Is the action of any interacting 6D CFT known? "6D super-conformal (2,0) CFT" is predicted to exist as an $AdS_7\times S_4/CFT_6$ dual of M-Theory.
My question is do we have any Lagrangians for a 6D CFT?
The conformal theories I know about are:
2D: Poisson equation in 2D
$$L_{2D}=\nabla\phi.\nabla\phi$$
3D: Chern-Simons theory?( not sure about this one.)
$$L_{3D} = A\wedge\partial A + \frac{2}{3}A\wedge A \wedge A$$
4D:
Yang-Mills:
$$L_{4D} = F^{\mu\nu}F_{\mu\nu}$$
"Conformal-Gravity" (Not a good theory as it has terms like $\partial^4$
$$L_{4D}=\sqrt{-g}C_{abcd}C^{abcd}$$
From the pattern it seems like there may be a 6D CFT symbolically of the form $A\partial^4 A$ where $A$ is a vector. Or $B\partial^2B$ where be $B$ is a 2-tensor.
Is it very difficult to find a 6D CFT? I would have thought it would be just a case of writing down a sum of all Lorenz invariant terms in 6D up to some finite power and checking for conformal symmetry? Would a computer search not find one?
(BTW even though GR is non-polynomial in terms of the metric tensor, it can be re-written in a polynomial form by using tensor-densities, in case that is an answer to why we can't look for 6D CFT by computer.).
 A: When it comes to conformal field theories, it is almost always useful to think about the low-lying scaling dimensions of their local operators.
If a CFT has an action, then it has a local operator that you are allowed to hit with $\int d^d \, x$ to get something dimensionless. In other words, it ought to have a scalar of dimension $d$. The main exception to this occurs with free CFTs because then there is an equation of motion causing many would-be operators to vanish on shell. So now there are two questions to ask.
First... can we write down a Lagrangian for a CFT in 6 dimensions which is free? Yes. But this would have no relation to the holographic dual of $AdS_7 \times S^4$. Second... can we write down a Lagrangian for a CFT in 6 dimensions which is interacting? Well to do this we would need scalar operators of dimension 6. And at least for $\mathcal{N} = (1, 0)$ and $\mathcal{N} = (2, 0)$ supersymmetry, the classification of unitary representations of the corresponding superconformal algebra rules out such operators.
This does not rule out the possibility of constructing a non-supersymmetric interacting 6d CFT with a Lagrangian but experience shows that an unwanted scale is generated whenever we try to do this. This is why the only item on your list which really is a conformal QFT is the free one ($L_{2D}$). Chern-Simons is a topological QFT, non-supersymmetric Yang-Mills famously has a running coupling and Weyl symmetry in gravity is a whole different beast.
All this is to say that the Lagrangian approach to conformal field theory is fairly limiting in general and this becomes especially true as you increase the spacetime dimension. By this point, most people suspect that a CFT in 5 or 6 dimensions can either be non-trivial or have an action but not both.
