Non-Abelian Monopoles in the Higgs Phase Muneto Nitta and Walter Vinci arXiv: 1012.4057; Nucl.Phys.B848:121-154,2011 Walter Vinci Research Fellow University of Minnesota Outline Introduction-Overview Coulomb Phase Non-Abelian monopoles Moduli spaces (Rational Maps) Higgs Phase Non-Abelian vortices and kinks Moduli spaces (Moduli Matrix) Kinks Monopoles ⇔ Non-Abelian” Monopoles and Kinks “ Moduli Matrix for Monopoles Motivations Non-Perturbative Dynamics of Gauge Field Theories Monopoles and Vortices Dual superconductivity model of confinement Nambu-‘t Hooft-Mandelstam Supersymmetric Gauge Theories Reliable control of strong coupling effects Seiberg-Witten Confinement, dualities String/Field theory interplay: D-branes/domain walls, strings/vortices, wall/vortex junction Monopoles in the Coulomb Phase Non-Abelian Monopoles Montonen, Olive, Weinberg et al. SU(N) Gauge theory: 1 1 2 2 = Tr (F ) + D Φ µν µ L 2g2 g2 | | µ . . . 0 1 N . . . 2 Φ = v/N, Φ = . . . , µ = 0 . . . i | | ⇒ ∞ ∞ i=1 0 . . . µ N SU (N ) Abelian case: all eigenvalues N 1 N 1 π = π (U (1) ) = . Z 2 1 − − U (1)N 1 are different − SU (N ) Non-Abelian case: some q 1 π = Z 2 − eigenvalues coincide S(U (n ) U (n ) U (n )) 1 2 q × · · · × Monopoles in the Coulomb Phase Non-Abelian Monopoles Bogomol’nyi completion: 1 1 3 2 E = d x Tr ( F D Φ) + ∂ ( ΦF ) ijk jk i i ijk jk 4g2 − g2 Topological term Bogomol’nyi equations: D Φ = F i ijk jk Monopoles in the Coulomb Phase Moduli Spaces (A , Φ) satisfying BPS equations and boundary conditions i (x) G : (0 , 0, x ) 1 , x . 0 0 3 3 G ∈ G → → ∞ SU(2) 3 1 = (A , Φ) / = S R framed i 0 framed M G M × Nham transorm, Spectral curves, Twistors, Rational maps... Monopoles in the Coulomb Phase Rational Map Construction Scattering data of the Hitchin equation along x 3 Donaldson ψ = (D + Φ)ψ = 0 H 3 ∇ µ 0 Φ = SU(2) case 0 µ ∞ − 0 m/g µ x ψ(x ) x e 3 x + 3 3 − 3 ∼ 1 | | → ∞ 1 0 m/g µ x m/g µ x x ψ(x ) b x e 3 + a x e 3 3 3 3 3 − −∞ ← ∼ 0 | | 1 | | [D , ] = 0 D ψ ∂ ψ = ∂ ψ = 0 z¯ H z¯ H z¯ H H z¯ ∇ ∇ ∼ ∇ ∇ a a(z), b b(z) Holomorphic functions → → Monopoles in the Coulomb Phase Rational Map Construction SU(2) Case: m 1 b(z) q z + + q 1 − m R (z) = · · · m ≡ a(z) zm + p zm 1 + + p 1 − m · · · m : Monopole charge Holomorphic (based) rational map to the projective complex line SU (2) 1 1 1 R : P P P = C C C m → U (1) R = (a(z), b(z)) m α 1 R = Fundamental SU(2) monopole : α = 0 i z z 0 − SU(2),k=1 3 1 = (z ) (α) S C C R 0 ∗ framed M × × Monopoles in the Coulomb Phase Rational Map Construction Jarvis General case: flag manifolds 1 R(z) : P lag C F n ,...,n → 1 q SU (N ) lag = F n ,...,n 1 q S(U (n ) U (n )) 1 q × · · · × Monopoles in the Coulomb Phase
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