Reinterpretation of Skyrme Theory
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Reinterpretation of Skyrme Theory
Y. M. ChoSeoul National Univ.
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Contents
I. Introduction & Overview
II. Skyrme Theory : A Review
III. Skyrme Theory & QCD
IV. Skyrme Theory & Condensed Matter Physics
V. Topological Objects in Skyrme Theory
VI. Physical Interpretation of Knot
VII. Discussions
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Reinterpretation of Skyrme Theory
I. Introduction & OverviewA) Skyrme theory has rich topological structures
1) monopole
2) baby skyrmion
3) skyrmion
4) knot
B) Skyrme theory is a theory of monopole, where all topological objects originates from monopoles.
C) Skyrme theory is a theory of confinement with a built-in Meissner effect, where the confinement scale is fixed at the classical level.
D) Skyrme theory is an effective theory of strong interaction which is dual to QCD. It confines monopoles, not the quarks.
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With
1) Skyrme Lagrangian
Reinterpretation of Skyrme Theory
II. Skyrme Theory : A Review
we have
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• Equation of motion
Reinterpretation of Skyrme Theory
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Reinterpretation of Skyrme Theory
2) Skyrmion
With
we have
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Reinterpretation of Skyrme Theory
and
With
we have the well-known skyrmion which has
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Reinterpretation of Skyrme Theory
• Baryon number
which represents the non-trivial homotopy .
It also has the magnetic charge
which represents the non-trivial homotopy .
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Reinterpretation of Skyrme Theory
we have
3) Skyrme-Faddeev Lagrangian
With
and
Monopole
Baby skyrmion
Knot
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III. Skyrme Theory & QCD
1) Reparametrization of Skyrme theory
Notice that
where is the “Cho connection”
Reinterpretation of Skyrme Theory
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Reinterpretation of Skyrme Theory
where
In general, we have
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Reinterpretation of Skyrme Theory
• Linear approximation
Near , we have
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Reinterpretation of Skyrme Theory
2) Abelian projection in QCD
Parallel transport
Under the gauge transformation,
we have
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Reinterpretation of Skyrme Theory
3) Dual structure of QCD
Notice that
and
so that
Restricted QCD
Extended QCD
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Reinterpretation of Skyrme Theory
4) Skyrme theory from QCD
where
we have
With
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Reinterpretation of Skyrme Theory
Furthermore, with
we have
where
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Reinterpretation of Skyrme Theory
IV. Skyrme Theory & Condensed Matter Physics
1) Gauge theory of two-component BEC
with
Consider
we have
where
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Reinterpretation of Skyrme Theory
2) Skyrme-Faddeev theory in BEC
With
we have
and
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Reinterpretation of Skyrme Theory
we have
In fact with
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Reinterpretation of Skyrme Theory
V. Topological Objects in Skyrme Theory1) Wu-Yang monopole
• Monopole charge
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Reinterpretation of Skyrme Theory
2) Helical baby skyrmion
Introduce the cylindrical coordinates
and let
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Reinterpretation of Skyrme Theory
Find
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Reinterpretation of Skyrme Theory
With the boundary condition
we obtain the non-Abelian vortex solution shown in Fig.1.
Helical Vortex
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Reinterpretation of Skyrme Theory
3) Meissner effect
The helical vortex has two helical magnetic fields
Find
and
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Reinterpretation of Skyrme Theory
• Supercurrent
With
we have
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Reinterpretation of Skyrme Theory
So we have two supercurrents and which generates
and .
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Reinterpretation of Skyrme Theory
4) Faddeev-Niemi knot
• Knot topology
• Knot quantum number
• Two different
Knot
Skyrmion
We can construct a knot by smoothly bending the helical baby skyrmion and connecting the periodic ends together.
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• Dynamical stability
Reinterpretation of Skyrme Theory
• Physical manifestation of knot
The supercurrent along the knot generates a net angular momentum which prevents the collapse of the knot.
The knot can be viewed as two magnetic fluxes linked together, whose linking number becomes the knot quantum number.
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Reinterpretation of Skyrme Theory
• Knot energy
Theoretically we have
where
Numerically one finds
up to
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Reinterpretation of Skyrme Theory
VI. Physical Interpretation of Knot
1) Knot in Skyrme theory
From
we have
we have
But from
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Reinterpretation of Skyrme Theory
2) Chromoelectric Knot in QCD
From
we find
From
• Decay width
Quantum
instability
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with
where
Reinterpretation of Skyrme Theory
we have
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VII. Discussions
A) The Skyrme theory is a theory of confinement where magnetic flux is confined by a built-in Meissner effect.
B) The Skyrme theory is an effective theory of strong interaction which is dual to QCD.
Notice that
but
C) The Skyrme theory, with the built-in Meissner effect, can play an important role in condensed matter physics.
Reinterpretation of Skyrme Theory
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Reinterpretation of Skyrme Theory
D) Knots in laboratory
1) Two-component BEC
2) Two-gap superconductor
3) Electroweak theory
4) QCD
5) Ordinary superconductor
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