Interpolating between a and F

Abstract

We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension d we define the quantity =sin(pi d/2) log Z, where Z is the path integral of the Euclidean CFT on the d-dimensional round sphere. 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 in various examples of unitary CFT that can be continued to non-integer dimensions, including free theories, double-trace deformations at large N, and perturbative fixed points in the I mu expansion. For all these examples is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate in the Wilson-Fisher fixed point of the O(N) vector model in d = 4 - I mu to order I mu (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 may be calculated exactly using an appropriate version of localization on S (d) . Our approach provides an interpolation between the a-maximization in d = 4 and the F-maximization in d = 3.