Soliton and black hole solutions ofsu(N)Einstein-Yang-Mills theory in anti-de Sitter space

Abstract

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On the stability of soliton and hairy black hole solutions of 𝔰𝔲(N) Einstein-Yang-Mills theory with a negative cosmological constant

“…The answer is affirmative: such solutions have been found numerically for gauge groups su(3) and su (4). 29 For the larger su(N) gauge group, the purely magnetic gauge field is described by N − 1 functions ω j (see Section II A). As in the su(2) case, there are continuous families of solutions, parameterized by the negative cosmological constant Λ, the event horizon radius r h (with r h = 0 for soliton solutions), and N − 1 parameters describing the form of the gauge field functions either on the event horizon or near the origin.…”

“…Writing the matrix Z in the form 24) where Z 11 is a symmetric N × N matrix, Z 12 is an N × (N − 1) matrix, and Z 22 is a symmetric (N − 1) × (N − 1) matrix, using (3.21) and (3.23) we find that the matrix V defined in (3.20) has the form 25) where (3.26) We are free to choose the matrix Z so as to simplify the form of V. We first make the choices (3.27) In this case, the form of V 12 simplifies to 28) which vanishes if we choose Z 12 such that 29) where r * ,min = 0 for equilibrium static soliton solutions and r * ,min = −∞ for equilibrium static black hole solutions. With this choice of Z 12 , it is straightforward to see that V 21 also vanishes.…”

“…29 Therefore in this section we very briefly describe some of the key features of the solutions which are required for our subsequent analysis.…”

“…Here we simply plot, in Figures 1 and 2, examples of nodeless soliton and black hole solutions for su(3) and su (4), respectively, referring the reader to Ref. 29 for further details of the phase space of solutions.…”

On the existence of soliton and hairy black hole solutions of {\mathfrak {su}}(N) Einstein–Yang–Mills theory with a negative cosmological constant

“…Recently soliton and black hole solutions of four-dimensional su(N) EYM theory with a negative cosmological constant have been found [8,14,15]. For purely magnetic solutions, the gauge field is described by N − 1 gauge field functions ω j .…”