Thanks for saying that it is the term “folklore” that put you off, and sorry for hearing that it did; it certainly wasn’t intended to be offending.

We did have a fair bit of discussion amongst us about finding the right word here, before we ended up settling for “folklore”, and not without including half a page of commentary on this choice of words. Maybe it helps if we share some of these thoughts in the open.

The issue we are facing is that we do want to discuss rigorous formalization and rigorous proof of statements that, when verbalized non-rigorously, are part of the body of statements about “M-theory” that are commonly accepted to hold among most practicing string theorists. But it is just as true that, as an actual mathematical theory, beyond that accepted web of informal statements,
an actual formulation of M-theory (hence also of M5-branes, in particular) did not and does not exist. Beyond being but a cause of concern for overly pedantic mathematicians, this is arguably the
glaring open problem behind modern string theory: to figure out what the actual, full, non-perturbative theory actually is.
This problem deserves attention.

For better or worse, this is not possible without disentangling, in one way or other, the web of partial results, plausibility arguments and consistency checks that constitute the existing body of thoughts about M-theory, from, on the other hand, a hard core of rules/axioms and rigorous step-by-step derivations based on these, which should eventually constitute an actual formulation of a fundamental physical theory based on mathematics.

Now the point of course is that the presently existing web of thoughts on M-theory is, while not rigorous, exceptionally good. There is little reason to doubt that an “M-theory” actually should exist, even if it hasn’t been formulated yet. So in presenting a proposal for an actual *axiom* of M-theory, and a list of theorems extracting consequences of this axiom, one needs, for communicating it, some word that delineates the fine line between what seems really clear to insiders and what has been rigorously derived and proven,
checkable in a mechanical way even by educated outsiders.

But, luckily, this situation is a familiar one in mathematics (and mathematical physics): Plenty of times it happens that an interesting but complex theory or theorem first emerges from discussion among insiders, who gradually become convinced of its truth, and who gradually begin to see strategies along which an actual proof might flow. Eventually this internal reasoning may become so convincing that practitioners become fully prepared to accept it as fact, shying away from the task of actually formulating and publishing a rigorous account not due to the doubt that it can be done, but because it may still be a tedious task left to do. In such cases the mathematics community speaks of “mathematical folklore”. And, sober as mathematicians are, this is is not meant pejoratively. Indeed, the facts that a mathematician values highest may often be mathematical folklore. But the word serves the purpose of carefully reminding the community that, while fascinating, useful, and widely regarded as fully plausible, the proof does remain to be laid out.

Famous examples of such folklore results are, or had been for a considerable time, for instance the construction of the spectrum of topological modular forms, or the cobordism hypothesis (see e.g. Stolz 14, p. vi for usage of “folklore” in this context). While it is undisputable that all available work on these matters is of the highest value,
there is further value in being fully clear about what has and what has not been proven at the full level of small-step rigorous argument, a point recalled recently in Barwick 17, Item 3.

Another example of mathematical folkore, possibly more analogous to the search for “M-theory”, is the search for a theory of the “field with one element”. Here, too, there is a collection of statements
extracted from extrapolating known results – extrapolations that, if there is any justice in the world, ought to become true in *some* theory, yet to be formulated – and connected by a tight web of interrelations and plausibility checks.
This $\mathbb{F}_1$-folklore is the most fascinating and valuable mathematics to many, and lots of interesting mathematics is generated motivated by it. But part of its fascination is precisely that it does remain folklore, and that there does remain an open question of formulating the actual theory, and then seeing it naturally produce,
by mechanical derivation from first principles, all the gems that people have previously laboured to unearth by ingenuity.

It is in this sense that we thought the word “folklore” in section 2 of arXiv:1904.10207 would serve a good purpose. As we tried to explain there, it is not intended at all to be pejorative, but just to explain the nature of the results we present in section 4, these being systematic derivations from an axiom of M-theory which we propose: “*Hypothesis H*”.

Given your reaction here, we have again had a discussion about what to do. From reasoning as above we still tend to feel that the word “folklore”, regarded logically and in its established use in mathematics, is appropriate and non-pejorative. But of course we are aware that communication has both a sender and a receiver, and if a word causes with the receiver a strictly different effect than intended by the sender, it is eventually not suited for communication.

Certainly there is no value in having words around which, even if unintentionally, cause bad feelings and thereby
contribute to distraction from the actual mathematics
instead of serving a sober focus on objectively open mathematical problems. We can easily change the word to something else in a revision of our article, if that does help remove tension. In any case we apologize for all bad feelings that our choice of word did
cause, unintentionally.

Maybe, while we are thinking about if and how to revise the wording,
it could be interesting to see what some of the bystanders here think about the issue of the “folklore” terminology, be it specifically in the case at hand, or be it more generally in mathematics and mathematical physics.

## Re: Twisted Cohomotopy implies M-Theory anomaly cancellation

Let me point out here for the record that this paper contains a lot of inaccuracies in its discussion of the global anomaly of the M5-brane.

Their condition (20), reading $[G^2] = 0$ on the worldvolume of the M5-brane (or in families on the family of worldvolumes) is automatically satisfied (i.e. no need to invoque any anomaly cancellation condition), because as the C-field sources the self-dual field, it is trivial. (I.e. even $[G] = 0$.)

They claim to find some problem with my proof of the cancellation of global anomalies on the M5-brane based on a misinterpretation of a term written $G_W^2$ in my paper. ($G_W$ does NOT coincide with their $G$ and may be non-trivial.) They claim it should contribute to the local anomaly, which is wrong.

They claim that my paper is incompatible with the paper of Harvey, Freed, Minasian, Moore, which checked the local anomaly cancellation. This is again wrong.

The formalism developed in my M5-brane paper (and other ones) was used in a recent paper with Greg Moore where we make very non-trivial precision check of global anomaly cancellation of 6d F-theory models. I can’t imagine how such checks would pass if there was something wrong in our way of computing global anomalies.

I am all the more annoyed because Urs contacted me to ask me questions about the paper. I took a lot of time to explain every of these points in detail, although it felt a bit like it was falling on deaf ears. As the thanks in the acknowledgement may lead people to think that I agree with their presentation of the existing knowledge on anomaly cancellation, let me record here that this is not the case, and that I never heard of this paper until now.

This has no bearing on the quality of the rest of the paper, which I haven’t studied in detail yet. Indeed, in the end their condition (20) does hold, albeit not for the reason they think it does.