New topological structures of Skyrme theory: baryon number and monopole number
Eur. Phys. J. C
New topological structures of Skyrme theory: baryon number and monopole number
Y. M. Cho 0 1 2
Kyoungtae Kimm 4
J. H. Yoon 3
Pengming Zhang 2
0 School of Physics and Astronomy, Seoul National University , Seoul 151-742 , Korea
1 Konkuk University , Administration Building 310-4, Seoul 143-701 , Korea
2 Institute of Modern Physics, Chinese Academy of Science , Lanzhou 730000 , China
3 Department of Physics, Konkuk University , Seoul 143-701 , Korea
4 Faculty of Liberal Education, Seoul National University , Seoul 151-747 , Korea
Based on the observation that the skyrmion in Skyrme theory can be viewed as a dressed monopole, we show that the skyrmions have two independent topology, the baryon topology π3(S3) and the monopole topology π2(S2). With this we propose to classify the skyrmions by two topological numbers (m, n), the monopole number m and the shell (radial) number n. In this scheme the popular (non spherically symmetric) skyrmions are classified as the (m, 1) skyrmions but the spherically symmetric skyrmions are classified as the (1, n) skyrmions, and the baryon number B is given by B = mn. Moreover, we show that the vacuum of the Skyrme theory has the structure of the vacuum of the SineGordon theory and QCD combined together, which can also be classified by two topological numbers ( p, q). This puts the Skyrme theory in a totally new perspective.
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The Skyrme theory has played an important role in physics.
It has been proposed as an effective field theory of pion
physics in strong interaction where the baryons appear as
the skyrmions, topological solitons made of pions [1–4]. This
view has been very successful, and the rich topological
structure of the theory has advanced our understanding of the
extended objects greatly [5–7].
The construction of skyrmions as nuclei has a long
history. A novel way to obtain non-spherically symmetric
multiskyrmions was developed based on the rational map, and
the solutions have been associated with and compared to
real nuclei [8,9]. And a systematic approach to construct
the skyrmions with large baryon number numerically which
have the shell strucutre has been developed [10–13]. This,
with the improved computational power has made people
construct skyrmions with the baryon number up to 108 [14].
With the new development the Skyrme theory have had a
remarkable progress recently. It has been able to provide a
quantitative understanding of the spectrum of rotational
excitations of carbon-12, including the excitation the Hoyle state
which is essential for the generation of heavy nuclear
elements in early universe [15–17]. And the spin-orbit
interaction which is essential for the magic number of nuclei is
investigated within the framework of Skyrme theory [18].
Moreover, a method to reduce the binding energy of skyrmions to a
realistic level to improve the Skyrme model has been
developed [19,20]. So by now in principle one could construct
all nuclei as multi-baryon skyrmions and discuss the
phenomenology of nuclear physics, although the experimental
confirmation of the theory is still in dispute.
But the Skyrme theory has multiple faces. In addition to
the well known skyrmions it has the (helical) baby skyrmion
and the Faddeev–Niemi knot. Most importantly, it has the
monopole which plays the fundamental role [21–23]. In this
view all finite energy topological objects in the theory could
be viewed either as dressed monopoles or as confined
magnetic flux of the monopole-antimonopole pair. The skyrmion
can be viewed as a dressed monopole, the baby skyrmion
as a magnetic vortex created by the monopole-antimonopole
pair infinitely separated apart, and the Faddeev–Niemi knot
as a twisted magnetic vortex ring made of the helical baby
skyrmion. This confirms that the theory can be interpreted
as a theory of monopole in which the magnetic flux of the
monopoles is confines and/or screened.
The fact that the skyrmion is closely related to the
monopole has been appreciated for a long time. It has been
well known that the skyrmions could actually be viewed as
the monopoles regularized to have finite energy [21–23].
In fact it has been well appreciated that the rational map
which plays the crucial role in the construction of the
multiskyrmions is exactly the π2(S2) mapping which provides the
monopole quantum number [10]. Nevertheless the skyrmions
have always been classified by the baryon number given by
π3(S3), not by the monopole number π2(S2). This was
puzzling.
The purpose of this paper is twofold. We first show that
the skyrmions have two topology, the baryon topology and
the monopole topology, so that they are classified by two
topological numbers, the baryon number B and the monopole
number M . Moreover, we show that the baryon number can
be replaced by the radial (shell) number which describes the
π1(S1) topology of radial excitation of multi-skyrmions. This
is based on the observation that the SU(2) space S3 has the
Hopf fibering S3 S2 × S1 and that (...truncated)