1. Localization of gravity on Higgs vortices with B. de Carlos Jesús M. Moreno IFT Madrid Hanoi, August 7th hep-th/0405144

2. Topological defects & extra dimensions The Higgs global string in D=6 Numerical solutions Weak and strong gravity limits A BPS system Conclusions Planning

3. d=5 domain wall d=6 vortex d=7 monopole, d=8 instanton the internal space of a topological defect living in a higher dimensional space-time Rubakov & Shaposhnikov ´83 Akama ´83 Visser ´85 Our D=4 world: Topological defects & extra dimensions

4. Solitons in string theory (D-branes): ideal candidates for localizing gauge and matter fields Polchinski ´95 REVIVAL: Gravity localized in a 3-brane DW in D=5 Graviton´s 0-mode reproduces Newton’s gravity on the brane Corrections from the bulk under control Need L bulk < 0 to balance positive tension on the brane Randall and Sundrum ´99 Topological defects & extra dimensions

5. Gravitational field in D=4  domain walls: regular, non static gravitational field (or non-static DW in a static Minkowski space-time) Vilenkin ´83 Ipser & Sikivie ‘84  strings: singular metric outside the core of the defect Cohen & Kaplan’88 Gregory ‘96  monopoles: static, well defined metric V x 0 Static DW, regular strings … (e.g. SUGRA models) Barriola & Vilenkin’89 Cvetic et al. 93 …. (non singular when we add time-dependence) V r 0 Topological defects

6.  Compact transverse space (trapped magnetic flux, N vortices) Sundrum ’99, Chodos and Poppitz ’00 Local string/vortex  Non-compact transverse space: local string (Abelian Higgs model) Gherghetta & Shaposhnikov ´00 Gherghetta, Meyer & Shaposhnikov ´01 Cohen & Kaplan ‘99 previous work: Wetterich’85 Gibbons & Wiltshire ‘87 Global string  Plain generalization to D=6 still singular  However, introducing L< 0 cures the singularity. Analytic arguments show that, in this case, there should be a non-singular solution Gregory ’00 Gregory & Santos ‘02 The string in D= 6

7. |f|  Matter lagrangian:  Global U(1) symmetry  Let us analyze this system in D=6 space-time The global string in D= 6

8. The action for the D=6 system is given by Metric: preserving covariance in D=4 compatible with the symmetries coordinates of the transverse space M(r), L(r) warp factors and we parametrize The global string in D= 6

9.  Equations  Gravity trapping The global string in D= 6

10. Equations  mn  qq  rr  eom (constraint) The global string in D= 6

11. F(0) = 0 L(0) = 0 L’(0) = 1 (no deficit angle) ( F(r) = f 1 r)

12. The global string in D= 6 QUESTION: Is it possible to match BOTH regions having a regular solution that confines gravity? ANSWER: YES! but for every value of v there is a unique value of L that provides such solution

13. Numerical method Initial guess ( 5 x N variables) RELAXATION ODE finite-difference equations (mesh of points) IterationImprovement The global string in D= 6

14. Boundary conditions F(0) = 0 L(0) = 0 m(0) = 0 F’(0) = 0 L’(0) = 0 In general, there will be an angle deficit L’(0) = 1 L = L c The global string in D= 6

15. Numerical solutions Scalar-field profile M 6 = V = 0.99577 V V (no l dependence) L = - 0.0671 V 6 Coincides with the calculated value

16. Numerical solutions Cigar-like space-time metric Asymptotically AdS 5 x S 1 Olasagasti & Vilenkin´00 De Carlos & J.M. ‘03

17. Numerical solutions Dependence on the Higgs scale

18. Numerical solutions Uniqueness of the solution: phase space Gregory ’00 Gregory & Santos ‘02 In the asymptotic region (far from the Higgs core) autonomous dynamical system

19. Numerical solutions Flowing towards difficult because is next to a repellor (AdS 6 ) Only one trajectory, corresponding to L c, ends up in which can be matched to a regular solution near the core 4 fixed points

20. Numerical solutions Plot + fit for small v values We find a good fit Gregory´s estimate (v a M 6 ) Numerical difficulties to explore the small v region

21. Numerical solutions -V(0) < L c < 0 Super heavy limit: ( v p M 6 )

22. Numerical solutions Region explored by the Higgs field in the super heavy limit

23. Numerical solutions Is it possible to generate a large hierarchy between M 6 and the D=4 Planck mass ? From the numerical solutions : the hierarchy is d a few orders of magnitue (e.g. 1000 for v = 0.7) (increases for smaller v values) Gregory ’00 Problem: fine tuning stability under radiative corrections

24. A BPS system Solving second order diff. eq. can be very hard and does not give analytical insight Is it possible to define a subsystem of first order (BPS-like) differential eqs. within the second order one? Carroll, Hellerman &Trodden ‘99

25. A BPS system BPS equations

26. A BPS system EXAMPLE No cosmological constant De Carlos & J.M. ’03

27. A BPS system E total = E grav + E kin + E pot 0

28. Conclusions We have analyzed the Higgs global string in a D=6 space time with a negative bulk L c trapping gravity solutions For every value of v there is a unique value of L c that that provides a regular solution. The critical cosmological constant is bounded by -V(0) < L c < 0 It is difficult to get a hierarchy between M 6 and M Planck Fine tuning, stability