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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 25th 2020
• (edited Feb 25th 2020)

Further towards homotopy-theoretic foundations for the M5-brane model in M-theory, we have a note observing emergence of twisted string structure (“non-abelian gerbe field”) on the M5-worldvolume, derived from twisted cohomotopy-theory, assuming Hypothesis H:

This amounts to observing one big homotopy-commutative diagram, the outer part of which we had derived earlier. The observation that the inner part thus factors through $B String^{c_2}$ follows immediately. The article walks a spotlight through this big diagram, explaining all its constitutents.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 25th 2020
• (edited Feb 25th 2020)

p. 3 ’Here we restrict attention to the subgroup … only for sake of exposition’ Is this because otherwise you’d have loads of $Sp(1)$ factors all over, and these are making little difference to the mathematics?

p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?

Typos

conisdering; Stringvstructure; quatization

and

In direct analogy with $Spin^c$-structure, is called $String^{c_2}$-structure…

missing ’it’.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 25th 2020
• (edited Feb 25th 2020)

Is this because otherwise you’d have loads of $Sp(1)$ factors all over, and these are making little difference to the mathematics?

Yes, exactly

p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?

I think now the combination of the two will be

$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8}$

hence the “super-exceptional Poincaré $String^{c_2}$ group”, if you wish.

On this, the Hopf WZ term is going to be a universal 5-gerbe connection

$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{\;\;\;\; \widetilde{\mathbf{\Gamma}}_7\;\;\;\;}{\longrightarrow} B^6 U(1)_{conn}$

whose curvature 7-form is the M5 super-cocycle

$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{h_3^{exc}\wedge \mu_{M2} + \mu_{M6}}{\longrightarrow} \mathbf{\Omega}^7_{cl}$

from FSS19c

and whose integral class is the

$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \to B String^{c_2}_{\chi_8} \overset{ \widetilde \Gamma_7 \;\coloneqq\; H_3^{univ} \wedge (\widetilde \Gamma_4 + \tfrac{1}{2}p_1) + 2\Gamma_7 }{\longrightarrow} K(\mathbb{Z},7)$

from FSS19b.

Something like this…

Typos

Thanks! Fixed now.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeFeb 25th 2020
• (edited Feb 25th 2020)

For what it’s worth, I think I’ve calculated that

$[S^5,B String^{c_2}(4)] \simeq (\mathbb{Z}/2)^3$

so that if your M5-brane is topologically a sphere, there are eight possible “sectors” for $B String^{c_2}(4)$-bundles on it. Does this show up in the physics?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 26th 2020
• (edited Feb 26th 2020)

The relation to “the physics” is this:

1) in the bottom part of the big diagram the slicing over $B Spin(8)$ is what enforces the relation between the C-field and the field of gravity, which is the anomaly cancellation conditions, such as that $[G_4 +\tfrac{1}{4}p_1]$ is integral.

2) the precomposition of that with the Borel-equivariant Hopf fibration encodes the relative shifted trivialization of the C-field on the M5-worldvolume

The commutativity of the outer diagram follows, and hence the map to $B String^{c_2}(4)$ (for instance as an element in $[S^5, B String^{c_2}(4)]$) is determined by these two physics conditions: anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.

So what you might want to check is this:

If we fix the worldvolume topology $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeFeb 26th 2020

anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.

hmm, that’s not as deep as I was thinking. :-)

If we fix the worldvolume topology $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!

Hmm, yes, that would be interesting. I guess you mean not just $X^8$ but the embedding $\phi\colon \Sigma \to \mathbb{R}^{2,1} \times X^8$? And maybe also $b$?

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeFeb 26th 2020

hmm, that’s not as deep as I was thinking. :-)

let me rephrase that. What I was thinking was that is in principle possible for someone to have done some kind of toy-model calculation for a spherical M5-brane on a specific target space, and applying the anomaly cancellation and shifted trivialisation conditions, found three independent configurations (corresponding to the three factors of $(\mathbb{Z}/2)^3)$. One might expect such configurations to have more concrete interpretations from a physics point of view. Or maybe I’m being too naive, and the sort of “realistic” target 8-manifolds don’t particularly support M5-branes of that shape.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 26th 2020

it is in principle possible for someone to have done some kind of toy-model calculation

Yes, since it hit the arXiv 11 hours ago.

Be the first.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 26th 2020

Isn’t the worldvolume of the M5-brane of dimension 6? Why the 5-sphere?

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeFeb 26th 2020

David R. is thinking of $\Sigma = \mathbb{R}^{0,1} \times S^5$.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeFeb 26th 2020

I would be happy to make some small contribution, but I fear that I would be missing any crucial physical that may rule out my simplistic approach. Also, I wouldn’t quite know how to put such an idea is the language that M-theorists would be used to. Also, having a sensible and motivated target space in which to place the spherical M5-brane, along the lines of something already in the literature. I guess at the least one would want $X^8$ to have nontrivial $\pi_5$, so as to wrap an M5-brane of this form around it, and better, have a factor of something like $\mathbb{R}^6 \setminus \{0\}$. But I know $X^8$ is usually something rather rich

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 26th 2020

On the contrary. The mathematical formulation allows to run without bothering about the folklore.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 26th 2020
• (edited Feb 26th 2020)

Just idling about, I came across Hitchin’s SL(2) over the octonions where he mentions (p. 13) a couple of fibrations arising for quaternionic reasons that I hadn’t seen yet

$S^3 \times S^3 \to Sp(2) \to S4$, $S^3 \to Sp(2) \to S^7$.

Feels like the second at least is in the vicinity.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeFeb 26th 2020
• (edited Feb 26th 2020)

Oh, but the second of these is the fiber of the projection map of the coset space realization $Sp(2)/Sp(1) \simeq S^7$; and the first composes that with the quaternionic Hopf fibration.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeFeb 26th 2020

Well, I can mock something up and run it by you, to see if you think it would pass muster in arXiv:hep-th

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 27th 2020

Re #14, oh yes, of course.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 28th 2020

Does that idea of Atiyah and Witten to look at M-theory on $G_2$-manifolds in relation to M-theory on the 8-manifold $\mathbf{H P}^2$, in the section on confinement, point perhaps to a broader relationship between M-theory on G2-manifolds and M-theory on 8-manifolds, something that could plug into the work you’ve done here?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMar 2nd 2020

An insightful discussion of the relation between 7d $G_2$-structure and 8d $Spin(7)$-structure is in arXiv:1405.3698

[ I am on vacation with family for a few days, back next week ]

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeMar 2nd 2020