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Twisted Cohomotopy implies M-theory anomaly cancellation
Abstract. We show that all the expected anomaly cancellations in M-theory follow from charge-quantizing the C-field in the non-abelian cohomology theory twisted Cohomotopy. Specifically, we show that such cocycles exhibit all of the following:
Along the way, we find that the calibrated $N =1$ exceptional geometries (${Spin}(7)$, $G_2$, ${SU}(3)$, ${SU}(2)$) are all induced from the classification of twists in Cohomotopy. Finally we show that the notable factor of $1/24$ in the anomaly polynomial reflects the order of the 3rd stable homotopy group of spheres.
What should we take from that discussion around p. 10 about the difference between global and local anomalies? That your twisted cohomotopy accounts for the 6 constraints above which are all local? If so, what would it take to approach global anomalies? The full global equivariant and differential Cohomotopy?
monpoles, p.43
What should we take from that discussion around p. 10 about the difference between global and local anomalies?
The point on p. 10 is to highlight that the issue being discussed in section 2.6 is present already at the level of just the rational/real/de Rham cohomology of the M5-brane anomaly, hence at the level of the de Rham class of the “local anomaly”, and to prepare the ground for solving that.
A wealth of extra complications arises when trying to understand the full “global anomaly” of the M5-brane, and nobody claims to have fully achieved that. But it’s also not necessary for seeing the issue, which is present already at the level of rational/real/de Rham cohomology, and which is what we show, later in 4.6, to be solved by charge-quantizing in twisted Cohomotopy.
Ok. I was wondering whether there’s a general account of anomalies so that where you say you are “Accounting for all anomaly cancellation” and then give “a whole list of subtle anomaly cancellation constraints on the C-field”, it might be possible to know the extent of the coverage of the list.
Is this a good introduction?
For the general idea I’d recommend the articles by Freed that we cite, for the specific situation of the M5-brane anomalies, the whole section 2.6 is a walk through the available literature.
Regarding the appearance of 4 versions of $S^7$ as a coset space, I see John Baez spoke about them once
The reals, complexes, quaternions and octonions all show up in the construction of these 4 squashed 7-spheres.
And
there are 4 different ways to see the 7-sphere as a homogeneous space G/H admitting a G-invariant metric. The obvious way is the usual ‘round’ 7-sphere, where the symmetry group G is as big as possible. The other 3 ways are less symmetrical, hence the name ‘squashed’.
This stuff is a beautiful spinoff of relations between the even Clifford algebras in 5, 6, 7, and 8 dimensions, all of which have 8-dimensional representations.
TWF195 explains the latter comment.
The section ncatlab.org/nlab/show/7-sphere#CosetSpaceRealization lists the original references on these coset realizations of the (squashed) 7-sphere. You could add references there. (Am on my phone and officially offline :-)
Will do.
made this a table-for-!include
: coset space structure on n-spheres – table
I am enchanted by that homotopy Cartesian diagram there, from our Sec. 3.4: This unifies all of
homotopy theory
exceptional geometry
twisted Cohomotopy in degree 7.
In the great network of the exceptional, is there a single source, driving all other manifestations, or should we expect merely an interconnected web? I mean, there are evidently close ties between the above and the existence of the octonions, and we know some have liked to look to them for answers, e.g., Boya’s Octonions and M-theory.
Elsewhere it was emergence from the superpoint that was running the show.
It’s still the bouquet emanating out of the superpoint, now beyond the rational approximation!
So a new brane bouquet is needed? For instance, the M-branes appearing in a single extension?
A thought: if ultimately the superpoint needs to be taken as $\mathbb{S}^{0|1}$, won’t we need that to appear in the cohesion - table as used currently to justify $\mathbb{R}^{0|1}$.
a new brane bouquet is needed?
No, this is still expanding on the brane bouquet as it was and is. The brane bouquet discovered the form of M-brane charge in the approximation of rational homotopy theory; and now this is about working out the consequences of assuming that this same form also applies beyond the rational approximation.
if ultimately the superpoint needs to be taken as $\mathbb{S}^{0|1}$
That was a speculation I had had, but so far it hasn’t led anywhere. In particular, it isn’t clear that it fits anywhere in the progression of the cohesive modalities. Maybe not.
If one reads “is there” as “is it possible to give one”, then the answer is: probably. If one reads “is there” as “has such been written up”, then the answer is: not really, but it will depend a lot on what you mean by “layman” and “all of this”.
In conclusion, it might be more rewarding to ask a slightly more specific question! And easier for me to know what might be helpful to say in reply.
How would you explain this…
What is “this”? The contents of this specific paper, the research project as a whole, the general use of higher structures in M-theory?
If the latter, it would make sense to look at Higher Structures in M-Theory. If the second, then look at The Rational Higher Structure of M-theory.
People here are very generous with their time, but you have to pose specific questions, such as about a particular passage in some paper.
In the commutative diagram towards the bottom of p.14 you have $B Spin(n)$ rather than $B Spin(n+1)$.
Re #14,
now this is about working out the consequences of assuming that this same form also applies beyond the rational approximation.
So I guess I’m wondering, if you had the full non-rational picture, how things might look. To what extent is it due to the simplifications of the rational approximation that it is possible to present as a series of extensions?
In the commutative diagram towards the bottom of p.14 you have $B Spin(n)$ rather than $B Spin(n+1)$.
Thanks again. Fixed now. Though this typo will be in the first arXiv version now…
How would you explain this to a room full of graduate mathematics/physics students?
I have some thoughts on how one could do this, should there be demand. My next task, though, is to prepare slides to explain our latest results to a room full of geometers and topologists. Meanwhile, feel invited to ask any specific questions you might have.
Another typo
Cororllary (p. 31)
I see it’s out today on the arXiv. There must be a whole range of directions to head out from here, such as global equivariance mentioned in the Outlook.
Is there any indication of a role for elliptic cohomology? Back in M Theory, Type IIA Superstrings, and Elliptic Cohomology Hisham was connecting anomaly cancellation to it:
We observe here that the vanishing W_7=0 of the Diaconescu-Moore-Witten anomaly in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2-branes and 5-branes in the M-theory limit.
And I guess somewhere out there is a non-perturbative F-theory.
Details need to be discussed, but one thing is clear: If Cohomotopy is indeed the fundamental cohomology theory of M-theory, then the chromatic tower applies to approximate its stabilization (represented by the sphere spectrum), in stages, by rational cohomology, then K-theory, then elliptic cohomology/tmf,…, complex cobordism.
p. 13
in §3.5 we observe that the classical Poincaré-Hopf theorem expresses twisted Cohomotopy in terms of the Euler characteristic, while in §3.6 we comment on how the classical Poincaré-Hopf theorem relates Cohomotopy to cobordism classes of submanifolds (branes) with normal structure.
The second ’Poincaré-Hopf’ should be ’Pontrjagin-Thom’.
Unknown #17, it may be nearly 7 years old, but Urs’s essay The use of non-abelian differential cohomology in fundamental physics might do for a general introduction to physics graduates.
How to expose this material depends on which people, with which background and interest, one is actually talking to.
Some may first want to get an idea of what string theory and M-theory is supposed to be in the first place. These might want to start with the string theory FAQ.
Others might sort of accept that and instead wonder how the hypothesis of Cohomotopy charge emerges from analysis of the super rational cohomology of super-spacetimes. There are detailed lecture notes on that at geometry of physics – fundamental super p-branes and at Super Lie n-algebra of Super p-branes (schreiber).
From here on, one may want to get an idea of what it means to go beyond the rational approximation of Cohomotopy to full Cohomotopy. This is the topic of the talks Equivariant Stable Cohomotopy and Branes (schreiber).
Is Remark 3.3 tempting us to consider lifts higher up the Whitehead tower, say, to $B String$? But to twist the cohomotopy there need to be an action on the spheres?
One beautiful aspect of Hypothesis H is that just requiring equivariance structure for the quaternionic Hopf fibration already implies Spin-structure, since the canonical inclusion $Sp(2)\cdot Sp(1) \hookrightarrow SO(8)$ factors through the canonical surjection $Spin(8) \to SO(8)$. Remark 3.3 just prepares the conceptual ground for dealing with that.
Next, twisted String structure appears from trivializing linear combinations of the 4-flux class, which involves summands of $\tfrac{1}{2}p_1$. The cohomotopical refinement of this is the homotopy pullback $\widehat X$ of the equivariant quanternionic Hopf fibration along the Cohomotopical C-field flux, shown in Def. 4.24. As explained in the following remark, around diagram (110), this exhibits a cohomotopical refinement of String-structure.
In a followup project we will discuss the Chern-Simons 2-gerbe on the modui space of degree-4 twisted cohomotopy
$S^4 \sslash Sp(2)\cdot Sp(1) \;\simeq\; B \big( Sp(1)\cdot Sp(1) \cdot Sp(1) \big) \overset{\widetilde \Gamma_4}{\longrightarrow} B^3 U(1)$which gives the approximation of the Cohomotopical C-field in ordinary integral cohomology. Under this approximation the above Cohomotopical twisted String-structure then reduces to the twisted String structures in M-theory that we have been discussing in earlier years.
Ok, thanks!
As explained in the following remark, around diagram (110), this exhibits a cohomotopical refinement of String-structure.
You mean exhibits this implicitly? There’s no direct mention of String-structure.
A couple of typos from around there:
classifyng; By direction inspection
The end of the article is now a little less abrupt with Remark 4.29, but couldn’t this form a concluding section?
Yeah, discussion of twisted string structure and relation to the earlier models of the C-field just in shifted integral cohomology should maybe go elsewhere, not to overburden this article.
We had brief discussion about that conclusion being a remark or a subsection. Maybe we’ll change our mind eventually.
Thanks for catching more typos! Have fixed them now.
Of any interest?
Thanks! Now that I open it, I remember that I looked at this before. But should be included in the citations, yes. Thanks again.
Just to say that now there is proof, in the second half of Section 4.6 (in v2) of the general fluxed tadpole cancellation condition.
It’s fairly amazing how triality of the symplectic subgroups of $Spin(8)$ (here) is at the heart of this. We had had the structure of the proof of the general fluxed tadpole cancellation laid out for weeks (now in the second half of Section 4.6, only in v2) but there kept being a prefactor of 1/2 where it shouldn’t be, bugging us. Today it finally fell into place. Turned out that I had introduced an error in the computation of the pullback of Pontryagin classes along the delooped triality automorphism. Correcting that error, it works like a charm (Lemma 4.28, that is).
If $Sp(1) \cdot \Sp(2)$ is isomorphic as an abstract group to $Spin(3) \cdot Spin(5)$ (Remark 3.17), can it also be the case as you have it in Prop 2.4 of SO(8) that $Sp(1) \cdot \Sp(2)$ is isomorphic to $SO(3) \times SO(5)$?
$Spin(3) \cdot Spin(5)$ can’t be isomorphic to $SO(3) \times SO(5)$, surely.
That’s another of these subtleties:
Triality acts by Lie group automorphism on $Spin(8)$, and hence it makes subgroups of $Spin(8)$ isomorphic to each other.
It does not however act by automorphisms on $SO(8)$. Rather, the triality of subgroups of $SO(8)$ is obtained by acting upstairs on subgroups of $Spin(8)$ and then projecting down to $SO(8)$.
And so, as a single subgroup of $Spin(8)$ gets moved around in there, it may happen that, depending on its position, it gets projected down to different subgroups of $SO(8)$.
Oh, I see.
I gave your paper a plug at MO - Mathematical/Physical uses of SO(8) and Spin(8) triality. I see there’s an existing answer concerning string theory.
Something is messed up here:
Next we characterize, in Prop. 4.31 below, the differential form data encoded in (123), For that we need the following two lemmas The statement of Lemma 4.29 is standard but rarely made fully explicit, we spell it out since it is crucial for our new result, Lemma 4.29.
in terms of punctuation, and Lemma 4.29 being crucial for Lemma 4.29.
Thanks for catching this! Fixed now.
Punctuation is still wrong:
…data encoded in (126), For that …
And being picky
…rarely made fully explicit, we spell it out…
either needs a connective, or start a new sentence at ’we’.
Thanks again. Fixing now.
Are you coining ’generalized cohomotopy’ for the purposes mentioned in Remark 3.22? The term has been used already for other purposes, e.g., here. I guess you’re using it because of the doubling up as in generalized complex geometry.
Presumably your version will classify pairs of framed submanifolds. Then there’s twisting to add in.
In the ungeneralized case, is it known what kind of framed submanifolds are classified by twisted cohomotopy?
Okay, thanks for the pointer. I have reworked (what is now) Remark 3.22 a little in order to clarify.
Regarding twisted PT:
For twisted Cohomotopy, it is straightforward to define the PT-construction in one direction: Pick one section of the spherical fibration as the base “point”, then for any other section = cocycle in Cohomotopy one gets a submanifold whose normal bundle is the restriction of the given twisting bundle, by regularizing and taking fibers at the chosen point, as usual for PT in this direction.
But in the twisted unstable case there does no longer seem to be a straightforward collapse map that would exhibit this construction as an isomorphism. A similar issue appears in the case of equivariant Cohomotopy. And of course in a stacky perspective equivariant and twisted Cohomotopy are two aspects of the same structure.
This needs more thinking. With David Roberts we started looking at the equivariant case. But this anomaly cancellation here got in the way and distracted me from that project. We’ll get to that next, I hope.
A vague thought: unstable twisted cohomotopy is a nonabelian twisted cohomology, and so about hom spaces occurring in an arrow category $\mathbf{H}^I$, as at twisted cohomology, which tells us of the approximations via jet toposes.
From Quantization via Linear Homotopy Types, we have $\mathbf{H}^I$ in example 3.8 as a model of linear HoTT, and
Examples 3.9, 3.10 and 3.11 are linearized variants of example 3.8.
In section 3 we hear of
The two hallmarks of quantum physics are the superposition principle and quantum interference
with summation in the examples there in $\mathbb{C}$ and $K U$.
So what happens when quantization encounters the nonabelian situation of unstable cohomotopy?
Yeah something interesting happens:
Consider, for definiteness, the simple special case of trivial twist and compactification to 4d Minkowski spacetime, in which case the non-abelian cocycle space of solitonic degree-4 cohomotopy is the unstable space of maps from spatially compactified spacetime to the $4$-sphere.
The corresponding motive of observables is the suspension spectrum of that mapping space.
By stable splitting of mapping spaces this is a wedge sum of motives of configuration spaces of calorons.
By the graph complex theorem this is the motive of Feynman diagrams for Chern-Simons/AKSZ-type field theory.
In three months from now Vincent will start his position here, and then we will finally write up this story of how nonperturbative thermal QFT arises as the cohomological quantization of the C-field charge-quantized in Cohomotopy theory.
Until then they are keeping Vincent distracted with lesser tasks. But okay, this gives us time to first do the writeup with David Roberts on demonstrating M5 tadpole cancellation in M-orientifolds by orbifold equivariant Cohomotopy.
I see. That material you were exploring here late last year.
I was thinking of suggesting a page on mathematical theories relating to string theory/ M-theory. We have applications to pure mathematics and a list of Fields medal work, but perhaps such a page would end up saying ’just about everything’.
perhaps such a page would end up saying ’just about everything’.
I think it will be better than that: it will single out the landscape of natural mathematics and discard the endless math-swampland of contrived mathematics. For instance the accidental theory of well-founded trees in material set theory will play no role, while the natural theory of intuitionistic depend types will (and does already).
But this is a question to come back to ten years from now. First to establish the theory now, then to loo back and review it.
I meant in the sense of significant mathematics:
Ce qui limite le vrai, ce n’est pas le faux, c’est l’insignifiant. (Thom)
I was trying to approach that natural/swampland distinction of yours in #49 many years ago in my Mathematical Kinds, or Being Kind to Mathematics.
Thanks for the pointer! Interesting. Maybe we should start an entry landscape of mathematics!
I can imagine that being rather contentious how to describe the landscape, but I would like to see an organised way of viewing mathematics through the eyes of physics, e.g., consequences of dualities in cascades of compactifications and dimension reductions of M-/F-theory.
Just putting off my marking, but I came across an instance of your 4 cosets for $S^7$ in an M-theoretic context in Figure 1, p. 24 of arXiv:1208.1262, which is an extended version of Figure 1 from Tri-Sasakian consistent reduction.
Thanks. Had not seen these. Will take a look.
For what it’s worth, here is now a set of slides for a talk on this stuff, tomorrow at IHP Paris, to an audience of mathematical physicsist:
One typo
quarge-quantized
Thanks! Fixed now.
Here is a followup result:
Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane
Abstract The full 6d Wess-Zumino-term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that the anomaly cancellation condition is implied by “Hypothesis H”: the proposal that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the full 6d Wess-Zumino-term with a twisted homotopy Whitehead integral formula, which we establish.
Am experimenting with producing a poster for Strings2019 in Brussels next week: here is what I have so far.
I’ve no idea what makes for a good poster format. Hopefully that pulls them in.
For the sake of uniformity, [JSSW19] needs arXiv.
Thanks, have fixed it in my local copy.
Another M-theoretic anomaly cancellation:
Twisted cohomotopy have anything to say?
Today we have a further followup, now the case of flat orientifolds, where twisted Cohomotopy becomes equivariant Cohomotopy, which now implies tadpole anomaly cancellation:
Is there any way to say when ’all’ cancellations have been dealt with, to know when some theory is anomaly-free? What of the cancellation of the time-reversal anomaly in #64?
Hi David,
now I finally found time to look at the article that you keep asking about. As far as I see, the result there is that a potential anomaly is not actually present, so no further cancellation condition needs to be invoked/postulated.
Is there any way to say when ’all’ cancellations have been dealt with, to know when some theory is anomaly-free?
This has a tautological answer:
Recall that physicist’s “anomaly” is mathematician’s “obstruction” read in reverse:
The mathematician first lays out design plans of what exactly they are about to construct, and then looks ahead to see if anything obstructs the path ahead to that goal.
The physicist instead storms ahead without further ado. When they hurt their nose running into that boulder lying in the way, they mumble something about “this is not what happened last time to me, this does’t seem normal – indeed, since we are used to lucking out, we have to say that the situation we find ourselves in is outright anomalous!”
In either case, an obstruction/anomaly just means that a certain would-be construction does not actually exist.
So:
If you really know what it is you are constructing, then you know you have lifted all obstructions, hence cancelled all anomalies, when you have actually constructed the darn thing.
If however, as here with M-theory, you do not know in advance what you are constructing, it’s a different story.
What is happening here is that the full theory itself is unknown but various limiting cases of it are prescribed and understood (close analogy: the would-be theory of the “field with one element”). So then at least those limiting cases should actually exist hence be consistent, hence be anomaly free.
But without knowing what thing the full theory actually is, there is, tautologically! no way to know if all its anomalies have been cancelled.
But the idea with working towards M-theory is the opposite: If a natural mathematical structure implies an indefinite but already long list of expected consistency checks/anomaly cancellations, then chances are that this mathematical structure is the elusive theory, and then, since that exists it is anomaly free.
Ok, thanks! This would make for an instructive philosophy of physics article.
Odd situation where you may be sitting on the nth (fourth?) superstring revolution but people haven’t caught on yet.
We have decided to bring the physics interpretation to the forefront of our previously purely representation-theoretic writeup on the image of the Burnside ring in the representation ring, for binary Platonic group.
Now it goes like so:
Lift of fractional D-brane charge to equivariant Cohomotopy theory
(pdf)
Abstract The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this $\,$ Hypothesis H, here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman’s comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox.
Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups (2T, 2O, 2I) and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.
We are preparing a, hopefully, more digestible concise survey (14 pages) of the implications of Hypothesis H for M-theory on 8-manifolds.
Looking good!
statemens; topolocal; amd; tiwted
In ’the vanishing point of algebraic topology’, you’re alluding to perspective painting?
In (13), should both $w_i$ be lower case?
What hangs on the decision on p. 6 to “restrict attention to G-structure” for $Sp(2)$? What would the full $Sp(1) \cdot Sp(2)$ give?
Thanks!, will fix when I have a moment.
Yes, “vanishing point” is suppossed to allude to perspective painting. Maybe it works better in Deutsch. Is it understandable?
Maybe some quote marks to indicate figurative usage. It comes rather out of the blue.
Typos fixed now, thanks again. The second $W_7$ in (13) is indeed capital, being the integral Stiefel-Whitney class, instead of the ordinary mod-2 class.
On page 12, in the next displayed maths under equation (25), there is a pair of arrows on the right (labelled “relations” and “conditions”) that seem to missing a domain object. It looks like perhaps they were meant to be stacked instead?
This is actually meant to be as diplayed – but if it is confusing it may need changing:
It’s trying to express that the set in question is a subquotient of the set of differential forms. The intermediate domain (the subobject before quotienting) is the one that is not displayed. It would look the same, just without the equivalence-relation subscript!
OK, in that case you could have some kind of anonymous symbol,. For instance, like the box in either of the answers here, or a \blacksquare
($\blacksquare$), or \bigbox
(from the stmaryrd
package), etc.
I see your point. How about this?
Replied there.
Would it even make sense to ask how far through all foreseeable consistency checks for Hypothesis H you are? Is there still a pile of things in the ’to do’ list?
Any sign of the ’community’ taking note?
Allow me to say: How would there not be a “pile of things” left to do – given what we are talking about. Compare Alejandro’s question here for perspective – and for a shared sense of impatience :-)
Next, we are in the process of preparing a note on a derivation, from Hypothesis H, of non-abelian gauge theory on D6/D8-brane intersections. Once it is at all in readable form, I can send you a private preview.
Regarding the community: Peer review is underway, of course. One referee asks, at the very end of a long report: What, incidentally, do you mean by cocycles in Cohomotopy? So it’s going to be a long process ;-) But that’s okay, the noise will flood in soon enough.
It’s not so much a sense of impatience for me as a difficulty in being able to judge the score so far. From my position, I may be in the strange situation of having been made aware, by the chance of our association, of an ongoing development whose importance will merit its authors a notable place in the history of physics, and being so aware before a great majority of leading physicists have taken in on board. Naturally, I want somebody I count a friend to succeed, especially someone with lively interests in philosophy. Success would also provide a window of opportunity for me to help push philosophy in a happier direction.
Ok, so maybe because of that last comment there is impatience, but still I’m also intrigued as an observer by the reception of new ideas. Just as I can blame some false moves in philosophy for meaning that I have a hard time getting a hearing for anything connected to category theory, we can imagine that a lack of interest in new thinking about recent geometry has deprived some physicists of the capacity to understand, say, what a cocycle in Cohomotopy is, rendering Hyp H rather opaque. But is that the main obstacle? Is there an X such that a mathematically-informed string theorist might reasonably say “Impressive results so far, but show me that Hyp H can achieve X and I’ll be properly sold on it?”
I remember a philosopher saying to me many years ago that until I could show that category theory could do something better than predicate logic had done for philosophy, e.g., than Russell’s treatment of definite description, then I would have a hard time persuading philosophers. I ignored this for years, but in a sense my new book is answering him, though I guess it heads in directions you won’t like, e.g., concerning natural language.
With the shift to non-abelian generalized cohomology can you still talk about orientations? You have orientations relate to anomalies at motivic quantization - Anomalies And Orientation.
The poking around this morning that prompted that question was driven by a thought I had that with the move to study
assignments of homotopy classes of maps into any coefficient space $A$,
one might wonder whether the term ’cohomology’ still does much work. It’s just maps in an $(\infty, 1)$-topos. Instead of degree-4 Cohomotopy cohomology theory, one might as well say maps into $S^4$ (and their 0-truncation).
That thought then paired up with another that abelianization/linearization must occur at some stage
Just as the idea of a category of motives is to constitute a “linearization” or “abelianization” of a category of spaces, so quantization is a process that sends (non-linear) spaces of field configurations to linear spaces of quantum states. This linearization by which quantum states may be added as elements of an abelian group encodes the superposition principle and hence quantum interference, the hallmark of quantum physics. (motives in physics)
Just to say that one is not to conflate
1) non-abelian gauge bundles, such as Yang-Mills fields $X \to B G$
2) prequantum bundles on the cocycle space of such non-abelian fields, which schematically are $Maps(X \to B G) \to B^n U(1)$
The latter must be linearized for quantization, not the former.
In particular, if/where the prequantum line bundle has trivial class, quantization is given essentially by passing to the cohomology of the cocycle phase space.
The attentive reader will have noticed me adding theorems to the nLab on what these quantized Cohomotopy cocycle phase spaces are like (hint: e.g. here :-)
We have now further streamlined the account of “Twisted cohomotopy implies M-theory anomaly cancellation” to make it more digestible. The whole discussion of the literature and motivation is now concisely on less than one page: last page here.
How does that read?
Re #86, so in terms of Hypothesis H, a characteristic class of $S^4$, $S^4 \to B^n U(1)$ might be interesting?
The characteristic class of interest is the generator $S^4 \longrightarrow K(\mathbb{Z},4) = B^3 U(1)$. This translates 4-Cohomotopy classes to the usual instanton calculus.
But in #86 I was talking about classes more generally on the cocycle spaces $Maps(X, S^4)$.
For instance, by Segal’s theorem we have that
$Maps(S^3, S^4) \;\simeq\; Conf(\mathbb{R}^3, S^1)$is a configuration space of points, and so the quantum observables on Cohomotopy phase spaces are given by the cohomology of configuration spaces of points. These in turn are given by cohomology of graph complexes and by weight systems and the like.
Ah, I see now, thanks. Some pieces have finally clicked into place.
Okay, good.
There is a subtlety in that Segal’s theorem gives un-ordered configuration spaces. Their cohomology is uninteresting.
But the ordered configuration space is a fiber product of un-ordered configuration spaces. Under Hypothesis H this means that interesting physics appears at brane intersections.
We have now given the proof of M5-brane anomaly cancellation from Hypothesis H (which was split off from the original version of the anomaly cancellation article) its stand-alone home, here:
If you are interested, please grab the latest pdf from behind that link. Comments are welcome.
We found one more mechanism, are finalizing a writeup here:
Abstract: We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Hořava-Witten’s proposal for the extension of the Green-Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in J-twisted Cohomotopy theory.
Comments are welcome! Please grab the latest version behind the above link.
Typos:
Diagram (11) should have $\mathbb{R}^8$ in the rightmost column.
Presumably ’Penrose fibrations’ (p. 6) should be singular.
ommutative
Thanks! Fixed now.
Thanks for sharing!
I noticed a typo in (80), where it should be $U(n,\mathbb{H})$ and not $U(2,\mathbb{H})$.
Thanks!! Fixed now.
low energy theories with the exact field content of the (minimally supersymmetric) standard model of particle physics (up to decoupled and ultra-heavy fields),
this sentence reads oddly to me. I know it might be standard idiom in hep-th, but specifying something to be exactly something else, it isn’t usually hedged around with parenthetical modifications, themselves not unique. Why not write something more like
low energy theories with the exact field content of the minimally supersymmetric standard model of particle physics, possibly with additional decoupled and ultra-heavy fields.
The first sentence seems like it wants to claim exactly the content of the SM, but then has to twice grudgingly admit that it’s that, plus some extra stuff. It’s like me claiming I can run (nearly) two kilometers in exactly ten minutes (plus some extra time).
No, the standard idiom in HEP is that phrase without these parentheticals. :-)
It’s the analog in maths of claiming that a limit is exactly X and the parenthesis gives the space in and conditions under which the limit is taken.
Prodded by a referee report, we have further beautified the article:
The new version makes more explicit that the vanishing of the Euler 8-class is a Fivebrane-structure on $Sp(2)$-manifolds (Example 3.2, Example 3.3) and how that brings out the universal avatar of the Hopf-WZ term as a universal cohomology class (Remark 4.3).
The proof of the culminating Theorem 4.8 remains just as beautiful as it was, but now the spaces previously called by pseudonyms $E^4$ and $E^7$ are called by their god-given names $S^4 \sslash \widehat{Sp(2)}$ and $S^7 \sslash \widehat{Sp(2)}$, respectively, where $\widehat{Sp(2)} \,:=\, Fivebrane\big( Sp(2)\big)$.
Comments are still welcome. If you look at it, please grab the latest pdf from behind the above link.