Conformal field theories that are not conformally invariant

Is it possible to construct a field theory that is Weyl-invariant but not conformally invariant? Any references?

• I think you would need gravitational anomaly for that. Conformal transformation may be regarded as a general coordinate transformation followed by a Weyl rescaling. – Peter Kravchuk Feb 6 '16 at 4:48
• So, only after quantization? – Kosm Feb 6 '16 at 4:56
• If by Weyl invariance you mean the invariance under local rescalings of the metric, and the metric enters the action in the way appropriate for ensuring general covariance, then definitely yes, only after quantization, if at all. – Peter Kravchuk Feb 6 '16 at 5:22
• I see, I'll keep that in mind. – Kosm Feb 6 '16 at 5:24
• If you agree that at quantum level Ward identity for Weyl symmetry will be $T^{\mu}_{\mu}=0$, then special conformal and dilatation currents are conserved and you get a CFT. In general, see e.g. hep-th/1302.0884 – Peter Kravchuk Feb 6 '16 at 5:28