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

2 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 & Overview A)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.

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

5 Equation of motion Reinterpretation of Skyrme Theory

6 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 we have 3) Skyrme-Faddeev Lagrangian With and Monopole Baby skyrmion Knot

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

11 Reinterpretation of Skyrme Theory where In general, we have

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 where we have With

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 with Consider we have where

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

19 Reinterpretation of Skyrme Theory we have In fact with

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 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.

28 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.

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

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

32 with where Reinterpretation of Skyrme Theory we have

33 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

34 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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