Holographic Interpolation between $a$ and $F$

An interpolating function $\tilde{F}$ between the $a$-anomaly coefficient in even dimensions and the free energy on an odd-dimensional sphere has been proposed recently and is conjectured to monotonically decrease along any renormalization group flow in continuous dimension $d$. We examine $\tilde{F}$ in the large-$N$ CFT’s in $d$ dimensions holographically described by the Einstein-Hilbert gravity in the AdS${}_{d+1}$ space. We show that $\tilde{F}$ is a smooth function of $d$ and correctly interpolates the $a$ coefficients and the free energies. The monotonicity of $\tilde{F}$ along an RG flow follows from the analytic continuation of the holographic $c$-theorem to continuous $d$, which completes the proof of the conjecture.