1. Takayuki Nagashima Tokyo Institute of Technology In collaboration with M.Eto (Pisa U.), T.Fujimori (TIT), M.Nitta (Keio U.), K.Ohashi (Cambridge U.) and N.Sakai (Tokyo Woman’s Christian U.). Dynamics of Vortex Strings between Domain Walls

2. Introduction Solitons in the Higgs phase of SUSY gauge theories 1/2 BPS solitons -- Domain walls, Vortices 1/4 BPS solitons -- Networks of domain walls, Vortex strings between domain walls, Monopoles with flux tubes …. Dynamics of 1/2 BPS solitons -- Well known. Dynamics of 1/4 BPS solitons -- Not understood. Today’s topic -- Vortex strings between domain walls. Vortex strings between domain walls Networks of domain walls

3. 1. From the original theory using moduli space approximation 2. From the effective theory on domain walls Dynamics of vortex strings between domain walls From different points of views Contents Coincide each other in some situations.

4. Model and It’s Vacua Model : d=3+1 N=2 SUSY U(1) gauge theory with Nf massive fundamental hypermultiplets with non-zero FI parameters. Domain walls preserve 1/2 SUSY. Zero modes (moduli parameters) are positions and phases of domain walls. Nf discrete vacua BPS equations for domain walls

5. Composite Solitons of Domain Walls and Vortices Vortices break further half SUSY. 1/4 BPS solitons. Zero modes are those of domain walls, and positions of vortices. BPS equations for domain walls and vortices Vortices ending on the domain walls

6. Vortex Dynamics 1.Vortex dynamics from the original theory by moduli space approximation 2.Vortex dynamics from effective theory on domain walls

7. Moduli Space Approximation Give the weak time dependence to moduli parameters. Becomes not a solution of equations of motions. Solve equations of motions up to. Substitute these solutions to the original action and integrate space coordinates. Obtain the non-linear sigma model whose target space is moduli space. Time evolution of moduli parameters (which are related to positions or phases of solitons). Geodesic motion on the moduli space

8. Configuration: [0,2,0] Focus on a configuration which has two domain walls and a pair of vortices in the middle vacuum. Z 0 : Relative position of vorticesZ 0 (t) Energy density in a plane containing vortices with various Z 0. Exact solution in the strong coupling limit.

9. Metric on the Moduli Space Moduli space approximation yields the metric on the moduli space near origin (small distance). Right-angle scattering in head-on collisions. Metric is nearly flat in terms of Z.

10. Metric on the Moduli Space Moduli space approximation yields the metric on the moduli space in asymptotic region (large distance). Tension of the vortex Typical length of the vortices Kinetic energy of two vortices (free motion).

11. Vortex Dynamics 1.Vortex dynamics from the original theory by moduli space approximation 2.Vortex dynamics from effective theory on domain walls

12. Effective Theory on a Domain Wall Effective theory on a domain wall Position and Phase of the domain wall as moduli fields Rescaling and Taking dual of the compact scalar field in d=2+1 We are interested in how vortices ending on the domain wall appear in the effective theory.

13. Vortices as Lumps or Charged Particles Vortices as lump solutions or Charged particles in dual. Example) Single lump at z=z 0. Logarithmic bending of the domain wall Phase winding or 1/r Electric field Vortex as particle with scalar charge and electric charge.

14. Effective Theory on N domain walls N positions and phases of domain walls as moduli fields. Taking dual of phases, it is U(1) gauge theory. Vortex has plus charge on the right domain wall or minus on the left. N We can extend this analysis to the case of multi domain walls.

15. Dynamics of Charged Particles Other particles as sources of scalar fields and electric fields. Well-known for monopoles in d=3+1. Let us focus on a particle positioned at with the velocity

16. Comparison of the Vortex Dynamics [0,2,0] Distances between Domain walls are large enough. Vortices are well-separated in z-plane. Asymptotic metric from dynamics of charged particles.

17. Summary We have investigated the dynamics of vortices between domain walls using the moduli space approximation. Vortices scatter with right-angle in head-on collisions. Asymptotic metric can be understood as kinetic energy of vortices. Vortices can be viewed as charged particles on the effective theory on domain walls. The asymptotic metric can be well reproduced by considering the dynamics of charged particles. Application of this work. Non-Abelian gauge theory on domain walls. Quantization of vortex strings. Similarities and differences from D-branes in string theory....