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.

We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in \( \mathcal{N}=4 \) SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the ’t Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS₂ string worldsheet. We also explain the connection of our results to the “generalized Bremsstrahlung functions” previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining their finite N generalization. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Q-functions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability.

Chern-Simons gauge theories coupled to massless fundamental scalars or fermions define interesting non-supersymmetric 3d CFTs that possess approximate higher-spin symmetries at large N . In this paper, we compute the scaling dimensions of the higher-spin operators in these models, to leading order in the 1/N expansion and exactly in the ’t Hooft coupling. We obtain these results in two independent ways: by using conformal symmetry and the classical equations of motion to fix the structure of the current non-conservation, and by a direct Feynman diagram calculation. The full dependence on the ’t Hooft coupling can be restored by using results that follow from the weakly broken higher-spin symmetry. This analysis also allows us to obtain some explicit results for the non-conserved, parity-breaking structures that appear in planar three-point functions of the higher-spin operators. At large spin, we find that the anomalous dimensions grow logarithmically with the spin, in agreement with general expectations. This logarithmic behavior disappears in the strong coupling limit, where the anomalous dimensions turn into those of the critical O(N ) or Gross-Neveu models, in agreement with the conjectured 3d bosonization duality.

In a conformal field theory with weakly broken higher spin symmetry, the leading order anomalous dimensions of the broken currents can be efficiently determined from the structure of the classical non-conservation equations. We apply this method to the explicit example of O(N) invariant scalar field theories in various dimensions, including the large N critical O(N) model in general d, the Wilson-Fisher fixed point in d = 4 − ϵ, cubic scalar models in d = 6 − ϵ and the nonlinear sigma model in d = 2 + ϵ. Using information from the d = 4 − ϵ and d = 2 + ϵ expansions, we obtain some estimates for the dimensions of the higher spin operators in the critical 3d O(N) models for a few low values of N and spin.