Maximally Non-Abelian Vortices from Self-dual Yang-Mills Fields - PowerPoint PPT Presentation

Maximally Non-Abelian Vortices from Self-dual Yang-Mills Fields Norisuke Sakai (Tokyo Woman’s Christian University) In collaboration with Nicholas Manton , Phys.Lett. B687 , 395-399,(2010) [arXiv:1001.5236] , Talk at YITP workshop 2010.7.21 Contents 1 Introduction 2 2 SO (3) Invariant Instantons 3 3 General SO (3) Invariant Gauge Fields 5 4 Maximally Non-Abelian Vortices 7 5 Conclusion 10

1 Introduction Non-Abelian Vortex : plays an important role in Dual Confinement Cosmic String Moduli gives Effective Fields on the soliton Moduli Space describes Dynamics of Non-Abelian Vortices Non-Abelian Vortices in U ( N ) gauge theory : Moduli Matrix Approach No Exact solutions Exactly Solvable Vortex : U (1) Vortex on a Hyperbolic Plane Equivalent to Instantons along a Line Dimensional Reduction of Instantons to Hyperbolic Plane → Vortices Witten, Phys.Rev.Lett. 38 , 121 (1977) Our Pourpose: Find Exactly Solutions of Non-Abelian Vortices 2

2 SO (3) Invariant Instantons Pure SU (2) Gauge Theory in Euclidean 4 dimensions Instantons as Solutions of Self-duality Equations F µν = 1 2 ϵ µνλρ F λρ Instantons along a line (Let’s call it τ axis) Invariant under Rotations SO (3) around τ axis ( SU (2) gauge transformations can be accompanied) Take spherical polar coordinates r, θ, φ for S 3 ds 2 = dτ 2 + dr 2 + r 2 ( dθ 2 + sin 2 θdϕ 2 ) SO (3) invariant configurations : functions of τ, r (independent of θ, ϕ ) Complex coordinates (Stereographic projection of S 2 ) y = tan θ 2 e iϕ z = τ + ir, ( ) 4 ds 2 = dzd ¯ z + (Im z ) 2 y ) 2 dyd ¯ y (1 + y ¯ 3

Conformally equivalent to hyperbolic plane and sphere Σ × S 2 ( ) ds 2 = (Im z ) 2 2 8 (Im z ) 2 dzd ¯ z + y ) 2 dyd ¯ y 2 (1 + y ¯ Yang-Mills Theory and Self-Duality is Conformally invariant SO (3) invariant Instantons are equivalent to U (1) vortices on a hyperbolic plane Σ Witten Ansatz : SO (3) of S 2 is embedded into SU (2) 0 = A 0 x a j = ϕ 2 + 1 ϵ jak x k + ϕ 1 x j x a A a A a r 3 ( δ ja − x j x a ) + A 1 r 2 , r 2 r Only U (1) ∈ SU (2) gauge symmetry is intact A gauge transformation gives ( A r = A 1 , H = − ϕ 1 − iϕ 2 ) ( 1 ) 0 A j = A i ( τ, r ) j = τ, r, , 0 − 1 ( ) ( ) ¯ − i ¯ 0 H ( τ, r ) − cos θ H ( τ, r ) sin θ A θ = , A ϕ = H ( τ, r ) 0 iH ( τ, r ) sin θ cos θ A i ( τ, r ) : 2 Dimensional gauge fields for U (1) ( I 3 of SU (2) ) 4

H ( τ, r ) : charged complex scalar field Self-Duality 1 1 1 F τr = r 2 sin θ F θϕ , F τθ = sin θ F ϕr , F rθ = sin θ F τϕ Reduces to BPS equations for Vortices on a Hyperbolic Plane 1 2 r 2 (1 − | H | 2 ) D τ H = iD r H, F τr = 3 General SO (3) Invariant Gauge Fields Metric on Σ × S 2 ( σ = 2 (Im z ) 2 , if Σ is the hyperbolic plane) 8 ds 2 = σ ( z, ¯ z ) dzd ¯ z + y ) 2 dyd ¯ y (1 + y ¯ Field configuration should be invariant under a combined spatial SO (3) rotation and gauge SO (3) rotation General Embedding of SO (3) into Non-Abelian Group G Isotropy generator SO (2) is mapped to an SO (2) generator Λ in G Most general SO (3) invariant gauge potential A z = A z ( z, ¯ z ) , A ¯ z = A ¯ z ( z, ¯ z ) 5

1 1 y (¯ A y = y ( − Φ( z, ¯ z ) − i Λ¯ y ) , A ¯ y = Φ( z, ¯ z ) + i Λ y ) 1 + y ¯ 1 + y ¯ SO (2) = U (1) invariance (generators are anti-hermitian matrix) [Λ , A z ] = [Λ , A ¯ z ] = 0 [Λ , ¯ Φ] = i ¯ [Λ , Φ] = − i Φ , Φ Self-Duality ( F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] ) 8 F z ¯ y = 0 , F ¯ zy = 0 , y ) 2 F z ¯ z = σ F y ¯ y (1 + y ¯ ( ) z = σ D z ¯ 2 i Λ − [Φ , ¯ Φ = 0 , D ¯ z Φ = 0 , F z ¯ Φ] 8 Finite Energy Solutions → Vacuum ( F z ¯ z = 0 ) at z → ∞ Vacuum value Φ 0 of Φ forms SO (3) algebra [Λ , ¯ Φ 0 ] = i ¯ [Φ 0 , ¯ [Λ , Φ 0 ] = − i Φ 0 , Φ 0 , Φ 0 ] = 2 i Λ Boundary Condition at r = Im z = 0 : Fields approach vacuum values 6

4 Maximally Non-Abelian Vortices Take SU (2 N ) gauge group : Λ can be taken in Cartan subalgebra   Λ 1 ∑   Λ 2 ...   Λ = i Λ α = 0  ,  Λ 2 N [Λ , Φ] = − i Φ → Λ β − Λ α = 1 if Φ αβ ̸ = 0 Maximally Non-Abelian case ( 1 N ) Λ = i 0 0 − 1 N 2 � SU (2 N ) → SU ( N ) × SU ( N ) × U (1) gauge symmetry SO (3) invariant gauge fields on Σ × S 2 ( A z ) ( A ¯ ) 0 0 z A z = , A ¯ z = � � 0 0 A z A ¯ z ( 0 0 ) ( 0 H † ) ¯ Φ = Φ = , H 0 0 0 7

A z ) : SU ( N ) ( � A z , ( ˜ SU ( N ) ) gauge field � H : A Higgs scalar in Bi-fundamental of SU ( N ) × SU ( N ) Bogomolny equations for non-Abelian Vortices on Hyperbolic Plane D z H † = 0 , D ¯ z H = 0 ( ) ( 1 N − HH † ) z = σ z = σ � − 1 N + H † H F z ¯ , F z ¯ 8 8 z H + � D ¯ z H = ∂ ¯ A ¯ z H − HA ¯ z , F z ¯ z = ∂ z A ¯ z − ∂ ¯ z A z + [ A z , A ¯ z ] Vacuum Solutions   1   1   � A z = 0 , A z = 0 H =  , ...  1 Unbroken local gauge symmetry : SU ( N ) d diagonal gauge group If SU (2 N ) → SU ( N 1 ) × SU ( N 2 ) × U (1) , N 1 ̸ = N 2 , F z ¯ z = 0 vacuum does not exist 8

Exact Vortex Solutions     ia (1) h (1) z ¯     1 0   z = − �   H = z =  , A ¯ A ¯ ... ...    1 0 Bogomolny equations reduce to ( f (1) z = ∂ z a (1) z a (1) − ∂ ¯ z ) z ¯ z ¯ ( − 1 + | h (1) | 2 ) z = σ z h (1) − 2 ia (1) z h (1) = 0 , if (1) ∂ ¯ ¯ z ¯ 8 = Witten’s equation for U (1) vortices on hyperbolic plane Exactly solved by mapping to the Liouville equation We found exact solutions in the diagonal U (1) N subgroup Genuine non-Abelian vortices (fractional U (1) and SU ( N ) winding) Solutions with complete orientational moduli remain to be worked out Moduli Matrix and Master Equations Solution of the first BPS equation � z = � z � z = S − 1 ∂ ¯ S − 1 ∂ ¯ A ¯ z S − ∂ ¯ z ψ 1 N , A ¯ S + ∂ ¯ z ψ 1 N 9

1 z ) � 2 ψ ( z, ¯ S − 1 ( z, ¯ H ( z, ¯ z ) = e z ) H 0 ( z ) S ( z, ¯ z ) Moduli matrix H 0 ( z ) , Master equations ( Ω ≡ SS † , � Ω ≡ � S � S † ) ( ) − 1 + 1 z ψ = σ Ω − 1 H 0 Ω H † N e ψ Tr( � 0 ) ∂ z ∂ ¯ 4 ( ) Ω − 1 H 0 Ω − 1 z Ω) = σ H † Ω − 1 H 0 Ω H † 0 � N 1 N Tr( � ∂ z (Ω − 1 ∂ ¯ 8 e ψ 0 ) ( ) 0 − 1 Ω) = − σ Ω − 1 H 0 Ω H † Ω − 1 H 0 Ω H † ∂ z ( � z � � N 1 N Tr( � Ω − 1 ∂ ¯ 8 e ψ 0 ) 5 Conclusion 1. SO (3) symmetric instantons of SU (2 N ) gauge group gives non- Abelian vortices on a hyperbolic plane . 2. Maximally non-Abelian case gives non-Abelian vortices in SU ( N ) × � SU ( N ) × U (1) gauge group. 3. The maximally non-Abelian vortices possess unbroken non-Abelian gauge symmetry SU ( N ) d . 4. Exact solutions of U (1) N subgroup are completely obtained, but the orientational moduli remain to be worked out. 10