Reinterpretation of Skyrme Theory

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


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


II. Skyrme Theory : A Review

we have

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• Equation of motion


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2) Skyrmion

With

we have

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and

With

we have the well-known skyrmion which has

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• Baryon number

which represents the non-trivial homotopy .

It also has the magnetic charge

which represents the non-trivial homotopy .

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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”


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where

In general, we have

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• Linear approximation

Near , we have

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2) Abelian projection in QCD

Parallel transport

Under the gauge transformation,

we have

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3) Dual structure of QCD

Notice that

and

so that

Restricted QCD

Extended QCD

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4) Skyrme theory from QCD

where

we have

With

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Furthermore, with

we have

where

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IV. Skyrme Theory & Condensed Matter Physics

1) Gauge theory of two-component BEC

with

Consider

we have

where

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2) Skyrme-Faddeev theory in BEC

With

we have

and

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we have

In fact with

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V. Topological Objects in Skyrme Theory1) Wu-Yang monopole

• Monopole charge

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2) Helical baby skyrmion

Introduce the cylindrical coordinates

and let

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Find

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With the boundary condition

we obtain the non-Abelian vortex solution shown in Fig.1.

Helical Vortex

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3) Meissner effect

The helical vortex has two helical magnetic fields

Find

and

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• Supercurrent

With

we have

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So we have two supercurrents and which generates

and .

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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


• 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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• Knot energy

Theoretically we have

where

Numerically one finds

up to

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VI. Physical Interpretation of Knot

1) Knot in Skyrme theory

From

we have

we have

But from

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2) Chromoelectric Knot in QCD

From

we find

From

• Decay width

Quantum

instability

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with

where


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.


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D) Knots in laboratory

1) Two-component BEC

2) Two-gap superconductor

3) Electroweak theory

4) QCD

5) Ordinary superconductor

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Reinterpretation of Skyrme Theory

Documents

theory of monopole

theory of confinement

knotskyrme theory

different knot skyrmion

wellknown skyrmion

nontrivial homotopy

nonabelian vortex solution

rich topological structures