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20#years#of#JNMP,#the#Sophus#Lie#Conference#Center,#12#June#2013�
References�
• #Journal#of#High#Energy#Physics,#JHEPB11(2009)124#• #Journal#of#Physics#A,##Vol.#43,#434014#(2010)#• #Acta#Polytechnica,#Vol.#51,#pp.47#B#pp.49#(2011)#####• #Physics#Review#D,#Vol.#85,#105006(2012)#• #Journal#of#Physics.:#Conf.#Ser.,#Vol.#411,#012014(2013)�
This#talk#is#based#on#the#above#papers.#�
The#SkyrmeBFaddeev#model�L.Faddeev, A.Niemi, Phys. Rev. Lett. 85, 3416 (2000)
The#soliton#soluWons#(Hopfions)#exhibit#the#
knotB,ringBstructures#
It#may#be#possible#as#a#low#energy#effecWve#
model#of#the#YangBMills#theory#
##########Hopfions##~#Glueball?�
R.A.Battye,P.M.Sutcliffe Phys.Rev.Lett.81,4798(1998)
Lagrangian�
StaWc#hamiltonian�
PosiWve#definite#for�
Hopf#charge#density�
#
Some#applicaWons#of#the#SF#model##
�• superconductors###########E.#Babaev:#PRL#(2002)##
• BoseBEinstain#condensate############E.#Babaev,#L.#D.#Faddeev#and#A.#J.#Niemi:#PR#B#(2002)#
##########T.#Kawakami,#T.#Mizushima,#M.#Nica#and#K.#Machida:##
#######################################################################################PRL#(2012)##
• DescripWon#of#a#lowBenergy#limit#of#the#pure#SU(2)#YangBMills#theory#(glueballs,#some#nonBtrivial#vacuum#
configuraWons)#B#ChoBFaddeevBNiemi#decomposiWon#
###########L.#D.#Faddeev#and#A.#J.#Niemi:#PRL#(1999)##
#
Modified#SkyrmeBFaddeev#model#
�• Recently#became#popular#some#modificaWons#
of#the#original#SkyrmeBFaddeev#model##
• The#result#obtained#by#H.#Gies#[Phys.#Rev#D#(2001)]#########in#calculaWon#of#the#Wilsonian#effecWve#acWon#
####of#the#SU(2)#YangBMills#theory#suggest#
####unavoidable#appearance#of#quarWc#term##
• The#extended#SkyrmeBFaddeev#model##given#
####by#L.#A.#Ferreira#[JHEP#(2009)]##
#
The#extended#SkyrmeBFaddeev#model#
�Lagrangian#�
Sterographic#project�
StaWc#hamiltonian�
L.A.Ferreira, JHEP05(2009)001, L.A.Ferreira, et.al., JHEP11(2009)124�
PosiWve#definite#for#�
EquaWon#of#moWon�
Zero#curvature#condiWon:�
Vortex#in#the#Extended#Skyrms#Faddeev#model�
L.A.Ferreira, JHEP05(2009)001�
AddiWonal#condiWon:�
SoluWons� Sta$c&vortex�
Traveling&wave&for&x3�
Ansatz�
(the#infinite#set#of)#conserved#currents#given#by##�
with#any#funcWon####�
One#can#see�
SoluWon#ansatz�
Boundary#condiWons�
Toroidal#coordinates:�
Hopfions#in#the#extended#Skyrme#Faddeev#model�
(Dimensionless)#staWc#energy�
Corresponds#to#the#zero##
curvature#condiWon#
Numerically-one-finds-:-�
The#soluWons#approximately#
saWsfy#the#zeroBcurvature#
(submodel#)condiWon!�
Topological#(Hopf)#charge#
Hopf#map�
####��������������
Lower#energy#bound#formula#
A.F.Vakulenko,L.V.Kapitanskii, Sov.Phys.Dokl. 24,433(1979) R.S.Ward, Nonlinearity 12,241 (1999)G�
(Sovolev#inequality)�
Vakulenko#
Ward�
EquaWon#for#g�
EquaWon#for#)�
EquaWons#of#moWon�
QH� &&(m,n)� """"""""a� E [M / |e|]�
E/Ebound��
<r2>1/2
[-1/|e|M]�
1� (1,1)� 0.53� 236.2� 1.18� 0.55�2� (1,2)� 0.79� 390.6� 1.16� 0.95�
(2,1)� 0.66� 466.7� 1.39� 1.04�3� (1,3)� 1.09� 548.7� 1.21� 1.45�
(3,1)� 0.80� 715.7� 1.57� 1.68�4� (1,4)� 1.41� 697.3� 1.23� 2.04�
(2,2)� 0.94� 765.7� 1.36� 1.76�(4,1)� 0.99� 965.7� 1.71� 2.40�
The#staWc#energies#and#the#radius�
Mean#square#radius�
The#energy#components#and#the#radius#of#tori�
(m,#n)#=#(1,#1)� (1,#2)� (2,#1)�
(m,#n)#=#(1,#3)� (3,#1)�
(m,#n)#=#(1,#4)� (2,#2)� (4,#1)�
Isosurfaces#of##Hopf#charge#density#for#the#soluWons#
#with#various#charges#up#to##################(Hopf#index)#�QH � 4
• Now#we#have#obtained#similar#results#of#
CPN-version-of#the#extended#SF#model.#
• We#are#invesWgaWng#some#physical#
applicaWons.#
#
RenormalizaWon#group#discussion#strongly#suggests#
that#�-and-e2-have-same-sign�
�� e2�
RenomalizaWon#group� ±� ±�
SkyrmeBFaddeev#(SF)� 0� s� succeeded�
GladikowskiBHellmund#(GH)� u� s� succeeded�
GH#with#same#posiWve#sign� s� s� fails�
Gies#model#with#2nd#derivaWve� s� s� poten>ally-succeeded�
Ours� u� u�