Quantum symmetries: $S$ or $Z$? Let $I$ be the action of some QFT (gauge-fixed and including all the necessary counter-terms); $S$ the associated scattering-matrix; and $Z$ the partition function (in the form of, say, a path integral). There are three notions of symmetries that are typically discussed,


*

*Action symmetries, that is, transformations of the form $\phi\to\phi'$ that leave $I$ invariant,

*$S$-matrix symmetries, that is, operators that (super)commute with $S$, and

*Quantum symmetries, that is, transformations of the form $\phi\to\phi'$ that leave the volume form $\mathrm e^{I[\phi]}\mathrm d\phi$ invariant.


It is a well-known phenomenon that the symmetries of $I$ need not agree with those of neither $S$ nor $Z$ (e.g., anomalies, SSB, etc.). What is not-so-clear is whether $S$ symmetries and $Z$ symmetries are equivalent. What I want to know is whether

For each symmetry of $S$ there is a symmetry of $Z$ and vice-versa

or a counter-example. If the equivalence is actually true, I would like to have a more-or-less precise statement, in the form of a theorem (to the usual level of rigour in physics textbooks).
No cheating please. Counter-examples are only valid for "real-life" QFT's (e.g. in a free theory everything commutes with $S$ but not everything leaves $Z$ invariant; this is not a valid counter-example because it is completely trivial). No TQFT's either. Thanks.


Someone mentioned in the comments the effective action $\Gamma[\phi]$, which is defined as the Legendre transform of $\log Z$. I didn't want to bring this object into the picture, because I wanted to leave it to the discretion of the rest of users whether to mention this object or not. In principle, I don't need answers to analyse the symmetries of this object, but they may if they believe it could be useful to do so. In any case, let me stress that $\Gamma$ is not the same object as $I$.

 A: People sometimes talk about on-shell symmetries: symmetries of the equations of motion or the S-matrix, which do not hold off-shell (i.e. at the level of the action, path integral, correlators, etc). Electric-magnetic duality transformations 
$$
\begin{aligned}
F &\rightarrow \cos(\theta)\,F - \sin(\theta)\,\star F \\
\star F &\rightarrow \sin(\theta)\,F + \cos(\theta)\,\star F
\end{aligned}
$$
are an example of this. These transformations leave the equations of motion and the S-matrix invariant (with a corresponding selection rule), so they would qualify as a symmetry. 
However, this symmetry is usually broken by non-perturbative effects (e.g. monopoles), to which the usual S-matrix is insensitive. In addition, it is well known that dualities are not symmetries of the full theory but instead a better understood as a change in our description of the physics. For instance duality transformations relate the partition function of the theory at different values of the coupling (see this paper for an example).
A: I think that a symmetry in Z implies perturbatively a symmetry in S since the perturbative S-matrix can be computed from the effective action (which is directly related to Z) in a straightforward way that leaves no room for breaking a symmetry. 
Whether the converse holds is questionable as it is not even clear whether the S-matrix determines the action.
