22 Pages·2012·2.92 MB·English

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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 ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 1/13 Outline 1 A brief history of Einstein-Yang-Mills 2 su(N) EYM with Λ < 0 3 Dyonic solutions 4 Conclusions and outlook ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 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? ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 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? ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 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? ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 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? ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 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? ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 3/13 su(N)EYMwithΛ<0 The model for su(N) EYM Einstein-Yang-Mills theory with su(N) gauge group 1 (cid:90) √ S = d4x −g[R −2Λ−TrF Fµν] µν 2 Field equations 1 R − Rg +Λg = T µν µν µν µν 2 D Fµ = ∇ Fµ+[A ,Fµ] = 0 µ ν µ ν µ ν Stress-energy tensor 1 T = TrF Fλ− g TrF Fλσ µν µλ ν 4 µν λσ ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 4/13 su(N)EYMwithΛ<0 Static, spherically symmetric, conﬁgurations Metric ds2 = −µ(r)σ(r)2dt2+[µ(r)]−1dr2+r2(cid:0)dθ2+sin2θdφ2(cid:1) 2m(r) Λr2 µ(r) = 1− − r 3 su(N) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ] Static, dyonic, gauge potential 1 (cid:16) (cid:17) i (cid:104)(cid:16) (cid:17) (cid:105) A dxµ = Adt + C −CH dθ− C +CH sinθ+Dcosθ dφ µ 2 2 N −1 electric gauge ﬁeld functions h (r) in matrix A j N −1 magnetic gauge ﬁeld functions ω (r) in matrix C j ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 5/13 su(N)EYMwithΛ<0 Static, spherically symmetric, conﬁgurations Metric ds2 = −µ(r)σ(r)2dt2+[µ(r)]−1dr2+r2(cid:0)dθ2+sin2θdφ2(cid:1) 2m(r) Λr2 µ(r) = 1− − r 3 su(N) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ] Static, dyonic, gauge potential 1 (cid:16) (cid:17) i (cid:104)(cid:16) (cid:17) (cid:105) A dxµ = Adt + C −CH dθ− C +CH sinθ+Dcosθ dφ µ 2 2 N −1 electric gauge ﬁeld functions h (r) in matrix A j N −1 magnetic gauge ﬁeld functions ω (r) in matrix C j ElizabethWinstanley (Sheﬃeld) Non-abeliandyonsinanti-deSitterspace BritGrav12,April2012 5/13

Non-abelian dyons in anti-de Sitter space Elizabeth Winstanley Consortium for Fundamental Physics School of Mathematics and Statistics University of She eld

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