1

Four lectures of Four lectures of solitonssolitons in in

Classical Field TheoryClassical Field Theory

KitasatoKitasato University, 4University, 4--5 5 DecemberDecember 20192019

YaYa ShnirShnir

BLTP, JINRBLTP, JINR

SkyrmeSkyrme modelmodel

Tony Hilton Royle Skyrme

QCD:QCD: LL == −− 1144FF aaµµννFF

aaµµνν ++ ¯̄ψψii[[iiγγµµDDµµ −−mmδδiijj ]]ψψjj

Low energy meson theory:Low energy meson theory:

LL ==ff 22ππ44TTrr ((∂∂µµUU∂∂

µµUU )) ++ .. .. ..

●● The idea of unifying bosons and fermions in a common framework

●● Consideration of localised field configurations instead of point-like particles

●● The desire to eliminate fermions from a fundamental formulation of theory

Skyrmes’ motivations (1962):SkyrmesSkyrmes’’ motivations (1962):motivations (1962):

Low energy QCD ($ 1000000 Problem) Low energy QCD ($ 1000000 Problem)

ΛQSB ~ 1 GeV

Perturbative QCD(Quarks & gluons)

Low-energy effective theory Hadrons

ΛQCD ~ 180 MeV

Weak definition of the confinement: There are no color states in physical spectrum

Strong definition of the confinement: The quarks in hadrons are binded by a linear potential

SkyrmeSkyrme modelmodel

The The SkyrmeSkyrme field:field:

Sigma-model term Skyrme term Potential term

= 186 MeV, = 136 MeVffππ mmππ

The topological charge:The topological charge: QQ ==11

2244ππ22εεiijjkk

��dd33xxTTrr

��((UU ††∂∂iiUU ))((UU ††∂∂jjUU ))((UU ††∂∂kkUU ))

��

Ri = (∂iU)U†The su(2) current:The su(2) current: QQ == −− 11

2244ππ22εεiijjkk

��dd33xxTTrr((RRiiRRjjRRkk))

Rescaling:

LL ==ff 22ππ44TTrr

��∂∂µµUU∂∂

µµUU ††��++

11

3322ee22TTrr

����UU ††∂∂µµUU,, UU

††∂∂ννUU��22��

++mm22ππff

22ππ

88TTrr ((UU −− II))

EE ==ffππ44ee

��dd33xx

��−− 1122TTrr ((RRiiRR

ii)) −− 111166

TTrr (([[RRii ,, RRjj ]]))22 ++mm22TTrr((UU −− II))

xxµµ →→ 22xxµµ//((eeffππ));; mm == 22mmππ//((ffππee))

UU :: SS33 →→ SS33UU ((��rr,, tt)) −−−−−−→→rr→→∞∞ II

SkyrmeSkyrme modelmodel

QQ ==11

ππ

FF ((rr)) −− ssiinn 22FF ((rr))

22

��∞∞

00

The boundary conditionsThe boundary conditions

FF ((00)) == ππ,, FF ((∞∞)) == 00 QQ == 11

LL == −−��dd33xx

1122 ((∂∂µµφφ

aa))22 −− 1144 [[((∂∂µµφφaa∂∂ννφφaa))22 −− ((∂∂µµφφaa))44 ]] ++mm22 ((11 −− φφ33))��

φφaa == ((σσ,, ππ11 ,, ππ22 ,, ππ33))UU ((rr)) == σσ ++ ππaa ·· ττ aa == ccooss FF ((rr)) ++ ii ˆ̂nn ·· ττ ssiinn FF ((rr))

Spherically symmetric Spherically symmetric skyrmionskyrmion::

(Hedgehog ansatz)UU ((rr)) == eexxpp [[iiττ aa ˆ̂rraaFF ((rr))]]

SkyrmionsSkyrmions

Q=1 Q=2 Q=3 Q=4

Q=5 Q=6 Q=7 Q=8

Rational map Rational map SkyrmionsSkyrmions

TheThe SkyrmeSkyrme fieldfield isis effectivelyeffectively a a mapmap UU: : SS33 →→ SU(2) ~ SSU(2) ~ S33

TheThe ideaidea of of thethe rational rational mapmap ansatzansatz: :

Separate the radial and the angular

dependence of the Skyrme field as as

Identify spheres SS22 with concentric

spheres in compactified RR33

Identify target space SS22with spheres

of latitude on SS33(N.S. Manton, (N.S. Manton, C.HoughtonC.Houghton && P.SutcliffeP.Sutcliffe, 1998), 1998)

zz == ttaann((θθ//22))eeiiϕϕStereographic Projection:

ˆ̂nnzz ==11

11 ++ ||zz||22��zz ++ ¯̄zz

11 ++ zz ¯̄zz,, ii

zz∗∗ −− zz11 ++ zz ¯̄zz

,,11 −− zz ¯̄zz11 ++ ¯̄zz

��

== ((ssiinn θθ ccooss φφ,, ssiinn θθ ssiinn φφ,, ccooss θθ))

Domain space Target space

ˆ̂nnZZ ==

��ZZ ++ ¯̄ZZ

11++ZZ ¯̄ZZ,, ii

¯̄ZZ −−ZZ11 ++ZZ ¯̄ZZ

,,11−−ZZ ¯̄ZZ11++ZZ ¯̄ZZ

��

UU == eexxpp {{iiff ((rr)) ˆ̂nnZZ ·· σσ}}

ZZ == PP ((zz))//QQ((zz))

Rational map approximation Rational map approximation

Static energy:Static energy: EE == 44ππ

�� ��rr22ff ′′

22++ 22QQ((ff ′′

22++ 11)) ssiinn22 ff ++WW

ssiinn44 ff

rr

��ddrr

4πQ=

� �1+|z|21+|Z|2

����dZ

dz

�����2

dzdz̄

(1+|z|2)2 WW==11

44ππ

�� ��11++ ||zz||2211++ ||ZZ||22

��������ddZZ

ddzz

����������44

ddzzdd¯̄zz

((11++ ||zz||22))22

The The holomorphicholomorphic maps of degree Q:maps of degree Q:

Q= 4:Q= 4:

Q= 7:Q= 7:

(Octahedral (Octahedral SkyrmionsSkyrmions))

((IcosahedralIcosahedral SkyrmionsSkyrmions))

ZZ((zz)) == zz44++22ii

√√33zz22++11

zz44−−22ii√√33zz22++11

Z(z) = z7−7z5−7z2−1z7+7z5−7z2+1

SkyrmionsSkyrmions

(N.Manton et al)

SkyrmeSkyrme crystalscrystals (Klebanov, Kugler, Shtrikman, Manton..)

Simple cubic BCC

FCC

½ Simple cubic

Knots and linksKnots and links

If there is a physical realisation?If there is a physical realisation?

Lord Kelvin: 1867: ”Vortex Atoms”

Construction of the Construction of the HopfionHopfion

d=3+1 d=3+1 φφaaφφaa == 11 φφ

aa == ((φφ11,, φφ22 ,, φφ33)) ∈∈ SS22 φφvvaacc == ((00,, 00,, 11))

φφ11 ++ iiφφ22 →→ ((φφ11 ++ iiφφ22))eeiiαα Residual SO(2) symmetry

Loops in domain space S3

��φφ :: SS33 →→ SS22

z

ϕρ

HomotopyHomotopy classclass pp33((SS22) = ) = ��

FFµµνν ==11

22εεµµννρρφφ

aa∂∂µµφφbb∂∂ννφφ

cc == ∂∂µµAAνν −− ∂∂ννAAµµ

φ1 + iφ2 = sinF (ρ, z)einϕ+iG(ρ,z)z; φ3 = cosF (ρ, z)

Target space S2

Topological charge:Topological charge:

(Linking number)(Linking number)QQ ==

11

88ππ22

��εεiijjkkAAiiFFjjkkdd

33xx

FF == 1122FFµµννddxx

µµ ∧∧ ddxxνν ;; ddFF == 00;; FF == ddAA

SolitonsSolitons of the of the FaddeevFaddeev--SkyrmeSkyrme modelmodel

Q=1 Q=2 Q=3

Q=4 Q=4 Q=4

1A1,1 2A2,1 33 ��AA33,,11

4 �A4,1 4L1,11,144AA22,,22

LL == 1122 ((∂∂iiφφaa))22 −− κκ2244

��εεaabbcc φφ

aa∂∂iiφφbb∂∂jjφφ

cc��22

SolitonsSolitons of the of the FaddeevFaddeev--SkyrmeSkyrme modelmodel

Q=5 5 �A5,1 Q=5 5L1,21,1 Q=6 6L

2,21,1 Q=6 6L3,11,1

Q=7 7K3,2 8L3,31,1Q=8Q=8 8 �A4,2 Q=8 8K3,2

Elastic rod approximationElastic rod approximation

D. Harland, J.M. Speight and P.M. SutcliffePhys. Rev.D83 (2011) 065008

Position curve

γγ((ss)) ∈∈ RR33

α(L) = α(0) + 2πN

Torsion Curvature κ(s)ττ ((ss))

Tubular coordinates:• Arclength parameter

• Polar coordinates in the diskss ∈∈ [[00,, LL]]

ρ, θ

FrenetFrenet frameframe

Preimage of

φ = (0, 0, 1)

Tangent vector �t(s)

frame vector �m(s)

Twisting function: αα((ss)) == ��tt ·· ��mm′′ ×× ��mm

Faddeev-Skyrme effective energy functional:

�m(s) = �n(s)sinα(s)+�b(s)cosα(s)

E =

� �A+Bκ2 + C(α′ − τ)2

�ds

d

ds

�t(s)�n(s)�b(s)

=

0 κ(s) 0

−κ(s) 0 τ(s)0 −τ (s) 0

�t(s)�n(s)(s)

SUSY

Branes and extra dims

Particle physics

Condensed matter

QCDComfinement

Topology

Differential Geometry

Astrophysics & Cosmology

QFTBlack holes

gg ????

MonopolesMonopoles

Petrus Peregrinus Petrus Peregrinus PolesPoles of a of a magnetmagnet 12691269

H. H. PoincarPoincaréé ClassicalClassical ee--g g interactioninteraction 18961896

P.A.M. P.A.M. DiracDirac Charge Charge quantizationquantization 19311931

H. HopfH. Hopf HopfHopf bundlebundle 19311931

P.A.M.DiracP.A.M.Dirac et alet al QEMDQEMD 19481948

G `t G `t HooftHooft, , A.PolyakovA.Polyakov NonNon--AbelianAbelian MonopolesMonopoles 19741974

T.T.WuT.T.Wu, , C.N.YangC.N.Yang Geometry & MonopolesGeometry & Monopoles 19751975

E.BogomolnyE.Bogomolny et alet al BPS MonopolesBPS Monopoles 19761976

C.MontonenC.Montonen and and D.OliveD.Olive Duality revisedDuality revised 19781978

S.MandelstamS.Mandelstam, , G`tG`t HooftHooft Dual Dual MeissnerMeissner effecteffect 1970s1970s

V.RubakovV.Rubakov, , C.CallanC.Callan Monopole catalysisMonopole catalysis 19811981

N.MantonN.Manton, , M.AtiyahM.Atiyah ModuliModuli spacespace 19821982

N.SeibergN.Seiberg and and E.WittenE.Witten N=2 SUSYN=2 SUSY 19941994

Magnetic Magnetic MonopolesMonopoles: Historical remarks: Historical remarks

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:

ϑϑϑϑ

cossin

;coscos

BEB

BEE

+→−→

ϑϑϑϑ

cossin

;coscos

geg

gee

+→−→

∇∇ ·· ��EE == 44ππee;;

∇∇ ×× ��EE ++ ∂∂ ��BB∂∂tt

== jjgg

∇∇ ·· ��BB == 44ππgg;;

∇∇ ×× ��BB −− ∂∂ ��EE∂∂tt

== jjee

mm dd22��rrddtt22

== ee[[��vv ×× ��BB]] == eeggrr33

��dd��rrddtt×× ��rr��

��JJ == [[��rr ××mm��vv]] −− eegg ��rrrr== ��LL −− eegg ˆ̂rr

L J

DiracDirac’’s monopole: Charge quantizations monopole: Charge quantization

��BB == gg ��rrrr33

== ∇∇ ×× ��AA;; ∇∇ ·· ��BB == 44ππgg ??��AA == gg

rr��rr××��nnrr−−((��rr··��nn))

- Dirac monopole potential

DiracDirac’’s string is invisible if the charge s string is invisible if the charge

quantization condition is imposed:quantization condition is imposed:

��BB == ��BBgg ++ ��BBssttrriinngg == gg��rr

rr33−− 44ππgg ��nn θθ((zz))δδ((xx))δδ((yy))

GGaauuggee iinnvvaarriiaannccee:: ��AA −−→→ ��AA ++∇∇λλ((��nn,, ��nn′′))

eegg == nn

Charge quantization for Charge quantization for dyonsdyons

Pair of Pair of dyonsdyons: : (e(e11,g,g11), (e), (e22,g,g22):): ee11gg22 −− ee22gg11 == nn,, nn ∈∈ ZZ

Electron Electron –– monopole pair (monopole pair (e,ge,g) )

There is an elementary electric charge

There is an elementary magnetic charge

((ee00,, 00)) ++ ((ee,, gg)) ==⇒⇒ ee00gg == nn

Pair of Pair of dyonsdyons: : (e(e11,g,g00), (e), (e22,g,g00):):

eg = n n ∈ Z

ee00 ==11

gg,, ee == nnee00

g =n

e0=

n

n0g0

ee11 −− ee22 ==nn

gg00==

nn

nn00ee00 == mmee00

eeii == ee00

��nnii ++

θθ

22ππ

��,, θθ ∈∈ [[00..22ππ]]

Most general charge quantization condition:

ee ++ iigg == ee00

��nn ++mm

θθ

22ππ

��++ ii

mm

ee00== ee00((mmττ ++ nn)) ττ ==

θθ

22ππ++

ii

ee2200

Charge quantization and SL(2,Charge quantization and SL(2,��) group) group

Transformations of SL(2,�): ττ →→aaττ ++ bb

ccττ ++ dd

Generators:

T : τ → τ + 1; S : τ → −1τ

NonNon--AbelianAbelian monopolesmonopoles

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 contain

monopole 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 mechanism

LL == 1122 TTrr FF22µµνν ++ TTrr ((DDµµΦΦ))

22 ++ VV ((||ΦΦ||))

MagneticMagnetic monopolesmonopoles

A.M.PolyakovA.M.Polyakov

*1945*1945

Gerard 't Gerard 't HooftHooft

*1946*1946

P.A.M.DiracP.A.M.Dirac

19021902--19841984DiracDirac monopolemonopole (1931)(1931) NonNon--AbelianAbelian monopolemonopole

WuWu--YangYang monopolemonopole (1975)(1975)

AANN == gg

11 −− ccooss θθrr ssiinn θθ

ˆ̂ee

AASS == −−gg 11 ++ ccooss θθ

rr ssiinn θθˆ̂ee

RegularRegular staticstatic configurationconfiguration

GaugeGauge groupgroup SU(2)SU(2)

MagneticMagnetic chargecharge isis thethe

topologicaltopological numbernumber: : Qg=n/2 Qg=n/2

TheThe monopolemonopole isis veryvery heavyheavy,,

M~m_vM~m_v/e/e

��BB == gg ��rrrr33,, ��EE == QQ ��rr

rr33

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 gg

MagneticMagnetic 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 field

Monopole solution mixes the Monopole solution mixes the spacialspacial and group rotations:and group rotations:

��ΦΦ →→ vv��rr

SS22 →→ SS22

��JJ == ��LL ++ ��TT ++ ��SS

DyonsDyons

Monopole has 4 collective coordinates: Monopole has 4 collective coordinates: RRkk aanndd χχ((tt))

Electric charge of a Electric charge of a dyondyon is is QQ == 44ππ ˙̇χχ

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.

QQ11gg22 −− QQ22gg11 == nn22

Relic MonopolesRelic Monopoles

Monopoles should have been produced in the very early Universe:

SSUU ((55)) →→ SSUU ((33)) ×× SSUU ((22)) ×× UU ((11)) →→ SSUU ((33)) ×× UU ((11))eemm

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(ΦΦ)) V(V(ΦΦ)) 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!

V(V(ΦΦ))

Experimental Experimental searchsearch forfor monopolesmonopoles

Accelerator Accelerator searchsearch

((FermilabFermilab, CERN, DESY, CERN, DESY……))

SuperconductingSuperconducting coilscoils

IndirectIndirect limits (limits (ParkerParker‘‘ss boundbound, ,

neutronneutron starsstars……))

SearchSearch forfor monopolemonopole catalysiscatalysis

((IceCubeIceCube, Berkeley, Stanford, IBM, Berkeley, Stanford, IBM……))

MonopolesMonopoles in in cosmiccosmic raysrays

((ScintillatorsScintillators and ionization detectors)and ionization detectors)

No No monopolemonopole detecteddetected yetyet!!((©©Picture by courtesyPicture by courtesy G.GiacomelliG.Giacomelli))

Fake monopolesFake monopoles

● „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 Monopoles

in 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 Monopoles

in the Spin Ice; Science, Vol. 326, 411-414, 2009)

A sum of nearest-neighbor Ising model term and long range dipolar interactions

HH == JJ��iijj

SSiiSSJJ ++ σσ��iijj

33((ˆ̂eeii·· ˆ̂rriijj ))((ˆ̂eejj ·· ˆ̂rriijj ))−−((eeii··eejj ))rr33iijj

(Mengotti et al, Nature Physics , 7 (2011) 68)

Where is the cheat?

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)

ΛΛχ χ ~ 1 ~ 1 GeVGeV ΛΛQCDQCD ~ 180 ~ 180 MeVMeV

HereHere wewe areare: : YangYang--MillsMills--HiggsHiggs TheoryTheory

´́t t HooftHooft--PolyakovPolyakov staticstatic sphericallyspherically symmetricsymmetric solutionsolution

SS == 1122��dd44xx {{ FFµµννFF µµνν ++ ((DDµµΦΦ))((DDµµΦΦ)) −− VV ((ΦΦ))}}

FFµµνν == ∂∂µµAAνν −− ∂∂ννAAµµ ++ iiee[[AAµµ ,, AAνν ]]DDµµΦΦ == ∂∂µµΦΦ ++ iiee[[AAµµ ,,ΦΦ]]

VV ((ΦΦ)) == λλ ((ΦΦ22 −− aa22))22

φφaa == rraa

eerr22HH ((eeaarr))

AAaann == εεaammnnrrmm

eerr22((11 −− KK ((eeaarr))))

‘‘t t HooftHooft--PolyakovPolyakov solutionsolution

“The fox knows many things, but

the hedgehog knows one big thing”

Archiolus

Topologically nontrivial asymptotic

φφaa −−→→rr→→∞∞

vvrraa

rr

Vacuum: ||φφ|| == aa

φφ :: SS22 →→ SS22

DDnnφφaa →→ 00 ∂∂nn

��rraa

rr

��−− eeεεaabbccAAbbnn

rrcc

rr== 00

AAaakk((rr)) −−→→rr→→∞∞

11

eeεεaannkk

rrnnrr22

;; BBaann −−→→rr→→∞∞

rraarrnneerr44

U(1) ‘t Hooft tensor:U(1) U(1) ‘‘t t HooftHooft tensortensor:: FFµµνν == ˆ̂φφaaFF aaµµνν ++11

eeǫǫaabbcc ˆ̂φφ

aaDDµµ ˆ̂φφbbDDνν ˆ̂φφ

cc

∂µ �Fµν = kν ; kµ =1

2εµνρσ∂

νF ρσ =1

2v3eεµνρσεabc∂

νφa∂ρφb∂σφc

∂∂µµkkµµ ≡≡ 00Topological current:Topological current:Topological current:

‘‘t t HooftHooft--PolyakovPolyakov ansatzansatz::

‘‘t t HooftHooft--PolyakovPolyakov solutionsolution

Magnetic charge:Magnetic chargeMagnetic charge:: gg ==

��dd33xx kk00 == −−

11

22eevv33

��dd22SSnn εεaabbccεεmmnnkk φφ

aa∂∂nnφφbb∂∂kkφφ

cc

local coordinates on S2 determinant of metric tensor on S2

==11

ee

��dd22ξξ√√gg ==

44ππnn

ee,, nn ∈∈ ZZ

φφaa ==rraa

eerr22HH((ξξ)) ,, AAaann == εεaammnn

rrmm

eerr22[[11 −−KK((ξξ))]] ,, AAaa00 == 00

Field equations:

ξ = aer

ξξ22dd22KK

ddξξ22== KKHH22 ++KK((KK22 −− 11)) ,, ξξ22 dd

22HH

ddξξ22== 22KK22HH ++

λλ

ee22HH((HH22 −− ξξ22))

DyonsDyons

AAaa00 ==rraa

eerr22JJ ((rr)) qq ==

11

vv

��ddSSnnEEnn ==

11

vv

��ddSSnnEE

aannφφaa ==

11

vv

��dd33xxEEaannDDnnφφ

aa

JJ ((rr)) →→ CCrr aass rr →→ ∞∞qq ==

44ππ

ee

∞∞��

00

ddξξdd

ddξξ

��ξξHH

dd

ddξξ

��JJHH

ξξ

��==

44ππCC

ee== CCgg

φφaa �� AAaa00

EE ==

��dd33xx

��11

22[[BBaannBB

aann ++ ((DDnnφφ

aa))((DDnnφφaa))]] ++ VV ((φφ))

==11

22

��dd33xx

11

22

��dd33xx ((BBaann −− DDnnφφaa))2

2++

��dd33xx BBaannDDnnφφ

aa ++

��dd33xx VV ((φφ))

Magnetic chargeNote:Note: The energy is minimal if

11 Ban = Dnφa

22 λλ == 00 ((VV ((φφ)) == 00))

Bogomol’nui-Prasad-Zommerfield equation

EE == QQ

BPS monopoleBPS monopole

Ban = Dnφa ξξ

ddKK

ddξξ== −−KKHH ,, ξξ ddHH

ddξξ== HH ++ ((11 −−KK22))

Note:Note: This system has an analytical solution:

KK ==ξξ

ssiinnhh ξξ,, HH == ξξ ccootthh ξξ −− 11

The BPS equation + Bianchi identity DDnnDDnnφφaa == 00

Homework: Prove it!

Long range asymptotic tail of the scalar field: φa → vr̂a − ra

er2

The energy is completely defined by the scalar field:

DDnnφφaaDDnnφφ

aa == ((∂∂nnφφaa))((∂∂nnφφ

aa)) ++ φφaa((∂∂nn∂∂nnφφaa)) ==

11

22∂∂nn∂∂nn((φφ

aaφφaa))

EE ==11

22

��dd33xx∂∂nn∂∂nn((φφ

aaφφaa)) ==44ππvv

ee

��ccootthh ξξ −− 11

ξξ

����11 −− ξξ

22

ssiinnhh22 ξξ

�� ������∞∞

00==

44ππvv

ee

Monopole catalysis of proton decayMonopole catalysis of proton decay

ThereThere areare zerozero--energyenergy solutionssolutions of of thethe DiracDirac equationequation forfor a a masslessmassless fermionfermion

coupledcoupled to a to a monopolemonopole; ;

Naive question:Naive question: What happened when a fermion collides with a monopole?

= + ?

SpinSpin--flip? Charge conjugation? flip? Charge conjugation? ChiralityChirality??

The ground state of a monopole becomes twoThe ground state of a monopole becomes two--fold degenerated:fold degenerated:

(i) (i) |Ω> |Ω> (no fermions; Q(no fermions; QFF= = --1/2)1/2) (ii)(ii) aa††|Ω|Ω> > (zero mode); Q(zero mode); QFF= 1/2)= 1/2)

There are nonThere are non--suppressed suppressed fermionicfermionic condensates on the monopole background:condensates on the monopole background:

��((ee++ee−− −− dd33 ¯̄dd33))(( ¯̄uu11uu11 −− ¯̄uu22uu22))))�� ∼∼ rr−−66

γγµµ((∂∂µµ ++ eeAAµµ))ψψ == 00;; ��JJ == ��LL ++ ��TT ++ ��SS ;; ��LL == 00,, ��TT ++ ��SS == 00

RubakovRubakov--CallanCallan effecteffect

Monopole could Monopole could catalysecatalyse

baryon number violating baryon number violating

processes like processes like

P + monopoleP + monopole →→ ee++ + mesons + monopole+ mesons + monopole

SU(5) model with SU(5) model with masslessmassless

fermions in sfermions in s--wavewave

Caution:Caution: the effect is modelthe effect is model--dependent! dependent!

Topology is power!

Even Abelian Dirac monopole has topological roots.

There are various monopoles in many theories.

Consistent description of monopoles is related with

topological arguments, it needs a non-Abelian theory

with spontaneousy broken symmetry.

Rubakov-Callan effect is strongly model-dependent.

Fake monopoles are just fakes

A monopole beyond the horizon?

To conclude:To conclude:

AcknowledgmentsAcknowledgments

Work done in collaboration withWork done in collaboration with::

C.Adam, P.Dorey, J.Kunz,

B.Kleihaus, U.Neemann, E.Radu

T.Romanzukiewicz, D.H.Tchrakian

and A. Wereszczynski

Other thanks toOther thanks to::

L.Ferreira, P. Forgacs,

T. Ioannidou, D.Harland,

B.Hartmann, O.Lechterfeld,

M.Speight, P.M.Sutcliffe

W.Zakrzewski and M.Volkov

Work in progress!Work in progress!