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Based on the AdS/CFT dictionary, global symmetries of the boundary theory are related to the gauge symmetries in the bulk theory, but I could not find a relation between gauge symmetries of the boundary theory with the symmetry structures of the bulk theory? Is there a similar correspondence for the gauge symmetries of the boundary theory?

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    $\begingroup$ I would like to see an answer too, but one aspect that might interest you has to do with electric-magnetic duality in ${\rm AdS}_4/{\rm CFT}_3$. In the standard story, if you have one abelian gauge field $A_\mu$ in the bulk, you have a ${\rm U}(1)$ current in the boundary associated to a global symmetry in the CFT. Now it seems that when you apply electric-magnetic duality in the bulk you end up adding one dynamical ${\rm U}(1)$ gauge field to the CFT. So there you have a situation in which the CFT has a gauge symmetry. See e.g arxiv.org/abs/2005.03667 and references therein. $\endgroup$
    – Gold
    Aug 3 at 22:00
  • $\begingroup$ @Interesting, thank you so much dear Gold. $\endgroup$
    – Arian
    Aug 4 at 7:17
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    $\begingroup$ There isn't going to be a relation like that because gauge symmetries are not an intrinsic property of a theory. Any time we call something a gauge theory, it is only the singlet sector that comprises the actual theory. Embedding this into something larger is often a helpful trick. But non gauge invariant observables are not part of the theory and therefore not dual to any sort of field in the bulk. $\endgroup$ Aug 9 at 14:08
  • $\begingroup$ Thanks , Although observables are gauge invariant and gauge symmetries are just redundancies ,I compared mathematical structure of the theories, so I expect a correspondence between them. $\endgroup$
    – Arian
    Aug 11 at 20:56

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