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Reinterpretation of Skyrme Theory

Reinterpretation of Skyrme Theory. Y. M. Cho Seoul National Univ. Contents. Introduction & Overview Skyrme Theory : A Review Skyrme Theory & QCD Skyrme Theory & Condensed Matter Physics Topological Objects in Skyrme Theory Physical Interpretation of Knot Discussions.

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Reinterpretation of Skyrme Theory

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  1. Reinterpretation of Skyrme Theory Y. M. Cho Seoul National Univ.

  2. Contents • Introduction & Overview • Skyrme Theory : A Review • Skyrme Theory & QCD • Skyrme Theory & Condensed Matter Physics • Topological Objects in Skyrme Theory • Physical Interpretation of Knot • Discussions

  3. Reinterpretation of Skyrme Theory • Introduction & Overview • Skyrme theory has rich topological structures • 1) monopole • 2) baby skyrmion • 3) skyrmion • 4) knot • Skyrme theory is a theory of monopole, where all topological objects originates from monopoles. • Skyrme theory is a theory of confinement with a built-in Meissner effect, where the confinement scale is fixed at the classical level. • Skyrme theory is an effective theory of strong interaction which is dual to QCD. It confines monopoles, not the quarks.

  4. Reinterpretation of Skyrme Theory II. Skyrme Theory : A Review 1) Skyrme Lagrangian With we have

  5. Reinterpretation of Skyrme Theory • Equation of motion

  6. Reinterpretation of Skyrme Theory 2) Skyrmion With we have

  7. Reinterpretation of Skyrme Theory and With we have the well-known skyrmion which has

  8. Reinterpretation of Skyrme Theory • Baryon number which represents the non-trivial homotopy . It also has the magnetic charge which represents the non-trivial homotopy .

  9. Reinterpretation of Skyrme Theory 3) Skyrme-Faddeev Lagrangian With we have and Monopole Baby skyrmion Knot

  10. Reinterpretation of Skyrme Theory III. Skyrme Theory & QCD 1) Reparametrization of Skyrme theory Notice that where is the “Cho connection”

  11. Reinterpretation of Skyrme Theory In general, we have where

  12. Reinterpretation of Skyrme Theory • Linear approximation Near , we have

  13. Reinterpretation of Skyrme Theory 2) Abelian projection in QCD Parallel transport Under the gauge transformation, we have

  14. Reinterpretation of Skyrme Theory 3) Dual structure of QCD Notice that and so that Restricted QCD Extended QCD

  15. Reinterpretation of Skyrme Theory 4) Skyrme theory from QCD With we have where

  16. Reinterpretation of Skyrme Theory Furthermore, with we have where

  17. Reinterpretation of Skyrme Theory IV. Skyrme Theory & Condensed Matter Physics 1) Gauge theory of two-component BEC Consider with we have where

  18. Reinterpretation of Skyrme Theory 2) Skyrme-Faddeev theory in BEC With we have and

  19. Reinterpretation of Skyrme Theory In fact with we have

  20. Reinterpretation of Skyrme Theory V. Topological Objects in Skyrme Theory 1) Wu-Yang monopole • Monopole charge

  21. Reinterpretation of Skyrme Theory 2) Helical baby skyrmion Introduce the cylindrical coordinates and let

  22. Reinterpretation of Skyrme Theory Find

  23. Reinterpretation of Skyrme Theory With the boundary condition we obtain the non-Abelian vortex solution shown in Fig.1. Helical Vortex

  24. Reinterpretation of Skyrme Theory 3) Meissner effect The helical vortex has two helical magnetic fields Find and

  25. Reinterpretation of Skyrme Theory • Supercurrent With we have

  26. Reinterpretation of Skyrme Theory So we have two supercurrents and which generates and .

  27. Reinterpretation of Skyrme Theory 4) Faddeev-Niemi knot We can construct a knot by smoothly bending the helical baby skyrmion and connecting the periodic ends together. • Knot topology • Knot quantum number • Two different Skyrmion Knot

  28. Reinterpretation of Skyrme Theory • Dynamical stability The supercurrent along the knot generates a net angular momentum which prevents the collapse of the knot. • Physical manifestation of knot The knot can be viewed as two magnetic fluxes linked together, whose linking number becomes the knot quantum number.

  29. Reinterpretation of Skyrme Theory • Knot energy Theoretically we have where Numerically one finds up to

  30. Reinterpretation of Skyrme Theory VI. Physical Interpretation of Knot 1) Knot in Skyrme theory From we have But from we have

  31. Reinterpretation of Skyrme Theory 2) Chromoelectric Knot in QCD From we find • Decay width From Quantum instability

  32. Reinterpretation of Skyrme Theory where with we have

  33. Reinterpretation of Skyrme Theory VII. Discussions • The Skyrme theory is a theory of confinement where magnetic flux is confined by a built-in Meissner effect. • The Skyrme theory is an effective theory of strong interaction which is dual to QCD. • Notice that • but • The Skyrme theory, with the built-in Meissner effect, can play an important role in condensed matter physics.

  34. Reinterpretation of Skyrme Theory • Knots in laboratory • 1) Two-component BEC • 2) Two-gap superconductor • 3) Electroweak theory • 4) QCD • 5) Ordinary superconductor

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