Takayuki Nagashima

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Dynamics of Vortex Strings between Domain Walls

• 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 Womans Christian U.).
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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.
• Todays topic -- Vortex strings between domain
  walls.

Networks of domain walls
Vortex strings between domain walls
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Contents
Dynamics of vortex strings between domain walls
From different
points of views
1. From the original theory using moduli space
approximation 2. From the effective theory on
domain walls
Coincide each other in some situations.
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Model and Its Vacua
Model d31 N2 SUSY U(1) gauge theory with Nf
massive fundamental hypermultiplets with non-zero
FI parameters.
Nf discrete vacua

• Domain walls preserve 1/2 SUSY.
• Zero modes (moduli parameters) are
  positions and phases of domain walls.

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Composite Solitons of Domain Walls and Vortices
Vortices ending on the domain walls

• Vortices break further half SUSY.
• 1/4 BPS solitons.
• Zero modes are those of domain walls, and
  positions of vortices.

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Vortex Dynamics

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

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Moduli Space Approximation
Time evolution of moduli parameters (which are
related to positions or phases of solitons).

• 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.

Geodesic motion on the moduli space
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Configuration 0,2,0
Focus on a configuration which has two domain
walls and a pair of vortices in the middle vacuum.
Exact solution in the strong coupling limit.
Energy density in a plane containing vortices
with various Z0.
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Metric on the Moduli Space
Moduli space approximation yields the metric on
the moduli space near origin (small distance).
Metric is nearly flat in terms of Z.
Right-angle scattering in head-on collisions.
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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).
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Vortex Dynamics

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

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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 d21
We are interested in how vortices ending on the
domain wall appear in the effective theory.
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Vortices as Lumps or Charged Particles
Vortices as lump solutions or Charged particles
in dual.

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

Vortex as particle with scalar charge and
electric charge.
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Effective Theory on N domain walls
We can extend this analysis to the case of multi
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
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Dynamics of Charged Particles
Well-known for monopoles in d31.
Other particles as sources of scalar fields and
electric fields.
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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.
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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... .