Symmetries in QFT preserving only combination of action and measure Could we list examples of symmetries that preserve only the combination of the measure $\mathcal{D}\phi$ together with $e^{-S}$ but not each on their own? (That is, symmetries which have no classical analogue).
 A: Liouville theory possesses conformal invariance on the quantum level, but its classical action isn't conformally invariant (because it contains dimensionful constants such as $\lambda'$).
A: I would like to submit a partial answer to my question.
In general, any examples where it is only the combination of the measure and action which are invariant would fall under the umbrella of `anomaly cancellation'. That is, any anomaly cancellation mechanism would give rise to such a phenomenon as I describe, and conversely, any such case can be thought of as arising from applying anomaly cancellation to an otherwise anomalous theory.
The Green-Schwarz `anomaly cancellation' mechanism (in 4d, 10d or whatever you like) is a perfect example.
The Liouville theory is another excellent example, where as the comment above describes, it can be thought of as a Polyakov action (in eg 0 dimensions) where there is a conformal anomaly. Adding the dilaton terms to make the full Lioville action would then be another example of anomaly cancellation.
Notes:
1) My definition of anomaly cancellation is `adding dynamical fields to an anomalous action in order to make it anomaly-free'.
2) All I am doing is giving the well-known name to the phenomenon I was thinking about, but at the time I did realise that these were the same concepts. So googling `anomaly cancellation' will bring up countless other examples presumably.
Hope everything I say is correct, otherwise please correct!
