Quoting from arXiv:0801.3471

Asymptotically flat BPS black hole solutions are known only for $d = 4,\,5$.

and I've seen this claim very timidly suggested in a couple of other places, but I couldn't find a proof.

We could define a supersymmetric black hole as such:

  • Solution to some supergravity theory
  • Asymptotically Minkowski, but not Minkowski
  • Has a Killing vector which is asymptotically time-like (stationary)
  • Has a Killing spinor
  • Regular down to horizon, including no diverging scalars (so type IIA D$0$ does not count)

Do they exist for $D>5$? Is there a proof of non-existence? Or have any counterexample been found in the last ten years?

Progress: I have managed to prove there are no SUSY black holes in $D=6$. The vector bilinear from a symplectic Majorana

$V^\mu \epsilon^{AB} = \bar\eta^A \gamma^\mu \eta^B$

is null because of a Fierz identity. Thus we have an always null Killing vector and combined with the other timelike Killing with a spatially compact Killing horizon it's enough to derive a contradiction. This is compatible with the classified 6D SUSY geometries I've found in, e.g., arXiv:hep-th/0306235.

Progress 2: it appears to me like arXiv:hep-th/0504080 provides a supersymmetric black hole in 7D. Though I still need to check whether the scalars don't diverge.

  • $\begingroup$ In my naivete I thought D-branes were all black-hole-like in sugra. Is this wrong? $\endgroup$ – Ryan Thorngren Sep 23 '18 at 16:58
  • 1
    $\begingroup$ p-branes with $p>0$ are only asymptotically flat in the orthogonal directions, and the D$0$ has a divergent dilaton so it's not really a regular solution of the classical sugra. $\endgroup$ – Riccardo Antonelli Sep 23 '18 at 17:03

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