Does the sphere free energy decrease along five-dimensional RG flows
In plain words
In three dimensions, a number computed by placing the theory on a sphere is proven to drop under coarse-graining (the F-theorem). Whether the analogous number drops in five dimensions is unknown.
Precise statement
For a unitary $d$-dimensional CFT define F~ = sin(pi d/2) log $Z$(S^d), with Z the renormalized partition function on the round sphere (Giombi and Klebanov 2014); in $d = 5$ this is F~ = log Z(S^5). Prove or refute F~_UV > F~_IR for every unitary RG flow between 5D CFTs. The 3D case was proven from strong subadditivity of entanglement entropy (Casini and Huerta 2012); that argument has not been extended to $d = 5$.
What would settle it
A general proof for unitary $5\mathrm{D}$ flows, or a unitary $5\mathrm{D}$ flow (for example a $5\mathrm{D}$ SCFT deformed into a gauge theory) with F~_UV <= F~_IR.
Status in the literature
Unverified note
Consistent with free-field, large-N and $\epsilon$-expansion examples (2014 onward); no general proof as of 2026.