There are several threads here on this topic, some quite old. The most prolific poster here used to be
@marcus, who was a tireless enthusiast for loop quantum gravity, and he started at least one thread on twistorial LQG. More recently
@kodama was asking about this topic.
Let me first state my own view of LQG. The precursor of LQG was the discovery of the "Ashtekar variables", a change of variables for general relativity (from a metric to a connection) in which the equations become polynomial and with a formal resemblance to Yang-Mills theory. That's a statement about the classical theory. Then one may ask, how do you construct quantum theory? LQG is a particular proposal for how to do this, which was partly inspired by Witten's work in 2+1 dimensions (his work on gravity as a gauge theory in 2+1 dimensions, and his Fields-medal-winning mathematical work on topological field theories in 3 dimensions).
I have seen two main streams of research in LQG. One is a top-down path in which the focus is on mapping the equations of the theory onto operator relations in a Hilbert space, the other is a bottom-up path analogous to concentrating on Feynman diagrams (e.g. for LQG, the "spin foams"). The first path is referred to as canonical by LQG theorists, but it actually has properties that deviate from how canonical quantization works for ordinary QFT.
The two real problems I recall for the first path, are (1) that the method of quantization destroys even the topology of space-time, and there's no reason to think that anything that looks geometric will reappear in the quantum theory (2) the method of implementing symmetries makes quantum anomalies impossible, yet some anomalies (the global ones) have empirical consequences (e.g. in pion decay). I don't think these criticisms have ever been written up systematically, but Urs Schreiber has posted about (1), and (2) comes up in a 2006 blog debate with string theorists and particle physicists called "The LQG Landscape".
Another problem that may be of note (related to the problem of the chiral anomaly) is fermion doubling in LQG, which Lee Smolin conceded might be a problem in that 2006 thread, and which he later wrote about in a paper with the former prodigy Jacob Barnett... A major reason that canonical LQG is quantized in a different way, is that the "gauge group" of gravity in Ashtekar variables ends up being complexified, which makes it non-compact, which causes problems for a Yang-Mills theory. For all these reasons, I regard canonical LQG as essentially a dead end, that I dust off only when there's some unusual reason to do so (e.g. Eric Weinstein's theory also has the problem of a non-compact gauge group).
It would be nice if someone did a retrospective analysis covering the historical motivations and the criticisms I just summarized, in a rigorous way (not least because the criticisms should be stated and argued for, with mathematical exactness). But as far as I know, it never quite happened. The LQG researchers moved on beyond the canonical approach, and the critics of LQG were mostly busy with string theory and confined themselves to informally stating why they rejected LQG. Perhaps it's something that a philosopher of physics could do.
Regarding the bottom-up approach to LQG, the spin foams, I think it's not so much that there is an obvious fatal problem, but rather that it never became predictive. You couldn't prove that the spin-foam path integral for notionally 4d quantum gravity would actually contain 4d metrics rather than degenerate ones of lower dimensionality, and the amplitudes for the spin foams should still contain infinitely many undetermined coupling constants, one of the reasons that the standard quantum gravity, based on gravitons derived directly from general relativity, is regarded as unpredictive. But I'm less sure about my comments here. Carlo Rovelli certainly claims that there's a good graviton scattering amplitude that has been obtained from the bottom-up approach to the theory; and in fact, if spin foam theories have something in common with standard perturbative quantum gravity, that might be a point in favor of the idea that they *are* equivalent theories in some limit.
Anyway, for a long time, all that was the entirety of my attitude towards the Ashtekar variables. But then, I think during discussions with
@kodama, I discovered an obscure Russian work from the 1990s, which quantized the Ashtekar variables in a more conventional way. Assuming that they did their work correctly, what they did really was just the standard quantization of general relativity - which is regarded as UV-incomplete, but valid as an effective field theory - but in new variables. So that's my current attitude towards Ashtekar variables - that LQG is ultimately a wrong turn, but that the Ashtekar variables themselves can be valid, at the level of the effective field theory.
Before I go on, you mentioned Witten's twistor string, and asked what came of that. The twistor string is actually an example of "topological string theory" (in twistor space). Topological string theory is not full string theory; even more than the usual string theory, it is an object of mathematical rather than physical study. But very curiously, Witten discovered that topological string theory in twistor space was capable of reproducing Yang-Mills theory. It was still a bit of a mess, because it was actually producing Yang-Mills theory coupled to conformal gravity, which is a non-unitary theory when quantized.
But as I understand it, the initial work on the twistor string soon gave rise to direct uses of twistor variables in Yang-Mills theory, which gave increasingly magical results and which didn't have the contamination by the problematic conformal gravity. The subfield of research arising for this, which is sometimes whimsically called "amplitudeology", you tend to hear about either in terms of miraculous simplifications of sums involving thousands of Feynman diagrams, or in the form of Arkani-Hamed's amplituhedron, which is supposed to be a perspective on physics in which space-time is no longer fundamental. The twistor string is arguably ancestral to both these developments, by way of a combinatorial formula (BCFW) that first arose in that context.
I feel that the real meaning of the twistor string has never quite been worked out. The field theorists took BCFW and the use of twistor variables and abandoned the string. One form of the twistor string, the "ambitwistor string", is still studied in itself, and I have a vague idea that it may have some relationship to the zero-tension limit of the superstring, but that it's not worked out.
I guess I should add that
twistor theory, as developed by Penrose's school, is actually distinct from all the above. It's best known for describing massless fields with "twistor diagrams", and then trying to describe mass and gravity with deformations of these that involve higher math like cohomology. Penrose always had his own idiosyncratic idea that something about quantum gravity would cause spontaneous collapse of the wave function, and even now he and his collaborators are still pushing those ideas ahead slowly. On the other hand, I do know that other people from his school, like Andrew Hodges, managed to repurpose some of the work on twistor diagrams, for the new era of mainstream twistor physics.
Now returning to your question as to whether twistors and LQG have ever been combined:
There is a paper or two by Simone Speziale about "twistor networks", which are something like spin foams in twistor space. In my framing of things, this is a work from the bottom-up approach to LQG and so has a chance of being part of a valid quantum theory.
Then, perhaps less noticed, Peter Woit's own theory actually combines Ashtekar variables and twistor variables. Woit is best known as a critic of string theory, but in recent years he began to advocate his own idea of "Euclidean twistor unification", and part of this involves obtaining gravity by interpreting a subset of his twistor variables as Ashtekar gravity variables. But I should note, this is not a "theory" in which any calculations can presently be performed, it's more an idea for a theory at present.
But if you follow the current mainstream, the best prospects for twistors in physics, are in constructs like the amplituhedron, that don't involve LQG at all. I must say that LQG is one of those theories that probably has more interest from the general public (from fans of theoretical physics, let us say) than from actual physicists, I think because it sounds like a quantization of space-time itself. Of course there are still people who work on it, but there is far more vitality in other research programs in quantum gravity. It wouldn't surprise me if, in the end, something about the correct approach to quantum gravity (whatever that turns out to be) is regarded as justifying some of the LQG dreams, but for now those dreams don't fly, because of specific technical problems with the theory.
One final comment. I have, in at least two places, seen it said that there is some deep connection between twistors and the Ashtekar variables. Just for reference, those two places are "The Sparling 3-form, Ashtekar Variables and Quasi-local Mass" by Mason and Frauendiener, and "Gravity, Twistors, and the MHV Formalism" by Mason and Skinner. (I see that Yannick Herfray has some papers that touch on it too.) The really important connection to make here may not be any particular theoretical hybrid, but rather some inherent conceptual connection between twistor space, chirality, and general relativity.
One final, final comment: for a big picture commentary on twistors by people who actually work in the field, see "Twistor theory at fifty" by Atiyah et al.
