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

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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