Characterize when scale and conformal symmetry uplift to Weyl invariance

Determine the full necessary and sufficient conditions under which, for Euclidean quantum field theories, flat-space scaling invariance enhances to flat-space conformal invariance and flat-space conformal invariance uplifts to Weyl invariance, beyond the currently known unitary Poincaré-invariant cases.

Background

The thesis explains that, in many cases, scale invariance is enhanced to conformal invariance, and conformal invariance can be coupled to curved space in a Weyl-covariant way. However, exceptions exist (e.g., certain non-unitary theories and specific couplings obstructed by geometry), and a general criterion is lacking.

The authors emphasize that while such enhancements occur in important classes of theories, a complete characterization applicable in continuous dimensions and beyond special cases has not been established.

References

Thus, we typically can find the following pattern of enhancement of invariance for unitary Poincaré-invariant field theories: Flat space scaling   Flat space conformal   Weyl. The full conditions for these uplifts are not known at present.

— Quantum field theories with many fields  (2603.04481 - Fraser-Taliente, 4 Mar 2026) in Subsection "Conformal invariance and Weyl invariance" (Chapter 2, Section 2.4)

At the quantum level, it would be important to go beyond the on-shell analysis given here, study possible anomalies, and determine whether the symmetry-based closure of the ultraviolet sector survives in a more complete treatment.

— Ultraviolet Closure in Asymptotically Weyl-Invariant Gravity  (2609.01047 - Coumbe, 1 Sep 2026) in Section 5, Discussion and conclusions