Abstract
We investigate the existence of black hole and soliton solutions to four dimensional, anti-de Sitter (adS), Einstein-Yang-Mills theories with general semisimple connected and simply connected gauge groups, concentrating on the so-called "regular" case. We here generalise results for the asymptotically flat case, and compare our system with similar results from the well-researched adS $\mathfrak{su}(N)$ system. We find the analysis differs from the asymptotically flat case in some important ways: the biggest difference is that for $\Lambda<0$, solutions are much less constrained as $r\rightarrow\infty$, making it possible to prove the existence of global solutions to the field equations in some neighbourhood of existing trivial solutions, and in the limit of $|\Lambda|\rightarrow\infty$. In particular, we can identify non-trivial solutions where the gauge field functions have no zeroes, which in the $\mathfrak{su}(N)$ case proved important to stability.
Highlights
Research into Einstein–Yang–Mills (EYM) theory, which concerns the coupling of gauge fields described by the Yang–Mills (YM) equations to gravitational fields described by Einstein’s equations, has become abundant in the literature in the last couple of decades
Connected to this is their stability: su(N ) purely magnetic solutions decouple into two sectors upon a linear perturbation, and spectral analysis shows that su(2) solutions possess n unstable modes in each sector, where n is the number of nodes of the gauge field; and in addition, these su(2) solutions must possess at least one node [14,15,16,17]
The traditional argument that has been used in this case is the ‘shooting argument’, which basically involves proving the existence of solutions locally at the boundaries, and proving that solutions which begin at the initial boundary r = rh (r = 0) near to existing embedded solutions can be integrated out arbitrarily far, remaining regular right into the asymptotic regime, where they will ‘meet up’ with solutions existing locally at r → ∞; and that these neighbouring solutions will remain close to the embedded solution
Summary
Research into Einstein–Yang–Mills (EYM) theory, which concerns the coupling of gauge fields described by the Yang–Mills (YM) equations to gravitational fields described by Einstein’s equations, has become abundant in the literature in the last couple of decades. The set of such so-called A1-vectors is finite, and have been tabulated by Dynkin [38] and Mal’cev [39] using what they call “characteristics”, which are in one-to-one correspondence with finite ordered sets of integers chosen from the set {0, 1, 2} These strings of integers represent the value of the simple roots on W0, the defining vector of the A1-subalgebra, chosen so that it lies in W ( ); and the tables of Mal’cev and Dynkin give us a classification of all possible spherically symmetric, purely magnetic EYM models which obey the correct regularity conditions asymptotically and at the centre, for any compact semisimple connected gauge group
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