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In higher spin currents, the boundary CFT is dual to an asymptotic $AdS$. I have heard that $dS$ is not quantizable. But I don't understand why we want it to be in the first place. Isn't Chern-Simons for a positive cosmological constant evaluated on a $SO(4)$, so can't we define the mathematical observables as elements of $SO(4)$?

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There are a number of related results to which you could be referring.

QFTs on a fixed de Sitter background

There are significant obstacles to formulating these in a way consistent with unitarity. A previous SE answer gets into this but there is another important reference which explains the issue. Essentially, IR divergences that would cancel in flat space inevitably appear when you try to define the S-matrix perturbatively. One possible way out is a non-perturbative formalism developed recently. Another has been a common belief that the problem would go away if full dynamical gravity were considered.

Asymptotically de Sitter quantum gravity

I think it is way too early to declare that this "can't be defined". Vafa and collaborators have conjectured that potential dS string backgrounds are in the swampland but satisfactory explanations are lacking and the claim could easily be a lamp post effect. If someone says dS quantum gravity "can't be defined" and then goes on to contrast the situation with AdS / CFT, this is probably just an overly bold way of saying that a powerful calculational framework is still being searched for.

Classical gravity with positive cosmological constant

There is no problem defining this. And indeed you have given a perfectly good definition in three dimensions. 3D gravity with $\Lambda > 0$, $\Lambda < 0$ and $\Lambda = 0$ can all be formulated as Chern-Simons theories with different gauge groups. The problem comes with trying to quantize it and there is a bit of confusing language around why it doesn't work. Of course, if we literally had no prejudice about any gravitational feature that should be robust, it would make no sense for us to say "the duality breaks". We would just define quantum Chern-Simons theory as being what we mean by "quantum gravity". Hopefully a previous answer can clear this one up too.

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