Interpolating between $a$ and $F$

TL;DR

Giombi and Klebanov introduce a continuous-dimension interpolation of the sphere free energy, $\tilde{F}=-\sin(\pi d/2)F$, connecting the $a$-anomaly in even dimensions to the $F$-theorem in odd dimensions. They compute $\tilde{F}$ across free theories, large-$N$ double-trace flows, and weakly coupled fixed points via the $\epsilon$-expansion, showing positivity and monotonicity under RG flow, and they extend the program to SUSY theories with a proposed $\tilde{F}$-maximization principle that unifies $a$- and $F$-maximization. The work yields quantitative estimates for the 3d Ising and Gross-Neveu models and demonstrates excellent agreement between perturbative results and exact localization in SUSY theories, suggesting a universal, all-dimension monotonic $\tilde{F}$-theorem. The framework provides a bridge between familiar dimension-specific theorems and a broader, continuous-dimension perspective with potential cross-checks against AdS/CFT dualities and conformal bootstrap data.

Abstract

We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension $d$ we define the quantity $\tilde F=\sin (πd/2)\log Z$, where $Z$ is the path integral of the Euclidean CFT on the $d$-dimensional round sphere. $\tilde F$ smoothly interpolates between $(-1)^{d/2}π/2$ 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 $ε$ 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 $d=4-ε$ to order $ε^4$. 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 $d=3$.

Figures (4)

• Figure 1: $\tilde{F}$ for free conformal scalar and fermion (the vertical axis is on a logarithmic scale, so we actually plot $-\log\tilde{F}$). $\tilde{F}$ is positive for all $d\ge 2$, and smoothly interpolates between $\frac{\pi}{2}(-1)^{d/2} a$-anomalies in even $d$ and $(-1)^{(d+1)/2} F$ in odd $d$. For example, the values $\tilde{F}=\pi/6$ and $\tilde{F}=\pi/12$ correspond respectively to central charges $c=1$ and $c=1/2$ for a free scalar and Majorana fermion in $d=2$.
• Figure 2: Plot of the ratio ${F_{\rm 3d\, O(N)}\over N F_{s}}$ as a function of $N$, showing a clear minimum around $N=3$.
• Figure 3: Plot of the result for $\tilde{F}_{\rm Ising}$ from the $\epsilon$-expansion to order $\epsilon^4$, normalized by the free scalar field value given in (\ref{['tFfree']}).
• Figure 4: Plot of $\tilde{F}$ for the "super-Ising" model with superpotential $W=X^3$, normalized by the free chiral superfield value $\tilde{F}_{\rm free~chir.}$. The solid line is the prediction of the localization proposal (\ref{['exact-F']}), while the dashed line is the result of the direct perturbative calculation (\ref{['tF-X3-ep']}) in $d=4-\epsilon$ to order $\epsilon^2$.