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We study the dimensional continuation of the sphere free energy in conformalfield theories. In continuous dimension $d$ we define the quantity $\tildeF=\sin (\pi d/2)\log Z$, where $Z$ is the path integral of the Euclidean CFT onthe $d$-dimensional round sphere. $\tilde F$ smoothly interpolates between$(-1)^{d/2}\pi/2$ times the $a$-anomaly coefficient in even $d$, and$(-1)^{(d+1)/2}$ times the sphere free energy $F$ in odd $d$. We calculate$\tilde F$ in various examples of unitary CFT that can be continued tonon-integer dimensions, including free theories, double-trace deformations atlarge $N$, and perturbative fixed points in the $\epsilon$ expansion. For allthese examples $\tilde F$ is positive, and it decreases under RG flow. Usingperturbation theory in the coupling, we calculate $\tilde F$ in theWilson-Fisher fixed point of the $O(N)$ vector model in $d=4-\epsilon$ to order$\epsilon^4$. We use this result to estimate the value of $F$ in the3-dimensional Ising model, and find that it is only a few percent below $F$ ofthe free conformally coupled scalar field. We use similar methods to estimatethe $F$ values for the $U(N)$ Gross-Neveu model in $d=3$ and the $O(N)$ modelin $d=5$. Finally, we carry out the dimensional continuation of interactingtheories with 4 supercharges, for which we suggest that $\tilde F$ may becalculated exactly using an appropriate version of localization on $S^d$. Ourapproach provides an interpolation between the $a$-maximization in $d=4$ andthe $F$-maximization in $d=3$.

Interpolating between $a$ and $F$