Monopoles and Skyrmions Ya. Shnir Quarks-2010 Коломна, Россия, 8 Июня 2010
Outline Abelian and non-abelian magnetic monopoles BPS monopoles Rational maps and construction of multimonopoles SU(2) axially symmetric multimonopoles Monopole-antimonopole chains Multimonopole moduli space Skyrmions Skyrme model Construction of multiskyrmions Axially symmetric multiskyrmions Rational maps and multiskyrmions Skyrmion-antiSkyrmion chains Multiskyrmions moduli space Summary and outlook
Yang-Mills Mills-Higgs Theory S= 1 2 d 4 x{f µν F µν + (D µ Φ)(D µ Φ) V(Φ)} F µν = µ A ν ν A µ +ie[a µ,a ν ] D µ Φ= µ Φ+ie[A µ,φ] V(Φ)=λ(Φ 2 η 2 ) 2 t Hooft-Polyakov static spherically symmetric solution φ a = ra er 2 H(eηr); A a n =ε amn rm er 2 (1 K(eηr)) R c ~m v -1
Bogomolny equations: BPS monopole mass: BPS monopoles λ =0, B k = D k F M = 4ph/ e Homotopy group π 2 (S 2 ) Long-range scalar field F ~ 1/r No net interaction between the BPS monopoles Analytical solution of the BPS equations: K= ξ sinhξ ; H=ξcothξ 1 Magnetic charge of a monopole is a topological number Φ: S 2 S 2 Sir M. Atiyah,, R. Ward (1977), P. Forgacs et al (1981), W. Nahm (1982), P. Sutcliffe (1996) and other
Self-dual monopoles vs non-self dual monopoles BPS monopoles: In the limit h=0 the energy becomes: E=Tr d 3 x { 1 4 (ε ijkf ij ±D i Φ) 2 1 2 ε ijkf ij D k Φ } The first order Bogomol'nyi equations yield absolute minimum: M =4πg B k =±D k Φ No net interaction between the BPS monopoles: the electromagnetic repulsion is compensated by the long-range scalar interaction. Non self-dual monopoles: They are solutions of the second order Yang-Mills equations: µ F µν =0 E>M BPS even if g=0 (deformations of the topologically trivial sector) The constituents are non BPS monopoles and/or vortices in a static equilibrium; separation is relatively small, there are no long-range forces
Rational map monopoles There is a transformation of a monopole into a rational map from the Riemanian sphere to inself: R: S 2 # S 2 (P. Sutcliffe, N.Manton et al) R(z)= a(z) b(z) = a 1z n 1 + +a n z n +b 1 z n 1 + +b n, z=x 1 +ix 2 Construction of the rational maps monopoles: Represent BPS equation in spherical coordinates r,z,z Impose a complex gauge Φ= ia r = i 2 U 1 r U, A z =U 1 z U, A z =0 Construct the monopoles using { U exp 2r 1+ R 2 ( R 2 1 2R 2R 1 R 2 )}
R = 1/z: one spherically symmetric monopole centered at the origin; R(z) = a 1z+a 2 z 2 +b 1 z+b 2 : two monopoles R(z)= i 3z 2 1 z(z 2 i : Tetrahedral monopoles (degree 3 map) 3) R(z)= z4 +2i 3z 2 +1 z 4 2iz 2 : Octahedral monopoles (degree 4 map) 3+1
Numerical Technique Energy minimization approach (static solutions in 2d and in 3d) supercomputers and grid systems Solution of system of PDE s obtained by imposing some symmetry conditions, boundary problem in in 2d and in 3d: Newton-Raphson iterative procedure (FIDISOL/CADSOL package); boundary problem in d1 (COLSYS package) Gradient flow method Dynamics of the solitons: : (pseudo( pseudo) spectral methods, symplectic methods.
Non-BPS axially symmetric monopoles MA pair: magnetic dipole (Taubes, Nahm, Rüber, R Kleihaus,Kunz & Shnir) A µ dx µ = ( K 1 r dr+(1 K 2)dθ ) τ (n) ϕ 2e nsinθ ( ) Φ=Φ aτa 2 = τ r H (n,m) τ 1 2 +H (n,m) θ 2 2. ( ) τ K (n,m) r 3 2e +(1 K 4 ) τ(n,m) θ 2e dϕ; Magnetic charge: Q= 1 2π S 2 (ΦdΦ dφ) = 1 2 n[1 ( 1)m ]
Monopole-antimonopole chains
Effective electromagnetic interaction } `t Hooft tensor: F µν ={ˆΦF µν 2eˆΦD i µˆφd νˆφ The electric and magnetic (topological) currents: µ F µν =4πjν; el µ F µν =4πjν mag The magnetic field if generated by the magnetic charges and electric currents j el r =jel θ =0; 4πjel ϕ = 2 1 ra ϕ +sinθ θ r 2 sinθ θa ϕ
Effective electromagnetic interaction `t Hooft tensor: ˆ Fµν Fµν = Φ Φ i ˆ ˆ Dν Φ ˆ Φ Dµ Φ Φ Φ 2e Φ The electric and magnetic (topological) currents: µ Fµν = 4π jνel ; µ F µν = 4π jνmag The magnetic field if generated by the magnetic charges and electric currents jrel = jθel = 0; 4πjϕel = r2 Aϕ + sin θ θ r2 s1in θ θ Aϕ
Skyrme field L= d 3 x Skyrmions U SU(2); R µ =( µ U)U 1 { } F 2 π 16 Tr(R µr µ ) 1 32e Tr([R 2 µ,r ν ][R µ,r ν ])+ m2 π F2 π 8 Tr(1 U) Nonlinear pion theory: U =φ 0 +iσ k φ k, φ 2 0 +φ2 k =1 Baryon charge is identified with the topological number U : S 3 S 3 B= 1 24π 2 d 3 xε ijk Tr(R i R j R k ) Homotopy group π 3 (S 3 ) B= 1: Spherically symmetric skyrmion U =exp{if(r)ˆn σ} ˆn=(sinθcosϕ,sinθsinϕ,cosθ) Boundary conditions: f ( r), with f (0) = πk, and The approximation : f ( ) = 0. f ( r) = 4 arctan[exp(r) ]
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 + 1 4 [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)
Axially symmetric multiskyrmions Trigonometric parametrisation: φ α =sinfsingn α ; φ 3 =sinfcosg; φ 4 =cosf; n α =(cosnϕ,sinnϕ) B= 2:
Stereographic projection: Rational map Skyrmions (N. S. Manton,C.Houghton and P. Sutcliffe) z=tan(θ/2)e iϕ ˆn z = 1 1+ z 2 (z+z, i(z z), 1 z 2 ) ˆn R : S 2 S 2 U =exp{if(r)ˆn R σ} ˆn R = 1 1+ R 2 (R+R, i(r R), 1 R 2 ) The holomorphic map of degree B: R=a(z)/b(z) R(z)= z4 +2i 3z 2 +1 B= 4: B= 7: (Octahedral Skyrmions) z 4 2i 3z 2 +1 B= 7:R(z)= z7 7z 5 7z 2 1 z 7 +7z 5 7z 2 +1 (Icosahedral Skyrmions)
R. A. Battye,, N. S. Manton, C.Houghton and P. Sutcliffe (1996,2004) Shell vs. Crystal Shell wins for m c 0.16
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
4 φ ( r, θ ) cos f ( r, θ ) (Skyrmion - antiskyrmion chains) Φ( r, θ ) 2 (MAM chains)
L= d 3 x Skyrmion-antiSkyrmion chains: mass term { } F 2 π 16 Tr(R µr µ ) 1 32e Tr([R 2 µ,r ν ][R µ,r ν ])+ m2 π F2 π 8 Tr(1 U) n=1 chains exist for m c 0.10
Skyrmions and instantons Skyrme field can be generated from the holonomy of a Yang-Mills instanton (M. Atiyah & N.Manton (1989), P. Sutcliffe (2010) { } U(x)=Pexp A U(x) : R 3 4 (x, x 4 )dx 4 SU(2) Baryon charge is equal to the Pontryagin charge: B= 1 24π 2 d 3 xε ijk Tr(R i R j R k ) N = 1 16π 2 Tr d 4 xf µν F µν Remarkable approximation to the exact solution of the Skyrme model (about 1%) Sakai-Sugimoto Sugimoto conjecture about a correspondence between Yang-Mills Mills-Chern- Simons instantons on a curved four-manifold and an extended Skyrme model (Skyrme( field and an infinite tower of massive vector mesons)
Summary and Outlook There is certain similarity between the monopoles and the Skyrmions. Rational maps approach works both for BPS monopoles and for Skyrmions. Axially-symmetric sphaleron solutions representing chains of solitons in alternating order exist in both models. Non-trival holonomy and Skyrmion-anti- Skyrmion chains? Skyrmed monopoles and rational maps? Gauged Skyrmions?