Do non-supersymmetric conformal manifolds exist above two dimensions
In plain words
Some scale-invariant theories come in continuous families, in which a parameter can be changed while scale invariance stays exact. Above two dimensions, all established interacting examples rely on supersymmetry, a symmetry between bosons and fermions, and whether any exist without it is unknown.
Precise statement
A conformal manifold is a continuous family of CFTs generated by exactly marginal operators (scaling dimension exactly d along the family). In $d = 3\ \text{or}\ 4$, determine whether a unitary interacting CFT without supersymmetry, at finite $N$, has an exactly marginal operator. Candidates come from holography: families of non-supersymmetric $\mathrm{AdS}_4$ S-fold solutions of type IIB supergravity argued to be perturbatively and nonperturbatively stable (Giambrone et al. 2021; Bobev, Gautason, van Muiden 2023).
What would settle it
A non-supersymmetric CFT with an exactly marginal operator established at finite N by field theory or bootstrap, or a theorem excluding one under stated assumptions.
Status in the literature
Holographic $\mathrm{AdS}4$ candidates (2021-2023) are argued stable, but no field-theory dual at finite $N$ has been identified.