Takayuki NagashimaTokyo 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

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

1. From the original theory using moduli space approximation2. 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.

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

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

Vortex Dynamics

1. Vortex dynamics from the original theory by moduli space approximation

2. Vortex dynamics from effective theory on domain walls

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 motionon the moduli space

Configuration: [0,2,0]Focus on a configuration which has two domain walls and a pair of vortices in the middle vacuum.

Z0: Relative position of vortices Z0(t)

Energy density in a plane containing vortices with various Z0.

Exact solution in the strong coupling limit.

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.

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).

Vortex Dynamics

1. Vortex dynamics from the original theory by moduli space approximation

2. Vortex dynamics from effective theory on domain walls

Effective Theory on a Domain Wall

Effective theory on a domain wallPosition 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.

Vortices as Lumps or Charged Particles

Vortices as lump solutions or Charged particles in dual.

Example) Single lump at z=z0.

• Logarithmic bending of the domain wall• Phase winding or 1/r Electric field

Vortex as particle with scalar charge and electric charge.

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.

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

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.

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... .