Effective Action of Domain Wall Networks

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Effective Action of Domain Wall Networks

• Takayuki Nagashima

Based on works in collaboration with M.Eto,
T.Fujimori, M.Nitta, K.Ohashi and N.Sakai
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Domain Wall Networks

• Networks of topological defects appear in various
  area of physics such as cosmology and condensed
  matter physics.
• Dynamics of these networks have been examined by
  numerical simulations so far because analytic
  solutions were lacking.
• Recently exact solutions of domain wall networks
  have been found in a supersymmetric gauge theory
  (2005 Eto at al.).

Examples of domain wall networks
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Introduction
Zero modes which are related to the sizes of the
loops are only possible normalizable modes.
Triangle loop
Plot of the energy density
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Model and Its Vacua
Model d31 8 SUSY U(Nc) gauge theory with Nf
fundamental hypermultiplets.
Vacua
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BPS Equations for Domain Wall Networks
All the moduli parameters
V-equivalence relation
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Single Triangle Loop
U(1) Nf4 model U(3) Nf4 model
Complex masses for the hypermultiplets
Shape of the domain wall webs
Vertex Edge Face
Face (vacuum) Edge (wall) Vertex (junction)
Dual diagram
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Moduli Parameters
We promote this normalizable mode to field.
We have to fix these modes.
When we consider the effective action
Normalizable mode
Non-normalizable mode
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Effective Action of Domain Wall Networks
It is well known that the effective action of BPS
solitons is obtained from the kinetic term of the
original Lagrangian.
Although the master equation is difficult to
solve, we can develop calculations to obtain the
general formula of the effective action without
using the solution of the master equation.
Substitute the explicit solution of the master
equation into this formula and perform the
integral.
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Explicit Solution
Strong coupling limit
Neglect the second term Remove the
moduli-independent divergence
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Kahler Potential in the Case of Small Loop
As long as
Scalar curvature
This is finite.
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In order to Know the Asymptotic Metric
Although the series have convergence region, we
can rewrite it as a sum of hypergeometric
functions. Therefore we can know the behaviour of
the function outside the convergence region using
the analytic continuation.
However we have not found the expansion of the
hypergeometric functions at large in any
literature.
Here we propose the another simple approach to
know the asymptotic (large loop) metric which we
call tropical limit.
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Asymptotic Metric (Large Loop)
Tropical limit
Pick up only the largest term in each region
Remove the moduli-independent divergence
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Tropical Limit and Vacuum
The configuration is SUSY vacuum in each region.
Independent of coupling constant
Vacuum configuration
It is expected that the tropical limit extracts
the leading contribution which is independent of
coupling constant.
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Another Contribution
We use this vacuum expectation value in each
region to evaluate the second term without using
explicit solutions.
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Sum of Two Contributions
U(1) gauge theory
The same configuration also appears in U(3) gauge
theory.
U(3) gauge theory
The only difference is the sign of the second
term.
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Understanding of the Asymptotic Metric
mass of the domain wall web masses of domain
walls energy localized at the
junctions (junction charge)
It is expected that the effective action can be
understood as the kinetic energy of internal
walls and junction charges.
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Kinetic Energy of Internal Walls
Kinetic energy of internal walls
Next we want to consider kinetic energy of
junction charges.
Let me explain two kinds of junction charges.
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Two Kinds of Domain Wall Junctions
It is known that the junction charge of the
Abelian junction is negative but that of the
non-Abelian junction is positive.
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Kinetic Energy of Junction Charges
U(1) Gauge Theory
U(3) Gauge Theory
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Shape of the Moduli Space
Finally, let us show the shape of the moduli
space. The metric is obtained by numerical
calculation and we embed it into the Euclidean
3-dimensional space.
Non-singular at the tip of the manifold with
positive curvature Geometry beween a cone and a
cigar.
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Summary

• We have succeeded in constructing the effective
  action of domain wall networks.
• The Kahler potential can be written as a finite
  sum of hypergeometric functions.
• The asymptotic metric can be obtained by the
  simple method which we call tropical limit.
• The asymptotic metric can be interpreted as the
  kinetic energy of internal walls and junction
  charges.
• This interpretation explains the sign flip of the
  asymptotic metric.