Introduction to Introduction to Solitons Solitons Institute of Theoretical Physics and Astronomy Institute of Theoretical Physics and Astronomy Vilnius, 2013 Vilnius, 2013 University of University of Oldenburg Oldenburg and BSU Minsk and BSU Minsk Ya Ya Shnir Shnir
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Introduction to Introduction to SolitonsSolitons
Institute of Theoretical Physics and AstronomyInstitute of Theoretical Physics and AstronomyVilnius, 2013Vilnius, 2013
University of University of OldenburgOldenburg and BSU Minskand BSU Minsk
YaYa ShnirShnir
SUSY
Branes and extra dims
Particle physics
Condensed matter
QCDComfinement
Topology
Differential Geometry
Astrophysics & Cosmology
QFTBlack holes
gg ????
MonopolesMonopoles
Electromagnetic duality and Dirac monopoleElectromagnetic duality and Dirac monopole
System of generalized Maxwell equations System of generalized Maxwell equations
is invariant with respect to the transformations of electromagnis invariant with respect to the transformations of electromagnetic duality:etic duality:
Classical motion in the monopole Coulomb magnetic field: Classical motion in the monopole Coulomb magnetic field:
Generalized angular momentum is conserved:Generalized angular momentum is conserved:
DiracDirac ’’s string is invisible if the charge s string is invisible if the charge quantization condition is imposed:quantization condition is imposed:
BreakBreak--through of 1974:through of 1974:`t t HooftHooft --PolyakovPolyakov monopole solutionmonopole solution
While a Dirac monopole could becould beincorporated in an Abelian theory, some non-Abelian models inevitably containinevitably containmonopole solutions
Non-Abelian monopole is a non-linear system of coupled gauge and scalar (Higgs)fields, its energy is finite and the fields areregular everywhere in space. The gauge symmetry is spontaneously broken via Higgs mechanis m
Properties of nonProperties of non--AbelianAbelian monopoles [SU(5)]monopoles [SU(5)]
Monopole has a Monopole has a corecore of of radiusradius rrmm ~ ~ mmxx--11 ~~ 1010--29 29 cm cm
Monopole Monopole isis superheavysuperheavy : : M M ~ ~ mmxx//α α ~ ~ 101017 17 GeVGeV ~~ 1010--7 7 ggMagneticMagnetic chargecharge of of thethe monopolemonopole has has topologicaltopological rootsroots ::
Electromagnetic subgroup is associated with rotations about Electromagnetic subgroup is associated with rotations about direction of the Higgs fielddirection of the Higgs fieldMonopole solution mixes the Monopole solution mixes the spacialspacial and group rotations:and group rotations:
Monopole has 4 collective coordinates: Monopole has 4 collective coordinates: RRRRRRRRkkkkkkkk aaaaaaaannnnnnnndddddddd χχχχχχχχ((((((((tttttttt))))))))
Electric charge of a Electric charge of a dyondyon is is QQQQQQQQ ======== 44444444ππππππππ ˙χχχχχχχχ
Charge quantization condition for a pair of Charge quantization condition for a pair of dyonsdyons ::
Consequence:Consequence: SpinSpin --statistic theorem admits both Bosestatistic theorem admits both Bose --Einstein and Einstein and FermiFermi --Dirac statistics.Dirac statistics.
As T < As T < TTcc ~ 10~ 1015 15 GeVGeV the Higgs field acquires a nonthe Higgs field acquires a non --zero zero v.e.vv.e.v ..
V(V(ΦΦΦΦΦΦΦΦ)) V(V(ΦΦΦΦΦΦΦΦ)) Predictions of the Big Bang scenario (adiabatic expansion)1 monopole per 101 monopole per 1044 nucleons!nucleons!
Inflation scenario: The potential is suffucientlyflat at Φ=0, the phase transition occurs at TTcc ~ 10~ 109 9 GeVGeV-- only a few monopoles may survive the inflation!only a few monopoles may survive the inflation!
(Castelnovo, C., R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature, Vol. 451, 42-45, 2008;D.J.P. Morris et al, Dirac Strings and Magnetic Monopolesin the Spin Ice; Science, Vol. 326, 411-414, 2009 )
„Monopoles“ in spin-ice crystal structures
(Castelnovo, C., R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature, Vol. 451, 42-45, 2008;D.J.P. Morris et al, Dirac Strings and Magnetic Monopolesin the Spin Ice; Science, Vol. 326, 411-414, 2009 )
A sum of nearest-neighbor Ising model term and long range dipolar interactions
Each dipole is replaced by a pair of Each dipole is replaced by a pair of equal and opposite magnetic chargesequal and opposite magnetic charges
Fake monopolesFake monopoles
„Monopoles“ and low energy QCD „Monopoles“ and low energy QCD
QCD confinement as dual Meissner effect: monopole condencation as a reason of formation of the chromoelectric flux tube and QCD is taking a form of the dual Ginzburg-Landau model (S.Mandelstam, G `t Hooft et al (1970s))
Where is the cheat?
There is no monopoles in QCD!There is no monopoles in QCD!
Perturbative QCD Low energy effective theory Hadrons(Quarks & gluons) (Pions and quarks)
LongLong--rangerange scalarscalar fieldfield F ~ 1/r
No No netnet interactioninteraction betweenbetween thethe BPS BPS monopolesmonopolesAnalyticalAnalytical solutionsolution of of thethe BPS BPS equationsequations::
Sir M. Sir M. AtiyahAtiyah , R. Ward (1977),, R. Ward (1977),P. P. ForgacsForgacs et alet al (1981), (1981), W. Nahm (1982), W. Nahm (1982), P. P. SutcliffeSutcliffe (1996) (1996) and and otherother
ThereThere isis a a transformationtransformation of a of a monopolemonopole intointo a rational a rational mapmap fromfrom thetheRiemanianRiemanian sphere to sphere to inselfinself : : R: SR: S22 ## SS2 2 (P. (P. SutcliffeSutcliffe , , N.MantonN.Manton et al)et al)
Construction of Construction of thethe rational rational mapsmaps monopolesmonopoles : :
Monopole catalysis of proton decayMonopole catalysis of proton decay
ThereThere areare zerozero --energyenergy solutionssolutions of of thethe DiracDirac equationequation forfor a a masslessmassless fermionfermioncoupledcoupled to a to a monopolemonopole ; ;
Naive question:Naive question: What happened when a fermion collides with a monopol e?
There is no vector (gauge) field There is no vector (gauge) field -- but gravity may be coupled to this system insteadbut gravity may be coupled to this system instead
TopologicalTopological defectsdefects & extra & extra dimensionsdimensions ::
d=5: domaindomain wall (kink in d=1 + D=4)wall (kink in d=1 + D=4)
d=6: vortex (in d=2) + D=4vortex (in d=2) + D=4
d=7: monopole (in d=3) + D=4monopole (in d=3) + D=4
d=8: instanton (in d=4) + D=4instanton (in d=4) + D=4
the internal space of a topological defect living in a higherdimensional space-time
OurOur D=4 worldD=4 world::
AcknowledgmentsAcknowledgments
Work done in collaboration withWork done in collaboration with::•P Dorey