Nobuyuki Sawado 1 Yuki Amari 1 Pawel Klimas

Nobuyuki Sawado 1 Yuki Amari 1, Pawel Klimas 2 Tokyo University of Science 1 Universidade Federal de Santa Catarina 2 At VI International Conference "Models in quantum field theory" 28 August, Saint Petersburg, Peterhof, Russia

0. The skyrmions Skyrmions are soliton solutions of several nonlinear s models The baby-skyrmions The classical solutions ü the topology (topological charge) ü the BPS (Bogomolny-Prasad-Sommerfeld) bound ü the Derrick’s scaling argument higher order derivative terms and (or) potentials The quantum aspects: ü the semiclassical (collective coordinate) quantizations ü the Wess-Zumino-Witten (Hopf) term ü the normalizable fermions (the Index analysis) (or Hopfions)

The statistical nature of the skyrmions Wilczek-Zee’s argument by linking number The skyrmion is adiabatically rotated F. Wilczek, A. Zee, ``Linking Numbers, Spin, and Statistics of Solitons, ‘’ Phys. Rev. Lett. 51, 2250 (1983) The spectral flow argument: The normalizable fermions fermion spectra coupld with the skyrmions exhibit the spectral flow of the localizing modes. The number is coincide with the topological charge of the skyrmions

1. The Skyrme model T. H. R. Skyrme, Proc. Roy. Soc. A 260 (1961) 127 The model The topological charge The left current “hedgehog”

The quantization and the spin-statistics Wess-Zumino -Witten term Possible WZ’s by The Hopf map or

A fermionic soliton based on the derivative expansion D. I. Diakonov, V. Yu. Petrov and P. V. Pobylitsa, Nucl. Phys. B 306 (1988)809 The partition function The expansion of the real part: Imaginary part: A Skyrme like action Wess-Zumino action

The spectral flow analysis N. S. and N. Shiiki, Nucl. Phys. A 739, 89 (2004) Dirac equation p Localizing modes ｆ（ｒ） X r Icosahedral skyrmion The Index Topological charge the index of the Dirac kernel

Y. Amari, P. Klimas, N. S. , Y. Tamaki, PRD 92, 045007(2015) The model Topology of the model A. D’Adda, M. Lucher, P. Di Vecchia, NPB 146 63 (1978) vortex can be formed

The equations of motion

The solution ansatz for the vortices The `twisted’ vortices The velocity is speed of light The light-cone coordinate Rescaling the coordinate and the field

The classical profile The Hamiltonian density

The collective coordinate quantization of the model Y. Amari, P. Klimas, N. S. , D 94, 025032(2016) The quantum energy The collective coordinate quantization The solitons are anyon

The partition function The derivative expansion:

The spectral flow Q=3 The quantum number Q=4

The normalizable modes Q=3 Q=4

Summary: The skyrmion and the CPN The skyrmion The classical solution The statistics The baby-skyrmion The CPN skyrmion

Summary Properties of spin-statistical nature of Skyrme-type theory are summerized. Their fermionic models are also considered. The statistical nature of the Skyrme model is defined via the Wess. Zumino-Witten term. It is topological invariant thus the anyon angle Is properly quantized. The normalizable fermions according to the index theorem are obtained. It is enable us to interpret the statistics as a number of localizing fermions.

At TUS Campus, Noda, JAPAN, 2018. 4 Спасибо!