) 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 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

  6. Cosmological solution in the higher dimensional theory String theory ・Low energy SUGRA ・Dynamical solution of Einstein equation Higher dimensional gravity ・ Inflation model ・ Stabilization of the scale of internal space complementary cosmological model Construction of the cosmological scenario

  7. Overview ☆ 4 dimensional Gravity ・ Reissner & Nordstrøm solution (Charged BH ) ・ Dynamical solution in Einstein-Maxwell theory(Kastor & Traschen) ★ Higher dimensional Gravity ・ p-branesolution of SUGRA (Horowitz & Strominger) ・ Dynamical p-brane solution (Gibbons, Binetruy, Kodama, Sasaki, Uzawa) ・ Dynamical solution of intersecting brane ・ Intersecting brane (Guven, Papadopoulos & Townsend, Ohta)

  8. ・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 1-brane 0-brane 2-brane

  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 heterotictheory ⇒ 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. [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-branebackground The dynamical background of the p-branecan be written by In the case, the field equations are reduced to ・ Internal and external space are Ricci flat.

  12. 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

  13. For example,

  14. ★ 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 can be obtained explicitly as 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

  15. ☆ Constants are nonzero ； For , the metric exists inside a domain ； bounded by the level set ★metric of 3-brane in ten dimension Small positive , is large. Large positive , is small. increases, domain shrinks

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

  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 where and 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. ☆ 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

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

  23. □ 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

  24. ★(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

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

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

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

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