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 $\tilde{F}=\sin (\pi d/2)\log Z$, where $Z$ is the path integral of the Euclidean CFT on the $d$-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 to non-integer dimensions, including free theories, double-trace deformations at large $N$, and perturbative fixed points in the $\epsilon$ expansion. For all these examples $\tilde{F}$ is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate $\tilde{F}$ in the Wilson-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 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 $\tilde{F}$ 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 $d=3$.

41 pages, 4 figures. v4: Eqs. (1.6), (4.13) and (5.37) corrected; footnote 9 added discussing the Euler density counterterm

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Cited by in corpus (6)

• A Cheap Alternative to the Lattice?
• Mutual information and the F-theorem
• Holographic Interpolation between $a$ and $F$
• RG flows on $S^d$ and Hamiltonian truncation
• Symmetries and Invariants in Higher-Spin Theory
• The AdS/CFT partition function, AdS as a lift of a CFT, and holographic RG flow from conformal deformations