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Title: Does the sphere free energy decrease along five-dimensional RG flows
Plain: 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: 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 $d = 5$.
Settled by: 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.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: Consistent with free-field, large-N and $\epsilon$-expansion examples (2014 onward); no general proof as of 2026.
Posed since: 2010