Non-abelian dyons in anti-de Sitter space Elizabeth Winstanley - PowerPoint PPT Presentation

Non-abelian dyons in anti-de Sitter space Elizabeth Winstanley Consortium for Fundamental Physics School of Mathematics and Statistics University of Sheffield United Kingdom Work done in collaboration with Ben Shepherd Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 1 / 13

Outline A brief history of Einstein-Yang-Mills 1 su ( N ) EYM with Λ < 0 2 Dyonic solutions 3 Conclusions and outlook 4 Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 2 / 13

Introduction A brief history of Einstein-Yang-Mills Asymptotically flat EYM studied for over 20 years Purely magnetic su (2) solitons and black holes found numerically 1989-90 Have no magnetic charge Non-abelian baldness of asymptotically flat su (2) EYM If Λ = 0, the only solution of the su (2) EYM equations with a non-zero (electric or magnetic) charge is Abelian Reissner-Nordstr¨ om Rules out dyonic su (2) asymptotically flat solutions [ Ershov and Gal’tsov, PLA 138 160 (1989), 150 159 (1990) ] Ershov/Gal’tsov result no longer holds for larger gauge group in asymptotically flat space What about asymptotically anti-de Sitter space? Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 3 / 13

Introduction A brief history of Einstein-Yang-Mills Asymptotically flat EYM studied for over 20 years Purely magnetic su (2) solitons and black holes found numerically 1989-90 Have no magnetic charge Non-abelian baldness of asymptotically flat su (2) EYM If Λ = 0, the only solution of the su (2) EYM equations with a non-zero (electric or magnetic) charge is Abelian Reissner-Nordstr¨ om Rules out dyonic su (2) asymptotically flat solutions [ Ershov and Gal’tsov, PLA 138 160 (1989), 150 159 (1990) ] Ershov/Gal’tsov result no longer holds for larger gauge group in asymptotically flat space What about asymptotically anti-de Sitter space? Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 3 / 13

Introduction A brief history of Einstein-Yang-Mills Asymptotically flat EYM studied for over 20 years Purely magnetic su (2) solitons and black holes found numerically 1989-90 Have no magnetic charge Non-abelian baldness of asymptotically flat su (2) EYM If Λ = 0, the only solution of the su (2) EYM equations with a non-zero (electric or magnetic) charge is Abelian Reissner-Nordstr¨ om Rules out dyonic su (2) asymptotically flat solutions [ Ershov and Gal’tsov, PLA 138 160 (1989), 150 159 (1990) ] Ershov/Gal’tsov result no longer holds for larger gauge group in asymptotically flat space What about asymptotically anti-de Sitter space? Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 3 / 13

Introduction A brief history of Einstein-Yang-Mills Asymptotically flat EYM studied for over 20 years Purely magnetic su (2) solitons and black holes found numerically 1989-90 Have no magnetic charge Non-abelian baldness of asymptotically flat su (2) EYM If Λ = 0, the only solution of the su (2) EYM equations with a non-zero (electric or magnetic) charge is Abelian Reissner-Nordstr¨ om Rules out dyonic su (2) asymptotically flat solutions [ Ershov and Gal’tsov, PLA 138 160 (1989), 150 159 (1990) ] Ershov/Gal’tsov result no longer holds for larger gauge group in asymptotically flat space What about asymptotically anti-de Sitter space? Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 3 / 13

Introduction A brief history of Einstein-Yang-Mills Asymptotically flat EYM studied for over 20 years Purely magnetic su (2) solitons and black holes found numerically 1989-90 Have no magnetic charge Non-abelian baldness of asymptotically flat su (2) EYM If Λ = 0, the only solution of the su (2) EYM equations with a non-zero (electric or magnetic) charge is Abelian Reissner-Nordstr¨ om Rules out dyonic su (2) asymptotically flat solutions [ Ershov and Gal’tsov, PLA 138 160 (1989), 150 159 (1990) ] Ershov/Gal’tsov result no longer holds for larger gauge group in asymptotically flat space What about asymptotically anti-de Sitter space? Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 3 / 13

su ( N ) EYM with Λ < 0 The model for su ( N ) EYM Einstein-Yang-Mills theory with su ( N ) gauge group d 4 x √− g [ R − 2Λ − Tr F µν F µν ] S = 1 � 2 Field equations R µν − 1 2 Rg µν + Λ g µν = T µν D µ F µ ν = ∇ µ F µ ν + [ A µ , F µ ν ] = 0 Stress-energy tensor ν − 1 T µν = Tr F µλ F λ 4 g µν Tr F λσ F λσ Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 4 / 13

su ( N ) EYM with Λ < 0 Static, spherically symmetric, configurations Metric ds 2 = − µ ( r ) σ ( r ) 2 dt 2 + [ µ ( r )] − 1 dr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � − Λ r 2 µ ( r ) = 1 − 2 m ( r ) 3 r su ( N ) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ] Static, dyonic, gauge potential A µ dx µ = A dt + 1 d θ − i � C − C H � �� C + C H � � sin θ + D cos θ d φ 2 2 N − 1 electric gauge field functions h j ( r ) in matrix A N − 1 magnetic gauge field functions ω j ( r ) in matrix C Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 5 / 13

su ( N ) EYM with Λ < 0 Static, spherically symmetric, configurations Metric ds 2 = − µ ( r ) σ ( r ) 2 dt 2 + [ µ ( r )] − 1 dr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � − Λ r 2 µ ( r ) = 1 − 2 m ( r ) 3 r su ( N ) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ] Static, dyonic, gauge potential A µ dx µ = A dt + 1 d θ − i � C − C H � �� C + C H � � sin θ + D cos θ d φ 2 2 N − 1 electric gauge field functions h j ( r ) in matrix A N − 1 magnetic gauge field functions ω j ( r ) in matrix C Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 5 / 13

su ( N ) EYM with Λ < 0 Field equations Yang-Mills equations �� � � � ω 2 � σ ′ � σ − 2 2 ( k + 1) k + 1 k − 1 k h ′′ h ′ = + h k − h k − 1 k k µ r 2 2 k 2 k r k �� � � ω 2 � 2 k k + 2 k k +1 + 2 ( k + 1) h k − 2 ( k + 1) h k +1 µ r 2 k + 1 � 2 �� � � σ ′ σ + µ ′ � k + 1 k − 1 ω k ω ′′ k + ω ′ 0 = + h k − h k − 1 k σ 2 µ 2 2 k 2 k µ � �� + ω k k + 1 1 − ω 2 ω 2 k − 1 + ω 2 � k +1 µ r 2 2 Einstein equations Give µ ′ ( r ) and σ ′ ( r ) in terms of ω k , h k and their derivatives Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 6 / 13

Dyons Solving the field equations Numerically solve the field equations for su (2) and su (3), with different values of Λ < 0 Soliton solutions Regular at the origin r = 0 Solutions parameterized by ω ′′ k (0) and h ′ k (0) Black hole solutions Regular event horizon at r = r h = 1 Solutions parameterized by ω k ( r h ) and h ′ k ( r h ) Electric functions h k are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ω k Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

Dyons Solving the field equations Numerically solve the field equations for su (2) and su (3), with different values of Λ < 0 Soliton solutions Regular at the origin r = 0 Solutions parameterized by ω ′′ k (0) and h ′ k (0) Black hole solutions Regular event horizon at r = r h = 1 Solutions parameterized by ω k ( r h ) and h ′ k ( r h ) Electric functions h k are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ω k Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

Dyons Solving the field equations Numerically solve the field equations for su (2) and su (3), with different values of Λ < 0 Soliton solutions Regular at the origin r = 0 Solutions parameterized by ω ′′ k (0) and h ′ k (0) Black hole solutions Regular event horizon at r = r h = 1 Solutions parameterized by ω k ( r h ) and h ′ k ( r h ) Electric functions h k are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ω k Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

Dyons Solving the field equations Numerically solve the field equations for su (2) and su (3), with different values of Λ < 0 Soliton solutions Regular at the origin r = 0 Solutions parameterized by ω ′′ k (0) and h ′ k (0) Black hole solutions Regular event horizon at r = r h = 1 Solutions parameterized by ω k ( r h ) and h ′ k ( r h ) Electric functions h k are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ω k Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

Dyons su (2) solitons, Λ = − 0 . 01 [ Bjoraker and Hosotani, PRL 84 1853 (2000), PRD 62 043513 (2000) ] Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 8 / 13

Dyons su (2) solitons, Λ = − 0 . 01 [ Shepherd and EW ] Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 9 / 13