An Abelian gauge symmetry can be spontaneously broken near a black hole horizon in anti-de Sitter space using a condensate of non-Abelian gauge fields. A second order phase transition is shown to separate Reissner-Nordström-anti-de Sitter solutions from a family of symmetry-breaking solutions which preserve a diagonal combination of gauge invariance and spatial rotational invariance.
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Steven S. Gubser (2008) studied this question.
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