Define non-perturbative QFTs in non-integer dimensions

Construct a non-perturbative definition of quantum field theories in non-integer spacetime dimensions that possess full SO(d+1,1) conformal invariance, extending beyond perturbative frameworks such as dimensional regularization.

Background

While conformal bootstrap and perturbative approaches (e.g., the 4 expansion and dimensional regularization) allow analytic continuation to non-integer d, a fully non-perturbative definition of QFTs in fractional dimensions is lacking.

The authors note that various appearances of fractional-dimension CFTs suggest they may exist as consistent theories, but emphasize that a rigorous definition beyond perturbation theory remains unknown.

References

These appearances suggest that these models may actually exist, though the definition of a field theory in non-integer dimension beyond perturbation theory remains unknown.

— Quantum field theories with many fields  (2603.04481 - Fraser-Taliente, 4 Mar 2026) in Subsection "Justification: continuous dimension" (Chapter 2, Section 2.6)