]]>prinicples; has be systematically obtained; gauged-of-gauge; cohomoloy; correspong; geoemtry

Thanks!

This, and a bunch of other little things, now improved (hopefully) in the latest version (here).

]]>Utterly trivial, but I keep tripping on your ’quest’ sentences when I start reading the article

In the quest to mathematically formulating M-theory,…

is grammatically awkward. Perhaps, “In the quest for the mathematical formulation of M-theory”

In the quest for shedding some light in that direction,…

the ’quest’ here is too weak. They have aims like the Holy Grail. Perhaps, “Looking to shed some light in that direction,…”

]]>Thanks! Fixed now.

]]>Looks intriguing. Here are some typos

]]>superymmetric; non-pertrubative; hmotopy; super-valume; cocyle; intersting; an hence; MO9-projectin

… and now with considerably strengthened main theorem (summary here)

]]>We are working on a followup:

]]>We are finalizing a review that sums up the whole rational story of this thread here:

Comments are welcome.

(This is going to appear as our contribution to the proceedings of the Durham symposium on the topic, a while back.)

]]>The supergeometric version is pretty immediate if one uses the equivalence between supermanifolds and suitable presheaves of ordinary manifolds on the site of superpoints. But somebody needs to sit down and write it out cleanly.

]]>Great stuff! Does establishing that extension to super-geometry come next? And then perhaps integrating the equations of motion, #147?

]]>Now the article alluded to in #139 above is out:

Daniel Grady,

*Cobordisms of global quotient orbifolds and an equivariant Pontrjagin-Thom construction*

The bulk of the article establishes a nice equivariant Pointryagin-Thom isomorphism between equivariant cohomotopy and cobordism classes of submanifolds inside orbifold fixed points.

In section 4 the application to the M5-brane locus, that I indicated in #139 above, is discussed.

So earlier in *Equivariant homotopy and super M-branes (schreiber)* we showed that *rational* equivariant cohomotopy sees the MK6-brane, but its only way to see the M5-brane domain wall inside the MK6 is by intersecing the latter with an orientifold, which cuts down supersymetry by half and hence yields the $\mathcal{N} = (1,0)$ M5-brane locus only.

Dan establishes that beyond the rational approximation, equivariant cohomotopy in $RO$-degree $V$ classifies cobordism classes of submanifolds inside the $G$-fixed points (inside the orbifold singularities) of codimension $dim(V^G)$ relative to the ambient fixed point locus.

This means that if we measure M-brane charge with the representation sphere $S^{\mathbb{H}_{adj}}$, where the space of quaternions $\mathbb{H}$ is equipped with the *adjoint* action by unit quaternions $\simeq SU(2)$ (as for the M2-brane!) and consider a D- or E-series finite subgroup $G \subset SU(2)$, then equivariant cohomotopy detects codimension-1 sub-branes inside the MK6.

This result should extend straightforwardly to super-geometry, where it should say that inside the $\mathbb{R}^{6,1\vert \mathbf{16}}$-supermanifold of the MK6, we *still* see a codimension-1 (hence codimension-$(1\vert 0)$!)-sub-supermanifold, hence an $\mathbb{R}^{5,1\vert \mathbf{8} + \mathbf{8}}$-submanifold, hence the $\mathcal{N}= (2,0)$-M5-brane locus.

Not in itself yet, as it is a (equivariantly-)topological statement. But once we bring in the SuGra equations of motion by requiring (equivariant) super-torsion freedom, something should happen.

]]>Re #139, does Dan’s result give us a glimpse of Theory X, or 6d (2,0)-superconformal QFT, then?

]]>Is there a particular open question this addresses?

Sure. Let me quote Moore 14:

]]>We still have no fundamental formulation of “M-theory” - the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. $[...]$. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

Re #139: thanks Urs! I think the last sentence was the kind of thing I was looking for! But could you maybe spell out the significance of this to a lay person? Is there a particular open question this addresses, for instance?

]]>Thanks, yes. Fixed now.

]]>Re #140, presumably dimensions should sum in $Y_{11} \simeq X_4 \times K_6$.

]]>Coo! So into some rich terrain:

Grothendieck predicted that the GT group is closely related to the absolute Galois group. Maxim Kontsevich later conjectured its action on certain space of quantum field theories and outlined its motivic aspects.

Back to physics meets number theory!

Just so I can see some earlier comments of yours in the same location:

]]>51, Hence we find that as we come to 10d from 11d, the twisted KU-coefficients are but one summand in the Goodwillie linearization of the dimensionally reduced M-brane coefficients. So there is more in the M-brane coefficients, in some sense. The big open question is: what is this beyond the rational approximation.

86, We are seeing the unstable spheres as the true coefficients in M-theory. From there we may think of passage to stabilization as just the first order approximation in the Goodwillie-Taylor tower, and then the identification of K-theory as just the first order approximation of

thatin the chromatic filration.

Regarding the last point I should add: Since this is about the full cocycle space $Maps(X_4, S^4)$, it does know about the cohomotopy of $Y_{11}$, too, as soon as we have a Kaluza-Klein-fibration or Cartesian product $Y_{11} \simeq X_4 \times K_7$, via $Maps(K_7, Maps(X_4, S^4)) \simeq Maps(X_4 \times K_7, S^4)$.

Curiously, this is the perspective on higher dimensional field theories as lower dimensional field theories “with values in” other low dimensional field theories as in the AGT correspondence.

]]>Let’s see, what would you like to know. The broad impact of the project, or the specific results of Dan and Vincent that I hinted at? The latter I feel I should not circulate before their main authors feel ready, but, roughly (and now it seems I am doing it anyway, shame on me):

Dan has established the nature of the isomorphism that Pontryagin-Thom collapse induces between equivariant cohomotopy and cobordism classes inside fixed point loci. His results imply in particular that full equivariant cohomotopy in degree 4 sees the $(2,0)$-supersymmetric M5 brane inside an MK6 singularity (while the rational shadow of it that we studied previously in 1805.05987 can only see the $(1,0)$-supersymmetric $\tfrac{1}{2}M5$ being the intersection of an MK6 with an MO9).

Vincent seems to have a proof that the Goodwillie-Taylor tower of the full cocycle space of equivariant cohomotopy in degree-4 on 4-manifolds $X$ produces the system of graph complexes on $X$ glued along the Ran space of $X$, and an unproven but reasonable argument that the operadic action of the Goodwillie derivatives of the identity on this give the action of the Grothendieck-Teichmueller group on the graph complexes. In summary this says something like that homotopy-theoretic perturbation theory applied to our hypothesized M-theoretic cohomology theory (“hypothesis H” last/first slides here) produces hallmark structures of perturbative interacting quantum field theory on 4d spacetimes.

]]>Hi Urs, sounds very exciting! For someone like myself who knows little to nothing about this, would it be possible to summarise the context of what might be being demonstrated here?

]]>Yes. There is an upcoming result by Dan Grady and one by Vincent Braunack-Mayer which show that the kernel is remarkable. Dan’s could become available by next week, or else real soon, Vincent’s will take a little longer to be ready for public consumption, but, if it works all out like it seems to, it will range real deep. Until these are officially public I should not say too much, though. But you’ll be the first to be alerted.

]]>Any sign yet of interesting M-theoretic degrees of freedom from the kernel of $\beta \circ \alpha$?

]]>In fact, if the differential $d_3$ in the AHSS for stable cohomotopy coincides on the relevant entry $H^4(X,\pi^0 = \mathbb{Z})$ with $Sq^3$ (as I suppose it must if it is non-vanishing at all), then the cokernel of $\beta$ will actually equal the cokernel of the index-2 subgroup in DMW00, section 4 of C-fields satisfying their “integral equation of motion”.

]]>