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Horowitz-Myers AdS soliton conjecture

A ``new" positive energy conjecture was conjectured back in 1999 by Horowitz and Myers [arXiv:hep-th/9808079], which somehow escaped my attention till Steve McCormick mentioned it to me in a tweet (X-post? sounds weird though). I will briefly outline what it is, but the technicalities will be left to the reader as an exercise. The idea is essentially to take the near-extremal p-brane solution and double Wick rotate it, so as to obtain the AdS soliton with periodicity. The interest then would be to work with metrics that look like 
g_{\mu \nu }=\tilde{g}_{\mu \nu }+h_{\mu \nu }
and fall-off conditions w.r.t r, where \tilde{g}_{\mu \nu } is the AdS soliton metric. Horowitz and Myers go on to propose two conjectures, which are as follows:
  1. SUGRA solutions: g_{\mu \nu }=\tilde{g}_{\mu \nu } is the only solution to D=10 type IIB SUGRA, taking \tilde{g}_{\mu \nu }\times S^{5}_{l} for which \mathcal{E}= 0, and \mathcal{E}\geq 0 in general.
  2. Geometric solutions: g_{\mu \nu }=\tilde{g}_{\mu \nu } is the only solution to D=5 field equations with \Lambda = -6/l^{2} so that \mathcal{E}=0, and \mathcal{E}\geq 0 in general.
Seemingly, these are still open problems. However, I believe that resolving at least some aspects of these conjectures will go a long way in making sense of the geometric description of AdS/CFT. 

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