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

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

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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 Elizabeth Winstanley

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SLIDE 1

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

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SLIDE 2

Outline

1

A brief history of Einstein-Yang-Mills

2

su(N) EYM with Λ < 0

3

Dyonic solutions

4

Conclusions and outlook

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 2 / 13

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SLIDE 3

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¨

  • m

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

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SLIDE 4

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¨

  • m

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

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SLIDE 5

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¨

  • m

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

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SLIDE 6

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¨

  • m

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

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SLIDE 7

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¨

  • m

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

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SLIDE 8

su(N) EYM with Λ < 0

The model for su(N) EYM

Einstein-Yang-Mills theory with su(N) gauge group

S = 1 2

  • d4x √−g [R − 2Λ − Tr FµνF µν]

Field equations

Rµν − 1 2Rgµν + Λgµν = Tµν DµF µ

ν = ∇µF µ ν + [Aµ, F µ ν ]

=

Stress-energy tensor

Tµν = Tr FµλF λ

ν − 1

4gµνTr FλσF λσ

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 4 / 13

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SLIDE 9

su(N) EYM with Λ < 0

Static, spherically symmetric, configurations

Metric

ds2 = −µ(r)σ(r)2 dt2 + [µ(r)]−1 dr2 + r2 dθ2 + sin2 θ dφ2 µ(r) = 1 − 2m(r) r − Λr2 3

su(N) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ]

Static, dyonic, gauge potential Aµ dxµ = A dt + 1 2

  • C − C H

dθ − i 2

  • C + C H

sin θ + D cos θ

  • dφ

N − 1 electric gauge field functions hj(r) in matrix A N − 1 magnetic gauge field functions ωj(r) in matrix C

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 5 / 13

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SLIDE 10

su(N) EYM with Λ < 0

Static, spherically symmetric, configurations

Metric

ds2 = −µ(r)σ(r)2 dt2 + [µ(r)]−1 dr2 + r2 dθ2 + sin2 θ dφ2 µ(r) = 1 − 2m(r) r − Λr2 3

su(N) gauge potential [ Kunzle Class. Quant. Grav. 8 2283 (1991) ]

Static, dyonic, gauge potential Aµ dxµ = A dt + 1 2

  • C − C H

dθ − i 2

  • C + C H

sin θ + D cos θ

  • dφ

N − 1 electric gauge field functions hj(r) in matrix A N − 1 magnetic gauge field functions ωj(r) in matrix C

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 5 / 13

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SLIDE 11

su(N) EYM with Λ < 0

Field equations

Yang-Mills equations

h′′

k

= h′

k

σ′ σ − 2 r

  • +
  • 2 (k + 1)

k ω2

k

µr2

  • k + 1

2k hk −

  • k − 1

2k hk−1

  • +
  • 2k

k + 1 ω2

k+1

µr2

  • k

2 (k + 1)hk −

  • k + 2

2 (k + 1)hk+1

  • =

ω′′

k + ω′ k

σ′ σ + µ′ µ

  • +

ωk σ2µ2

  • k + 1

2k hk −

  • k − 1

2k hk−1 2 + ωk µr2

  • 1 − ω2

k + 1

2

  • ω2

k−1 + ω2 k+1

  • Einstein equations

Give µ′(r) and σ′(r) in terms of ωk, hk and their derivatives

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 6 / 13

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SLIDE 12

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 = rh = 1 Solutions parameterized by ωk(rh) and h′

k(rh)

Electric functions hk are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ωk

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

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SLIDE 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 = rh = 1 Solutions parameterized by ωk(rh) and h′

k(rh)

Electric functions hk are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ωk

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

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SLIDE 14

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 = rh = 1 Solutions parameterized by ωk(rh) and h′

k(rh)

Electric functions hk are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ωk

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

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SLIDE 15

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 = rh = 1 Solutions parameterized by ωk(rh) and h′

k(rh)

Electric functions hk are monotonically increasing Colour-code solution space by number of zeros of magnetic gauge functions ωk

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 7 / 13

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SLIDE 16

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

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SLIDE 17

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

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SLIDE 18

Dyons

su(3) solitons - part of the solution space for Λ = −0.01

[ Shepherd and EW ]

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 10 / 13

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SLIDE 19

Dyons

su(2) black holes, Λ = −0.01, rh = 1 [ Shepherd and EW ]

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 11 / 13

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SLIDE 20

Dyons

su(3) black holes Λ = −3, rh = 1, ω1(rh) = 1.3, ω2(rh) = 1.2

[ Shepherd and EW ]

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 12 / 13

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SLIDE 21

Conclusions

Conclusions and outlook

Dyons in su(N) EYM in adS

Black hole and soliton solutions with both electric and magnetic gauge fields: su(2) dyonic solutions found previously by Bjoraker and Hosotani Examination of larger part of solution space shows a very rich menagerie of solutions New su(3) dyonic solutions Solutions in which magnetic functions ωk have no zeros for larger |Λ|

Open questions

For sufficiently large |Λ| Prove analytically the existence of dyonic solutions for all N Stability when ωk have no zeros?

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 13 / 13

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SLIDE 22

Conclusions

Conclusions and outlook

Dyons in su(N) EYM in adS

Black hole and soliton solutions with both electric and magnetic gauge fields: su(2) dyonic solutions found previously by Bjoraker and Hosotani Examination of larger part of solution space shows a very rich menagerie of solutions New su(3) dyonic solutions Solutions in which magnetic functions ωk have no zeros for larger |Λ|

Open questions

For sufficiently large |Λ| Prove analytically the existence of dyonic solutions for all N Stability when ωk have no zeros?

Elizabeth Winstanley (Sheffield) Non-abelian dyons in anti-de Sitter space BritGrav12, April 2012 13 / 13