## April 24, 2019

### Twisted Cohomotopy Implies M-Theory Anomaly Cancellation

#### Posted by David Corfield

The latest instalment of Urs’s march towards M-theory is out on the arXiv today, Twisted Cohomotopy implies M-Theory anomaly cancellation, (arXiv:1904.10207).

A review of the program to date appears in the recent:

In the latest paper the authors are moving beyond the approximation of rational cohomology to explain from first principles the folklore about anomaly cancellations in M-theory. The generalized non-abelian cohomology theory known as cohomotopy theory, and in particular its twisted version, appears to have all the answers, accounting for six conditions on the cocycles corresponding to the C-field.

En route we get to have a fresh look (p. 5 and sec 3.4) at why the exceptional coset realisations of the 7-sphere, which John told us about back in 2007 in Rotations in the 7th Dimension, are important. (See also TWF 195 from “Speaking of M-theory and the like…”.)

Why am I following this? That it’s even possible for an amateur to follow is quite something. But then it is completely in the spirit of the founding principle of this blog that cutting-edge ideas are there to be understood.

So many important pieces of pure mathematics arise from string-theoretic structures and their dualities and compactifications (see cascades of Kaluza-Klein reductions), that chasing back to the M-theoretic source like this should have very important consequences for mathematics itself.

And there’s more to come:

While here we explicitly consider only the plain topological sector of the C-field, hence its charge quantization in plain homotopy theory, the natural form of the charge quantization formulation… immediately generalizes to global equivariant and differential Cohomotopy. (p. 6)

Anyone, as I am, interested in modal homotopy type theory and following the story of (differential, super) cohesion and its modalities arising from various adjoint quadruples, will be interested to learn that global equivariance relies on a further such quadruple.

As for the importance for physics itself, that’s beyond my pay grade, but who wouldn’t like Einstein’s God to choose such a fundamental cohomology theory as cohomotopy, represented by the spheres, when selecting a gauge field theory?

Posted at April 24, 2019 11:11 AM UTC

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### 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.

1. 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 $[G] = 0$.)

2. 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.

3. 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.

4. 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.

Posted by: Samuel Monnier on April 24, 2019 3:59 PM | Permalink | Reply to this

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

I see. I’ll pass that on.

Posted by: David Corfield on April 24, 2019 5:24 PM | Permalink | Reply to this

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

Hi Samuel,

as you know, the whole point of considering spacetimes that are 4-spherical fibrations is that the singular M5-brane locus has been removed from spacetime, which is key to having a clean mathematical formulation of the situation. So the entire argument, yours as well as ours, happens away from the M5-brane locus, and there the cohomology class of the 4-flux by itself does not need to vanish – unless of course you demand it to vanish by hand.

Whether or not there is a problem with your proof is hard to tell for me, because you don’t give a solid argument for why that spurious term should vanish. But all is good, because we prove that what we call “Hypothesis H” rigorously does imply this condition, so everything seems to be falling nicely into place and we are left with a very clean and picture.

I apologize if you feel the relation of your work to that of FMM is misrepresented, I will be happy to adjust the wording on this, if it matters. But I thought highly of the approach of your article (based, incidentally, on removing the M5-brane locus from spacetime :-) as improving in mathematical precision on the suggestion made in FMM. I’d think we don’t need to fall back to FMM after this achievement.

Posted by: Urs Schreiber on April 25, 2019 8:49 AM | Permalink | Reply to this

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

Hi Urs,

In the first paragraph, I don’t know what is meant by “away from the M5-brane”. Everything happens in a tubular neighborhood of the M5-brane of vanishingly small radius. The C-field is singular because the M5-brane is a magnetic source. If the normal bundle is trivial, we can define an “effective C-field on the M5-brane”, taking simply the tangent component. If the normal bundle is non-trivial there is no canonical way of defining a “tangent component” and the prescription given in my paper explain what the “effetive C-field on the M5-brane” is. You can see this prescription as the mathematically precise way of defining the coupling of the C-field with the M5-brane degrees of freedom when the normal bundle of the M5-brane is non-trivial. It is this effective C-field on the M5-brane that sources the self-dual field and that is trivialized by the latter. Are you saying that this is not the C-field you’re speaking about in (20) of your paper? You can’t possibly be saying that anomaly cancellation on the 6-dimensional locus of the M5-brane gives you topological constraints on the C-field over the whole 11-dimensional spacetime, right?

In the second paragraph, it depends which term you’re speaking about. If it’s the $G^2$ term you mention in your paper, which I take as the square of the effective C-field field strength on the M5-brane in a 2-parameter family of M5-brane so that the total dimension is 8, then it vanishes because as the C-field sources the self-dual field, it is exact so its field strength vanishes. If you’re speaking about the $G_W^2$ term in my paper, it is neither “spurious” nor vanishing. It is absolutely necessary for the consistency of the global anomaly. I explained to you numerous times why it does not contribute to the local anomaly. To compute the local anomaly, you can consider a fibration of M5-brane worldvolume over a disk and $G_W$, in this case, can be taken to be the effective C-field’s field strength on the M5-brane, which vanishes for the reason just explained. If you are computing the global anomaly associated to a 7-dimensional mapping torus bounded by an 8-dimensional manifold, which is the setting in my paper, then $G_W$ is a relative form on the 8-manifold. Integral of relative forms on manifolds with boundaries are topological, so this is really a topological term contributing only to the global anomaly. Probably the 6-7th time I’ve explained that, but at least now it’s public.

Concerning your 3rd paragraph. It’s not a question of “adjusting the wording”. I listed two statements that you make that are plain wrong. The least you can do is correcting those. I don’t know what on earth “improving in mathematical precision on the suggestion made in FMM” means. FHMM proved the cancellation of local anomalies in M-theory backgrounds with non-intersecting M5-branes. I proved the cancellation of global anomalies in the same backgrounds. My proof has a single caveat, which is that the formula for the global anomaly for the self-dual field may in principle be off by a 4th root of unity, due to a shortcoming of its proof in a previous paper (which also has to do with the impossibility of a Lagrangian definition of the theory of a single self-dual field). Yet, after the number of consistency check this formula has passed, it’s as good as true.

I hope you understand why I am upset and why this comment is maybe not as courteous as it could be:

1. You contact me asking questions about my paper, not really listening to the answers but rather trying to find a supposed “mistake” due to a misunderstanding of my paper.

2. You publish a paper claiming I’m wrong thanking me in the comments but without even showing me a draft beforehand.

3. When I point out the exact wrong statements that you made, you’re still not listening, but rather persist with the same misunderstanding about the supposed “spurious” term, softening your comment with condescending compliments about how my paper helped “improve” on FHMM. As mentioned above, the point of the paper is not to improve the formalism of FHMM: they prove the cancellation of the local anomaly, I prove the cancellation of the global anomaly.

This is my last comment about this. Everything is public, people can judge what’s right and what’s wrong if they take the time to read the papers carefully.

Posted by: Samuel Monnier on April 25, 2019 6:58 PM | Permalink | Reply to this

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

Hi Samuel,

to be concrete, and maybe to recall for bystanders, the issue under discussion is the following one, which is rather straightforward (and involves no relative forms):

Under consideration is a 4-spherical fibration

$\array{ X \\ \big\downarrow{}^{{}_{ \mathrlap{\pi} } } \\ X_{bas} }$

associated to a rank-5 vector bundle $N$ on $X_{bas}$, to be thought of as a family of spacetimes of 11-dimensional supergravity, which have a 5-brane locus “threading” through the 4-sphere fibers removed.

For bystanders: A simple example is the trivial fibration $X = AdS_7 \times S^4$ which is the near-horizon geometry of a single 1/2 BPS M5-brane. More generally the base is of the form $X_{bas} = Q_{M5} \times \mathbb{R}_{\gt 0} \times U$ (as a smooth manifold, not necessarily isometrically), where $Q_{M5}$ is a 6-manifold representing the 5-brane worldvolume, $\mathbb{R}_{\gt 0}$ parametrizes the positive distance away from it (while the 4-sphere fibers are the unit sphere around it) and $U$ is any finite-dimensional manifold over which we are parametrizing the situation, in families.

Further under consideration is a differential 4-form on $X$ with unit flux through the 4-sphere fibers

$G_4 := \tfrac{1}{2}\chi_4(\pi) + \pi^\ast(G^{basic}_4) \,.$

Then the anomaly inflow in real/de Rham cohomology as on your p. 15. is the fiber integration

$\pi_\ast \big( -\tfrac{1}{6} G_4 \wedge G_4 \wedge G_4 + G_4 \wedge I_8 \big)$

which, by the nature of the Euler form $\chi(\pi)$, comes out to be

$\cdots = I_8 - \tfrac{1}{24} p_2(N) - \tfrac{1}{2}G_4^{basic}\wedge G_4^{basic}$

Now:

1. the first summand $- I_8$ is the bulk anomaly inflow as it was traditionally understood, which however cancels the worldvolume anomaly of the 6d field theory on a would-be M5-brane potentially threading through our 4-sphere fibers, only up to a remainder $\tfrac{1}{24} p_2(N)$ (Witten’s (5.7))

2. while the second summand $- \tfrac{1}{24} p_2(N)$ cancels just that previously “puzzling” remainder, thus exibiting this formulation of anomaly inflow via fiber integration over the 4-spheres around the 5-brane as a clean version of the suggestion of FMM 98 for how this term comes about;

3. but this nice state of affairs comes at the cost of yet one more term, $\tfrac{1}{2}G_4^{basic}\wedge G_4^{basic}$.

The issue under discussion is why this third term should go away.

In your (3.7) you display the negative of this term as an extra summand of the M5-brane worldvolume anomaly. This way the two summands cancel each other. But there is not suppoosed to be such an extra summand in the M5-brane worldvolume anomaly, it is certainly not there in Witten’s (5.4).

When I asked you about this you pointed me to extra terms in the global anomaly, hence beyond the approximation of real/de Rham cohomology, discussed in section 4 of your arXiv:1110.4639.

I studied that section, and would have some comments or questions, but in either case the term $\tfrac{1}{2}G_4^{basic}\wedge G_4^{basic}$ is present already in real/de Rham cohomology, so that no element in a torsion subgroup of cohomology can affect this. In the end we’d of course hope the global anomaly to vanish as a cocycle in full differential cohomology, including torsion subgroup effects, but for that it is necessary that it vanishes at least in real/de Rham cohomology in the first place, and as long as that is under discussion, it doesn’t help to turn to torsion effects.

Then you said that relative cohomology needs to be invoked. You can say that, but it wasn’t part of the issue under discussion and it remains vague enough that I don’t see how it addresses the above rather straightforward issue. But if you feel like writing it out in detail, with assumptions, claims and proofs cleanly laid out, that could help to make the conversation less frustrating.

Posted by: Urs Schreiber on April 26, 2019 10:57 AM | Permalink | Reply to this

### families

The source of confusion, here is that you are working with a fixed 6-manifold, $X_{\text{bas}}$, whereas (in his (3.7) and thereabouts) Samuel is working with a family $\begin{matrix}X_{\text{bas}}\to&W\\&\downarrow\\&B\end{matrix}$ His $G_W$ is a 4-form on $W$ whose restriction to any fiber (the thing you call $G_4^{\text{basic}}$) is cohomologically trivial.

Of course, if $[G_4^{\text{basic}}]=0$, then it follows trivially that $[G_4^{\text{basic}}\wedge G_4^{\text{basic}}]=0$.

Posted by: Jacques Distler on April 26, 2019 7:10 PM | Permalink | PGP Sig | Reply to this

### Re: families

Hi Jacques,

the $X_{bas}$ in my comment above (and in our preprint) may be the total space of any fibration/family of 5-brane worldvolumes (times a radial ray); I had mentioned the case of a product fibration as an illustrative example.

Restriction to a single worldvolume fiber is not really relevant for the discussion of anomaly polynomials, as they all vanish fiberwise, already by degree reasons.

But mainly I’d re-iterate that in the setup under discussion (our sections 2.5 and 4.5), with (families of) 4-spherically fibered spacetimes, the singular M5-brane locus itself is removed, present are only its copies a positive distance away from the singularity; and hence there is no $H_3$ trivializing the 4-flux.

(And if we included the singular locus, it would become unclear what to make of data supported at the singularity, which is the reason for removing it in the first place.)

One may consider a different setup, where 5-brane worldvolume flux $H_3$ is present, but no singular 5-brane, where $d H_3 = G_4$ does make sense and does hold. This is a different discussion altogether, now the topic of the second half of section 4.6 here in v2 (this didn’t make it before Easter…)

Posted by: Urs Schreiber on April 27, 2019 2:25 PM | Permalink | Reply to this

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

Hi Urs,

I don’t get notified of the replies here for some reason.

Ok I thought randomly about that again and I realize I did say something incorrect. While the G \wedge G is topologically trivial whenever you consider a fibration of M5-brane wolrdvolume, for the reasons explained before, it can indeed contribute to the local anomaly. To compute the local anomaly, you consider a fibration over a disk and even if G \wedge G is exact, you do get a boundary contribution.

This term is indeed present in the formula for the self-dual field anomaly. The self-dual field has a gauge anomaly when the background field that sources it (here G) is non-zero. This local anomaly can be identified with the curvature of the theta line bundle of which the partition function of the self-dual field is a section, as explained in the Witten paper you cite. (The gauge field sourcing the self-dual field (G) has Wilson lines identified with the intermediate Jacobian of the 6-manifold. The intermediate Jacobian is itself the fiber of a bundle over the space of metrics modulo diffeomorphisms. The partition function is the section of a line bundle over the whole fiber bundle. The component of the curvature along the intermediate Jacobian is the local gauge anomaly.)

Witten doesn’t have it in (5.4) because he’s checking the cancellation of gravitational anomalies. He says he already checked the cancellation of the gauge anomalies in Section 3. You’ll see the G^2 term there. He explains quite clearly that the local gauge anomaly of the self-dual field is cancelled by a contribution from the M-theory Chern-Simons term. My derivation generalizes this cancellation to global anomalies, where the G^2 term can now be non-trivial over an 8-manifold with boundary.

I hope it’s a bit more clear now. I may meet Hisham this week and hopefully I can clear up any remaining misunderstanding with him.

Posted by: Samuel Monnier on May 13, 2019 12:23 AM | Permalink | Reply to this

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

Hi Samuel,

no problem.

But the term in question, as in the concrete review of the issue above, is not a gauge anomaly term as in section 3 of Witten’s arXiv:hep-th/9610234 (e.g. there is no gauge parameter in that discussion), but does concern the cancellation of the gravitational anomalies, left open in Witten’s section 5.

(Incidentally, twisted Cohomotopy does also imply cancellation of the gauge anomaly of the M5-brane, in that it implies the required half-integral flux quantization, as per Prop. 4.12 in our arXiv:1904.10207.)

But if we cannot find agreement, then, reiterating what I said at the end of the above, it would be useful if you could lay out your perspective in a more formalized fashion – maybe in a little LaTeX-ed note – with clear statement of i) assumptions, ii) claim and iii) proof steps. That could help get the issue straightened out and would give us something concrete to react to, if necessary.

Finally, I still think there is no need for strong emotions: the math is straightforward and checkable step-by-step; your contribution is undoubted and acknowledged; us adding an observation on top of that does not diminish it; and the picture we all arrive at, thereby, is beautiful. :-)

Posted by: Urs Schreiber on May 13, 2019 8:33 AM | Permalink | Reply to this

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

Are you saying Witten is not describing a gauge anomaly? Above (3.7) I read for instance: “…the failure of gauge invariance was described in…”. And again, we’re looking at the anomaly line bundle over the intermediate Jacobian parametrizing the Wilson lines of the C-field. If that’s not a gauge anomaly I don’t know what is.

I understand that the term you describe comes from the CS term. That’s a gauge anomaly under the gauge transformation of the C-field near the M5-brane. $G_4^2$ has a chance of being non-zero only if the C-fields changes along the base of the anomaly fibration. This is the exact analogue of the fact that the usual 3d CS action is not gauge invariant on a manifold with boundary.

Concerning your comment about your equation (4.12), I’m not sure. As far as I am aware, the half-integral quantization is necessary to cancel a global, discrete gravitational anomaly on the M2-brane. This is at least how it was derived by Witten on his paper on the M-theory effective action. The cancellation of the gauge anomalies on the M5-brane is precisely the content of Section 3 of the Witten paper we are speaking about. Your proposition 4.12 looks like a flux quantization condition, not an anomaly cancellation condition. The flux quantization is certainly necessary for the consistency of the system, but it is not sufficient for the cancellation of gauge anomalies. The M5-brane has a gauge anomaly even when $G$ is properly quantized, which is canceled by the term that is bothering you.

“it would be useful if you could lay out your perspective in a more formalized fashion, maybe in a little LaTeX-ed note, with clear statement of i) assumptions, ii) claim and iii) proof steps.”

? Everything is latexed and formalized in my paper. Assumptions, claim and proof of what exactly? Really, that sounds like another evasive maneuver. You spelled out quite clearly what is troubling you: the fact that the $G^2$ term arises in the inflow from the M-theory CS term, but, according to you, not in the anomaly formula for the self-dual field. I pointed out precisely where Witten explains that this term actually arises in the anomaly of the self-dual field and how it exactly cancels the M-theory CS term. The fact that this term arises is also explained differently in Belov-Moore and in my papers about the global anomaly of the self-dual field. I don’t know what more I could latex.

Concerning strong emotion, I already explained why they arose. I can add that you calling your colleagues work “folklore” doesn’t help either.

This said, to come back to Witten’s paper, there is a very subtle point that may be the source of the confusion. If you look at Witten’s (3.6), you see an $x^2$ term, which is really our $G^2$ term ($x$ is $G$ up to factors of $2\pi$). The second term is absolutely needed, because else the Chern-Simons action is ill-defined. But on the other hand, when considering things globally, the $\frac{1}{8} L(TM)$ term is also ill-defined, exactly in the same way. Here Witten makes the gauge CS term well-defined by adding the $\frac{1}{8} \lambda^2$ (putting a factor of $2\pi$ in front of the action). For consistency, an exact same term should be added to the gravitational part. This is confusing, because if you take $\lambda$ to be $\frac{1}{2} p_1$, then this term depends on the metric and introduces a gravitational anomaly in the gauge CS term. A more elegant way of doing that is, instead of adding $\frac{1}{8} \lambda^2$, to add $\frac{1}{8} \sigma_X$ (where $\sigma_X$ is the signature of the 8d bounding manifold like in Witten). This has the same effect of making the CS action well-defined and splits the gauge and gravitational anomalies cleanly, without introducing any spurious gravitational anomaly into the gauge CS term. (I could explain why those two terms both make the action well-defined if you’re interested, although I think I already did it in a private email.) In any case, the CS theory describing both the gravitational anomaly and the gauge anomaly reads $\frac{1}{8} L(TM) - \frac{1}{2} G^2$, as stated in my paper on the M5-brane and proven (up to a small caveat mentioned above) in my paper about the global gravitational anomaly of the self-dual field.

That’s really the best I can do.

Posted by: Samuel Monnier on May 13, 2019 1:26 PM | Permalink | Reply to this

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

Maybe just to add something concerning the last paragraph. This was definitely the source of my confusion, when I was claiming that $G_4^2$ does not contribute to the local anomaly. There is a way of constructing Witten’s Chern-Simons action without invoquing a bounding 8-manifold $X$ bounding its 7d spacetime, explained in: https://arxiv.org/abs/1607.01396 .

There, if the 7d-manifold happens to bound, you get a formula similar to Witten’s, except that $\lambda$ is not $\frac{1}{2} p_1$ or anything simple. It is a relative class, say $\lambda_0$, on the 8-manifold that can be constructed out of a Wu structure on the 7d-manifold. For some strange reason I was putting $\frac{1}{2} G^2 = \frac{1}{2} \left(\frac{\lambda_0}{2}\right)^2$ in my mind during most of our discussion. With this substitution everything I said is true, but not really relevant to the $G^2$ term indeed.

Posted by: Samuel Monnier on May 13, 2019 2:32 PM | Permalink | Reply to this

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

Hi Samuel,

sorry to see you taking the suggestion of making a formal note on your proposed resolution in bad spirit, it was just meant to help achieve clarity.

To wrap this up, we have now added a line to p. 12 of v2 saying this:

“Later, [Monnier 19, private communication] interprets the vanishing of this term as an incarnation of the cancellation of gauge anomalies according to [Wi96b, Sec. 3].”

Posted by: Urs Schreiber on May 18, 2019 3:59 PM | Permalink | Reply to this

### on “folklore”

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.

Posted by: Urs Schreiber on May 18, 2019 4:22 PM | Permalink | Reply to this

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