Does an interacting unitary O(N) CFT exist in five dimensions
In plain words
In five dimensions, a model of N-component spins may have a scale-invariant point that appears only at short distances. Whether this point is a consistent quantum theory for realistic N is unknown.
Precise statement
The $O(N)$ model in $4 < d < 6$ has a UV fixed point at large N, equivalently described by the cubic theory of Fei, Giombi and Klebanov. Determine whether for any finite N in $d = 5$ it is a unitary CFT, or whether it is non-unitary (complex or metastable) for all finite N through instanton effects; if unitary fixed points exist, find the smallest such N.
What would settle it
Numerical bootstrap islands or exclusions for the 5D O(N) CFT as a function of N, consistent with large-N data, or a proof that nonperturbative instability persists at every finite N.
Status in the literature
Large-$N$ and $6 - \epsilon$ expansions support existence at large $N$; bootstrap kinks are weak and instanton arguments (2019) suggest the fixed point is at best metastable.