> let's be careful keep an open mind...
I may need to explain in a little more depth, as things just jump off the page.
It is as if I posted a photo of my "bee yard", and talked of my "honey harvest", but the photo showed a row of paper wasp nests.
Let's start at the end - take the phrase "...bees may also be perceiving certain quantum fields..."
Well, there are quantum fields, but they are very likely impossible to perceive in any way except very very very indirectly. One physical effect that proves that at least one flavor of "quantum field" exists is the "Casmir Force". Take a look at Pg 6 in the linked paper for the state-of-the-art custom technology used to place two plates very close together in as good a vacuum as possible in the late 1990s, which was needed to exclude the traditional electromagnetic fields.
See http://web.mit.edu/kardar/www/research/seminars/Casimir/PRL-Lamoreaux.pdf
https://tinyurl.com/93ddh4ct
How close? 0.1 to 6 µm. A human red blood cell is about 5 µm in diameter for perspective. And this was not detecting the field itself, this was just showing that there is a force in a very good vacuum, where there should be no force at all unless it is Casmir. This was EVIDENCE FOR the existence of a quantum field, nothing more. The mainstream view at present is that there are really only 2 tangible "quantum fields", the electromagnetic field and the electron field. They interact, and one can see the outcomes of this interaction. Particles exist as a result of an "excitation of the underlying quantum field".
I view this in terms of the ocean - assume a choppy sea. All the white caps are possible particles. If one of the whitecaps rises high enough to splash into your dingy, then a "particle" has separated from the sea and come into existence as a particle, rather than falling back into the sea and not becoming a particle. The chop is the "excitation of the underlying quantum field", and "a quark" just soaked you from knees to toes. THAT'S what a quantum field is. We just are not small enough to be able to sail on a quantum field, or fast enough to see it happen. But this is how all matter exists.
We may thus safely conclude that such fields are neither perceptible to bees, nor useful to them in their "dance", as they are also not small or fast enough.
The above is USE OF TERMS WHILE NOT RESPECTING OR KNOWING THE DEFINTION OF THE TERMS
Taking a step back, we have:
> "The mathematics has a quantum mechanical interpretation that suggests that the bees may also be perceiving certain quantum fields..."
There is no "quantum mechanical interpretation" of ANY type of math. Math says what it says, and the very nice thing about math is that you DON'T need to "interpret" it. While one might say that we "attempt to describe quantum mechanics with math", we would never attempt to describe math with quantum mechanics. "String Theory" does not have an "interpretation of Math", nor does "Solid-State Physics".
The above is STATEMENTS THAT CANNOT POSSIBLY MAKE SENSE.
One more - "...the types of singularities that arise in a completely integrable system known as the Toda lattice..."
A "Toda lattice", a one-dimensional construct. The only thing that this has in common with the other buzzwords being tossed about here is that they were all hot topics in the mid-1960s. (Quantum Fields, Toda lattices, and the honey bee dance language were all mid-1960s hot topics in the press.)
Now, let's step back and look at the whole thing - what sort of premise starts out in pure Topology ("The geometry of momentum mappings on generalized flag manifolds...") and ends up in Quantum Electrodynamics, but then veers off into BEES?
The above is an attempt to ASSEMBLE BITS AND BUZZWORDS FROM WIDELY DIVERGENT FIELDS into a "Something New".
I will bet $20 that we can find all of these buzzwords featured in the 1965-1970 editions of a mass-market magazine like "Popular Science" or "Analog", likely found years later in a used bookstore and assembled into the work presented by a less-than complete understanding of the basic concepts.
I am certainly NOT saying that this was deliberate hucksterism - such work is done by very sincere people. It is similar to the "single-point energy" and "perpetual motion" problem. People want to believe, and learn just enough to become a hazard to navigation.
We could slog through the whole thing, but topology is tedious - a "flag manifold" is just a Grassmannian - it is "any vector space in the plane" that has a dimension one that is a straight line passing through the origin. So it kinda looks like a pennant on a stick to a topologist, hence the name "flag".
The place to start with the topology of Grassmannians would be with "Principles of Algebraic Geometry", by Griffiths and Harris, here's the 2011 edition:
https://drive.google.com/file/d/1eDJ0RloSe73anGFvnSJ-RkDyvErnuKC-/view?usp=sharing
https://tinyurl.com/k33ct832
And Murray Gel-Mann (the guy who called such things "Quantum Flapdoodle") would be tickled if more people read any of his books. He passed away in 2019, and his books were his pride and joy, after his native American pottery collection. Here is Quark/Jaguar, which he revised several times:
https://drive.google.com/file/d/1Aq2YpEL_mHYLY9fCtd-oRp--mGk_0JCa/view?usp=sharing
https://tinyurl.com/3wsmfmju
Both books will stay up for a month or so, but torrent files are sure to exist for both, as both are "classics".
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