Skyrme model U : S 3 S 3. d 3 x. The Skyrme field: The topological charge: Q = 1. Rescaling: Sigma-model term Skyrme term Potential term.

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1 Skyrme model The Skyrme field: U(r, t) r I U : S 3 S 3 L = f2 π 4 Tr µ U µ U e 2 Tr U µ U,U ν U 2 f π = 186 MeV, m π = 136 MeV + m2 πf 2 π 8 Tr (U I) Sigma-model term Skyrme term Potential term The topological charge: Q = 1 24π 2 ε ijk d 3 xtr (U i U)(U j U)(U k U) The su(2) current: R i = ( i U)U Q = 1 24π 2 ε ijk d 3 xtr(r i R j R k ) Rescaling: E = f π 4e x µ 2x µ /(ef π ); m = 2m π /(f π e) d 3 x 1 2 Tr (R ir i ) 1 16 Tr ([R i,r j ]) 2 +m 2 Tr(U I)

2 Rational map Skyrmions The Skyrme field is effectively a map U: S 3 SU(2) ~ S 3 (N.S. Manton, C.Houghton & P.Sutcliffe,, 1998) ˆn z = Stereographic Projection: 1 1+ z 2 z = tan(θ/2)e iϕ z + z 1+z z, z iz 1+z z, 1 z z 1+ z = (sinθcosφ,sinθsinφ,cosθ) The idea of the rational map ansatz: Separate the radial and the angular dependenceof theskyrmefield as Identify spheres S 2 withconcentric spheres in compactified R 3 Identifytargetspace S 2 with spheres of latitudeon S 3 ˆn Z = U = exp{if(r)ˆn Z σ} Z+ Z 1+Z Z, i Z Z 1+Z Z, 1 Z Z 1+Z Z Z = P(z)/Q(z) Domain space Target space

3 Static energy: 1+ z 2 4πQ= 1+ Z 2 Rational map approximation E = 4π dz 2 dzd z dz r 2 f 2 +2Q(f 2 +1)sin 2 f +W sin4 f r (1+ z 2 ) 2 W= 1 4π 1+ z 2 1+ Z 2 dz dz 4 dr dzd z (1+ z 2 ) 2 The holomorphic maps of degree Q: Q= 4: Z(z) = z4 +2i 3z 2 +1 z 4 2i 3z 2 +1 (Octahedral Skyrmions) Q= 7: Z(z) = z7 7z 5 7z 2 1 z 7 +7z 5 7z 2 +1 (Icosahedral Skyrmions)

4 Skyrmions Q=1 Q=2 Q=3 Q=4 Q=5 Q=6 Q=7 Q=8

5 Skyrmions N.Manton et al

6 Crystalline structure of nucleons R. A. Battye,, N. S. Manton, C.Houghton and P. Sutcliffe (1996,2004) Shell vs. Crystal Shell wins for m c 0.16

7 Skyrmion s interaction There is no self-dual skyrmions: E í B A single Skyrmion is approximated by a triplet of orthogonal dipoles Field equations: Asymptotically The dipole-dipole interaction energy Attractive channel: µ R µ [Rν,[R ν R µ ]] = 0 φ k d k r k 4πr 3 + O(r 3 ); φ 0 1 R ñn There are 6 zero modes of the B=1 Skyrmion: 3 translations + 3 rotations E int = 2d2 3πR 3 (cosα 1)[1 3( R n)] Translations: U iu 1 2( k U)U U 1 2 d 3 xtr( i U i U 1 2 Rotations (Adkins, Nappi & Witten (1983)); rigid body approximation U A(t)UA(t) 8

8 BPS Skyrmions C.Adam, J.Sánchez-Guillén and A.Wereszczynski L = L 2 +L 4 +L 0 +L 6 E ±Q L 2 = f2 π 4 Tr U µ UU µ U U L 4 = 1 32e Tr 2 µ U,U ν U 2 L 0 = µ 2 V L 6 = λ2 24π 2Tr ε µνρσ U µ UU ν UU ρ U 2 = Bµ B µ π 2 Q±µ V = 0 BPS BPS equation: Tr ε µνρ U µ UU ν UU ρ U =±C L 0 Hedgehog parametrization: U(r) = exp[iτaˆr a F(r)] z = 2µr 3 3λ λ 2 sin 2 F sin 2 FF Field equation: 2r 2 r r 2 =L 0 cos 3 F 2 =±3 4 (z z 0)

9 BPS Skyrmions - Compactons Field equation: cos 3 F 2 =±3 4 (z z 0) F(z) = 3z 2arccos 3 4 0, z 4 3, ; z [0, 4 3 ] More compactons (non-bps): L = L 0 +L 4 +L 6

10 Skyrmion-antiSkyrmion chains (P. Sutcliffe, S.Krusch, Y.Shnir, T.Tchrakian ) φα = sin f sin g nα ; φ3 = sin f cos g ; φ4 = cos f ; θ=π Q = 12 n cos(mθ ) θ=0 = 12 n [1 ( 1)m ] f ( 0) = π, f ( ) = 0 r g (0) = 0, g ( ) = mπ

11 Knots and links If there is a physical realisation? Lord Kelvin: 1867: Vortex Atoms

12 Construction of the Hopfion φ a φ a = 1 φ a = (φ 1,φ 2,φ 3 ) S 2 φ vac = (0,0,1) φ 1 +iφ 2 (φ 1 +iφ 2 )e iα Residual SO(2) symmetry d=3+1 Loops in domain space S 3 Target space S 2 φ : S 3 S 2 Homotopy classp 3 (S 2 ) = φ 1 +iφ 2 = sinf(ρ,z)e inϕ+ig(ρ,z)z ; φ 3 = cosf(ρ,z) F µν = 1 2 ε µνρφ a µ φ b ν φ c = µ A ν ν A µ F = 1 2 F µνdx µ dx ν ; df = 0; F = da Q = 1 8π 2 ε ijk A i F jk d 3 x Topological charge: (Linking number) z ϕ ρ

13 Construction of the Hopfion φ 1 +iφ 2 = sinf(ρ,z)e inϕ+img(ρ,z)z ; φ 3 = cosf(ρ,z) n,m Z F = 1 2 F µνdx µ dx ν ; F = da A = ncos 2 F 2 dg+msin2 F 2 dϕ A F = nmcos 2 F 2 df dg dϕ Q = nm Rational map: Z : S 3 CP 1 Step 1: R 3 C 2 (Z 1,Z 0 ) = Z = φ 1 +iφ 2 1+φ 3 x+ix2 sinf,cosf +i sinf r r x 3 Step 2: input configuration Z = P(Z 1,Z 0 ) Q(Z 1, Z 0 ) Axially symmetric hopfion QA n,m Z = Zn 1 Z m 0

14 Faddev-Skyrme model Torus knot: Q = αb+βa QK a,b Z = Zα 1 Z β 0 Z1 a +Zb 0 Links: QL α,β p,q Z = Q = p+q +α+β Z α 1 (Z 1 Z 0 ) a + Z β 1 (Z 1 +Z 0) b ϕ a = (0, 0, 1) at infinity (in any direction) ϕ a = (0, 0, 1) on the ring x 2 + y 2 = 1, z = 0 (vortex core) Vakulenko-Kapitansky topological bound: E cq 3/4 Faddeev-Skyrme Lagrangian L = 1 2 ( iφ a ) 2 κ2 4 εabc φ a i φ b j φ c 2

15 Solitons of the Faddeev-Skyrme model 1A 1,1 2A 2,1 3 A 3,1 Q=1 Q=2 Q=3 4A 2,2 4 A 4,1 4L 1,1 Q=4 Q=4 Q=4 1,1

16 Solitons of the Faddeev-Skyrme model Q=5 5A Q=5 5L 1,2 5,1 1,1 Q=6 6L 2,2 1,1 Q=6 6L 3,1 1,1 Q=7 7K 3,2 Q=8 8A Q=8 8L 3,3 4,2 Q=8 8K 3,2 1,1

17 Buckled, linked and knotted hopfions 5 A 5,1 4L 1,1 1,1 1A 1,1 Axially-symmetric 8K 3,2 Buckled 6L 3,1 1,1 Linked 4A 2,2 Knotted 7K 3,2 9L 2,2,2 1,1,1 9K 3,2

18 Position curves and linking numbers 4A 4,1 8 A 4,2 8K 3,2 6L 1,1 3,1

19 Elastic rod approximation Tubular coordinates: s [0,L] ρ,θ Arclength parameter Polar coordinates in the disk Tangent vector t(s) D. Harland, J.M. Speight and P.M. Sutcliffe Phys. Rev.D83 (2011) Torsion τ(s) Curvature κ(s) Frenet frame Position curve γ(s) R 3 Preimage of φ = (0,0,1) d ds t(s) n(s) = b(s) Twisting function: frame vector m(s) m(s)=n(s)sinα(s)+ b(s)cosα(s) 0 κ(s) 0 t(s) κ(s) 0 τ(s) n(s) 0 τ(s) 0 (s) α(s) =t m m α(l) = α(0)+2πn Faddeev-Skyrme effective energy functional: E = A+Bκ 2 +C(α τ) 2 ds

20 Hopfions in NLS system Non-linear Schrödinger equation: iψ t + ψ +2σ ψ 2 ψ = 0 σ(ρ) = e ρ2 /2 ψ 2 [0, [ Iψ R ψ ψ : S 3 S 2 /N ψ = F(z,ρ)e iµt e imφ Chemical potential Vorticity m=0 m=1 m=2

21 Monopoles Particle physics SUSY Condensed matter Branes and extra dims Black holes g?? Astrophysics & Cosmology QFT QCD Comfinement Topology Differential Geometry

22 Magnetic Monopoles: : Historical remarks Petrus Peregrinus Poles of a magnet 1269 H. Poincaré Classical e-g interaction 1896 P.A.M. Dirac Charge quantization 1931 H. Hopf Hopf bundle 1931 P.A.M.Dirac et al QEMD 1948 G `t Hooft, A.Polyakov Non-Abelian Monopoles 1974 T.T.Wu, C.N.Yang Geometry & Monopoles 1975 E.Bogomolny et al BPS Monopoles 1976 C.Montonen and D.Olive Duality revised 1978 S.Mandelstam, G`t Hooft Dual Meissner effect 1970s V.Rubakov, C.Callan Monopole catalysis 1981 N.Manton, M.Atiyah Moduli space 1982 N.Seiberg and E.Witten N=2 SUSY 1994

23 Electromagnetic duality and Dirac monopole System of generalized Maxwell equations E = 4πe; E + " B t = j g B = 4πg; B " E t = j e is invariant with respect to the transformations of electromagnetic etic duality: E B E cosϑ B cosϑ; E sinϑ + B cosϑ e ecosϑ g cosϑ; g esinϑ + g cosϑ Classical motion in the monopole Coulomb magnetic field: m d2 "r dt = e[v B] = eg 2 r 3 d"r dt r Generalized angular momentum is conserved: J = [r mv] eg "r r = L egˆr L J

24 Dirac s s monopole: Charge quantization B = g "r r 3 = A; B = 4πg? A = g r "r "n r ("r "n) B = B g + B string = g r 4πg n θ(z)δ(x)δ(y) r3 Gauge invariance: A A+ λ(n,n ) - Dirac monopole potential Dirac s s string is invisible if the charge quantization condition is imposed: eg = n