In response to #23:
First, some AdS/CFT basics. You have a (conformal) quantum field theory on the AdS boundary. This gives rise to a variety of fields in the AdS bulk, according to what is called the AdS/CFT "dictionary". For each scalar quantity you can build out of the fields in the CFT, there's an emergent scalar field in the bulk. For each conserved current in the CFT, there's a bulk gauge field. The energy-momentum or stress-energy tensor of the CFT gives rise to the bulk graviton. Other combinations of CFT fields give rise to heavy bulk fields, excited string states, and so on.
One thing this makes clear is that the bulk fields aren't completely independent of each other. They are generated by different combinations drawn from the same set of ingredients, the fields in the boundary CFT.
The question is raised, what's the boundary dual of Ashtekar gravity, or of Ashtekar gravity coupled to an SU(2) gauge field. Well, Ashtekar gravity (at least classically) is supposed to be just general relativity with new variables, so it should still be dual to the CFT stress-energy tensor; and an SU(2) gauge field in the bulk, should be dual to an SU(2) current in the boundary CFT.
On the other hand, Ashtekar variables express general relativity as a kind of Yang-Mills theory, albeit one with no predefined metric. So you might anticipate that Ashtekar gravity also has to come from a conserved current on the boundary? Really, this is the kind of problem where you can't guess the answer in advance. You have to do the hard work of constructing the theory if you can, and then see what's left when you get to the end. The logic of AdS/CFT definitely implies that Ashtekar gravity in AdS should come from the stress-energy tensor of the boundary theory, but there may be some extra algebraic property implied by the existence of the Ashtekar formulation of gravity (in terms of connection variables rather than a metric).
The same goes for the common origin of SU(2)L and SU(2)R in a gauged SO(4), in chiral graviweak unification - AdS/CFT principles imply that it would come from an conserved SO(4) current on the boundary, but something is going to work differently, e.g. one is not starting with a predefined metric in the bulk, so perhaps one needs to consider a "topological AdS/CFT" which can generate a bulk topogical field theory, part of which then has a phase in which a metric emerges.
kodama said:
is there any possibility given gravity is gauge theory squared that 5 dimensional gravity in Ashtekar variables, joined with SU(2) weak force, is dual to 4 dimensional QCD squared via AdS/CFT with QCD written as a 4 dimensional CFT
Ashtekar variables are quite specific to 3+1 dimensions. Generalizing to higher dimensions, you lose some of their properties. So you would need to take care that your 5-dimensional variables, when reduced back to 4 dimensions by Kaluza-Klein compactification, still had all the properties that you want in your d=4 Ashtekar gravity.
The same applies to embedding SO(4) chiral graviweak unification in higher dimensions. Woit doesn't get the extra U(1) in this way, instead he wants to get an extra U(3) from a different source, that will provide both the extra U(1) of electroweak unification, and the SU(3) of QCD.
How any of this relates to double copy relations ("gravity is gauge theory squared"), is again something I can't guess without doing a lot more work. There has been some work on the double copy in AdS space, in which double copy relations between gauge field and gravity in the bulk, map onto relations among the dual operators on the boundary. The ingredients for a double copy relation still exist on the boundary: the dual of the bulk gauge field is a vector current, and a vector is a one-index tensor; meanwhile the dual of the gravitational field is the stress-energy tensor, which is a two-index tensor. So it makes sense that you can make a two-index tensor out of two copies of a one-index tensor. But double copy relations (from my reading) have a lot of finicky details, there isn't a simple template that you can apply to every situation.
But in this case, we are asked to think about double copies involving Ashtekar gravity, and that's the immediate challenge; coming up with a form of double copy relations that can apply to Ashtekar variables at all, never mind all the other details.
One last comment, QCD isn't a conformal theory, so it won't produce a simple AdS dual. When people talk about holographic QCD, they usually mean a construction (Sakai-Sugimoto-Witten) that isn't AdS at all. Though I think there is a "bottom-up holographic QCD" in which the dual is an AdS that is truncated or cut off.