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Anomalies usually occur when a classical symmetry ceases to be a symmetry of the theory when quantized. Are there quantum systems with certain symmetries which cease to exist when you take classical limits? In other words, are there systems which when quantized gets a symmetry enhancement?

I would prefer an answer where the symmetries or the symmetry breaking are realised at the level of action (or algebra if that makes sense), because for example, a particle residing in the minima of a classical double well potential breaks the $\mathbb{Z}_2$ symmetry of the problem, but quantum mechanically due to tunneling, the symmetry is somewhat preserved anyway. In a nutshell, I am asking whether there is a formalization of the conceptual inverse of anomalies.

Hat tip: This question was originally asked by Silly Goose in the hbar chatroom, but this interested me so much that I posted it on the main site.

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    $\begingroup$ If helpful, the original motivation for this question is from Chern-Simons theory. Via path integral quantization of a classical Chern-Simons theory, one obtains a larger set of gauge transformations which leaves the dynamics of the quantum theory invariant (at least to my understanding). $\endgroup$
    – Silly Goose
    Commented Jun 12 at 6:33
  • $\begingroup$ Linked. $\endgroup$
    – Cosmas Zachos
    Commented Nov 7 at 22:32

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There are a few examples I can think of where quantum effects enlarge the symmetry group compared to the symmetries of the (classical) action, e.g. dual superconformal symmetry of 4d $\mathcal{N} = 4$ super Yang-Mills theory or supersymmetry enhancement to $\mathcal{N} = 2$ for some 4d $\mathcal{N} = 1$ super QCD models at renormalisation group fixed points (https://arxiv.org/abs/1707.04751), but in none of the examples the quantum symmetry group is realised at the level of the action. I am not sure if this is the answer you wanted.

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  • $\begingroup$ Don't most QFTs gain extra symmetries to become CFTs at RG fixed points? (modulo subtleties regarding scale invariant theories $\ne$ CFTs...) $\endgroup$
    – Sanjana
    Commented Jun 19 at 5:30
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    $\begingroup$ That is correct, but also the Lagrangian for the model tends to become meaningless when RG flowed to fixed points since the theory becomes strongly coupled and perturbative expansion does not work. 4d $\mathcal{N}=4$ SYM is an exception though. $\endgroup$
    – Dexter Kim
    Commented Jun 20 at 6:13
  • $\begingroup$ The reason RG flow was mentioned is because RG flows to the IR are sometimes thought of as quantum corrections being piled onto the UV description of the theory, usually described by a UV Lagrangian. A theory having a Lagrangian description has an obvious classical limit. $\endgroup$
    – Dexter Kim
    Commented Jun 20 at 6:20
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Look up the phenomenon of Many-Body Localization. In some cases, presence of disordered (quantum) interactions can enhance (or worsen) integrability of the system, where "integrability" refers to the presence of local integrals of motion. Here is a friendly lecture on this topic.

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