PUPT-2472 Interpolating between a and F

We study the dimensional continuation of the sphere free energy in conformal field the-ories. In continuous dimension d we define the quantity F ̃ = sin(pid/2) logZ, where Z is the path integral of the Euclidean CFT on the d-dimensional round sphere. F ̃ smoothly interpolates between (−1)d/2pi/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 F ̃ in various examples of unitary CFT that can be continued to non-integer dimensions, including free theories, double-trace defor-mations at large N, and perturbative fixed points in the expansion. For all these examples F ̃ is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate F ̃ in the Wilson-Fisher fixed point of the O(N) vector model in d = 4 − to order 4. We use this result to estimate the value of F in the 3-dimensional Ising model, and find that it is only a few percent below F of the free conformally coupled scalar field. We use similar methods to estimate the F values for the U(N) Gross-Neveu model in d = 3 and the O(N) model in d = 5. Finally, we carry out the dimensional continuation of interacting theories with 4 supercharges, for which we suggest that F ̃ may be calculated exactly using an appropriate version of localization on Sd. Our approach provides an interpolation between the a-maximization in d = 4 and the F-maximization in d = 3. ar X i