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Are there any papers which directly tackle the question of whether or not there exists interacting CFTs in spacetime dimensions higher than 6?

It has been proven that there do not exist any superconformal field theories in spacetime dimensions higher than 6. This question is about conformal field theories in general, without supersymmetry.

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  • $\begingroup$ There is a lore about $\mathrm{O}(N)$ universality classes in odd dimensions $d = 2k-1$, which can be reached by epsilon expansion from $d = 2k$. There are known examples up to 6$d$ however. See e.g. arxiv.org/abs/1806.02340 $\endgroup$ – MannyC Mar 8 at 5:45
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There are no papers in the literature that address the question directly. If you're interested, you can have a look at hep-th/1305.0004 by Fitzpatrick, Kaplan and Poland which addresses a certain large-$D$ limit for conformal four-point functions, but they don't make any definite statements about whether such CFTs exist or not.

Let me make an edit, since some comments mentioned some known examples in $D>6$. There are indeed various examples of interacting CFTs in $D>6$ that obey the axioms of CFT except for unitarity. The idea is that you take some field, let say a scalar field $\phi$, with a kinetic term $$ L \sim \phi \Box^\alpha \phi $$ where $\alpha > 1$, either integer of fractional. (You can make rigorous sense of the operator $\Box^\alpha$ with some work.) The scaling dimension of $\phi$ is then $D/2-\alpha < D/2-1$, so the famous unitarity bound $\Delta \geq D/2-1$ is violated. Now for $\alpha$ sufficiently large and $n$ a given integer, you can write down interactions of the form $$ \delta L = g \phi^n $$ where $g$ is classically marginal for some $D = D_*$, and then you can repeat the usual RG story (do an expansion in $D = D_* - \epsilon$ dimensions etc.) The same extends to fermions and higher-spin fields. Such theories are interesting but probably not quite what OP is looking for.

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