Can a theory gain symmetries through quantum corrections? It is well known that not all symmetries are preserved when quantising a theory, as evinced by renormalising composite operators or by other means, which show that quantum corrections may alter a conservation law, such as with the chiral anomaly, or 'parity' anomaly of gauge fields coupled to fermions in odd dimensions.
However is the reverse possible: can a theory after quantisation gain a symmetry? Or if not, can it gain a 'partial symmetry'?
(For example invariance under $x\to x+a$ for any $a$ is translation symmetry, and invariance under $x\to x+2\pi$ would be said to be a partial symmetry. My question concerns whether a theory can gain a full symmetry, or a partial one at least through being quantised.)
 A: Maybe not the answer you are looking for but, remember that (Wilsonian) QFTs are defined at a certain scale $\mu$.For example we can take Yang-Mills theory with various matter fields added with a certain set of coupling constants/masses $a_i$. Classically this theory can be made to have conformal symmetry by choosing the couplings in such a way that all the coupling constants are dimensionless. For concreteness let us take $SU(N)$ Yang-Mills theory with 6 scalars in the adjoint representation with a general quartic potential and 4 Dirac fermions with general Yukawa couplings. It is well known that conformal symmetry is broken by quantum effects generically. But it is also known that at a point in parameter space, $\partial_\mu a_i=0$, this theory is superconformal at the quantum level. So it can indeed happen that quantum/loop corrections conspire among each other to enhance symmetries. Another example is ABJM which appears to only have $SU(2)\times SU(2)$ flavor symmetry but actually has $SU(4)$ or even $SO(8)$ symmetry depending on the ranks of the gauge group.
