Interpolating between a and F

Abstract

Abstract: We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension d we define the quantity (Formula presented.) =sin(πd/2) log Z, where Z is the path integral of the Euclidean CFT on the d-dimensional round sphere. (Formula presented.) smoothly interpolates between (−1)d/2π/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 (Formula presented.) 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 ϵ expansion. For all these examples (Formula presented.) is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate (Formula presented.) in the Wilson-Fisher fixed point of the O(N) vector model in d = 4 − ϵ to order ϵ4. 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 (Formula presented.) 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.