Existence of topological hairy dyons and dyonic black holes in anti-de Sitter 𝔰𝔲(N) Einstein-Yang-Mills theory

Abstract

We investigate dyonic black hole and dyon solutions of four-dimensional su(N) Einstein-Yang-Mills theory with a negative cosmological constant. We derive a set of field equations in this case, and prove the existence of non-trivial solutions to these equations for any integer N, with 2N − 2 gauge degrees of freedom. We do this by showing that solutions exist locally at infinity, and at the event horizon for black holes and the origin for solitons. We then prove that we can patch these solutions together regularly into global solutions that can be integrated arbitrarily far into the asymptotic regime. Our main result is to show that dyonic solutions exist in open sets in the parameter space, and hence that we can find non-trivial dyonic solutions in a number of regimes whose magnetic gauge fields have no zeros, which is likely important to the stability of the solutions. C 2016 AIP Publishing LLC.

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

“…46 Recently the existence of dyonic soliton and black hole solutions of the su(N) field equations has been proven. 47 The existence of stable su(2) dyonic solutions has been proven very recently 48 and it would be interesting to investigate whether our results in this paper on the existence of stable purely magnetic solitons and black holes in su(N) EYM in anti-de Sitter space can be extended to dyonic solutions.…”

Non-Abelian clouds around Reissner-Nordström black holes: The existence line

“…The spherically symmetric solutions have k ¼ 1, in which case one can find both solitons and black hole solutions [33]; for k ¼ 0; −1, one finds only "topological" black holes (see Ref. [34] for recent results on such configurations).…”

On the existence of topological dyons and dyonic black holes in anti-de Sitter Einstein-Yang-Mills theories with compact semisimple gauge groups

“…these solutions are 'purely magnetic', referred to as 'zero magnetic charge models' [19,35]. These purely magnetic solutions have been extended to asymptotically adS space for a general gauge group in the spherically symmetric case [36], and to su(N) in the case of 'topological symmetry' [16,26,33] (which we shall outline). However since there are no similar restrictions on the electric gauge field for adS solutions, we will extend the model to cover the cases of dyonic solutions with topological symmetry.…”

“…This is in contrast to the asymptotically flat "no magnetic charge" case [17][18][19] where the electric sector must be trivial for asymptotic regularity. Since globally regular dyonic solutions are only generally supported in adS space due to the closed geometry, such solutions are less common in the literature; but monopole and dyon solutions have been found [20], and notably for us in the dyonic su(2) [21,22] for which stability is proven [22], and the dyonic su(N) [33] case. Moreover, these solutions are 'nodeless' in the sense that the magnetic gauge functions possess no zeroes, which has been a necessary result for stability in previous cases [7,[22][23][24].…”

“…We shall do this using the following general method. We shall construct our topological ansätze before deriving the field equations themselves, reducing them down to the 'regular' case [17,33]. We shall identify various embedded solutions whose existence has been proven elsewhere -these are crucial to our proof.…”