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Are there names for $image(ch \cdot \beta)$ and $cokernel(ch \cdot \beta)$?
You might call image and cokernel of the composite maps, all the way from M-theory, through string theory, to effective field theory, the “real non-perturbatively corrected” Landscape and Swampland.
As you know, Tom Banks’ recent rant was all about how that “real non-perturbaitive” Landscape/Swampland should be entirely different from the naive EFT picture of the Landscape/Swampland (corresponding here to image/cokernel of just $ch$ itself). But I don’t recall that he used any particularly catchy phrase for the distinction.
added pointer to today’s
Thanks for your energy. Maybe we could try to make the formulations of these statement a little more precise so that they become closer to what more mathematically minded people would understand under a conjecture (namely a well-formed statement, whose formulation is clear and unambiguous, only its truth remaining open).
First and foremost I would suggest to preface these statements with a sentence on what, in the following, is going to be meant by a “state”, even if just roughly.
Because even at a rough level (not beginning to try to ask for definitions of state spaces of quantum gravity theories), there are a variety of meaning here. On the one hand there is “solutions of gravitational EOMs”, and I think this is meant for the most part. But every now and then what is meant are “elementary particle excitations” and then there is maybe some tacit hope that for BPS situations these two notions agree, or at least “match”.
For instance, conjecture 2.5 postulates that, for a given electric charge, there exists an “electrically charged state” whose mass is bounded above by that charge, in suitable units. But since electrons exist, which have finite electric charge and no mass (fundamentally, before symmetry breaking), this “conjecture”-statement is empty unless one says how the intended meaning of “state” rules out ordinary massless charged particles.
By the way, our Instiki parser does not understand \begin{itemized}
, it wants you to type a naive bullet list, like this:
* first item goes here
* second item goes here
Thanks for your comment, I completely agree with you: these statements need a reformulation (clarification) to fit in nLab. I would like to do that, since would also like to see what these conjectures mean in a more formal setting. If you prefer, I can always remove the section, for now, and re-add it in future when it is more detailed!
It’s good that you added the section, since it records the folklore that the swampland community is currenty talking about, so the section certainly has journalistic value.
If it’s too hard to add a sentence on what some of the terms are really meant to mean, then let’s just add a general sentence that orients the reader as to what nature of content they are about to see. Maybe something as follows:
The following list of statements have come (around 2020-2021) to be widely discussed in a sector of the string phenomenology community which associates with the swampland imagery. They are commonly referred to and known as “conjectures” but many terms used remain vague/undefined and may change their intended meaning with context (not the least the notion of “state in quantum gravity”). However, some of these statements have been interpreted in special cases as more precise statements about Calabi-Yau moduli spaces, and in this form they may be closer to conjectures in the sense of mathematics.
I have edited a little the section on the “swampland cobordism conjecture” (here):
added pointer to McNamara & Vafa 17
where it said, without further ado “cobordism group must vanish” (with a pointer to the usual cobordism group definition) I changed it to read: “some kind of ’quantum-gravity version’ of a cobordism group must vanish”, for otherwise it’s not even what McNamara-Vafa have in mind.
before the symbols “$\Omega^{\mathrm{QG}}_d = 0$” I replaced an optimistic “i.e.” with a more honest “In heuristic symbols:” (or maybe “informal symbols”, in any case, the symbol $\Omega^{QG}$ does not refer to an object that has been defined, not even by physics standards).
Finally, I took the liberty of adding the following remark after the “swampland cobordism conjecture”:
A rigorous discussion of a possible role of cobordism cohomology in M-theory, assuming Hypothesis H, is in Sati & Schreiber 21a; for relation to discussion of Conj. 2.16 see p. 83 there.
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